Spherical tube hypersurfaces

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris

For further volumes: http://www.springer.com/series/304

2020

•

Alexander Isaev

Spherical Tube Hypersurfaces

123

Prof. Alexander Isaev Australian National University Mathematical Sciences Institute 0200 Canberra Aust Capital Terr Australia [email protected]

ISBN 978-3-642-19782-6 e-ISBN 978-3-642-19783-3 DOI 10.1007/978-3-642-19783-3 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011925542 Mathematics Subject Classification (2011): 32-XX c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

In this book we consider (connected) smooth real hypersurfaces in the complex vector space Cn+1 with n ≥ 1. Specifically, we are interested in tube hypersurfaces, i.e. real hypersurfaces of the form

Γ + iV, where Γ is a hypersurface in a totally real (n + 1)-dimensional linear subspace V ⊂ Cn+1 . From now on we fix the subspace V and choose coordinates z0 , . . . , zn in Cn+1 such that V = {Im z j = 0, j = 0, . . . , n}. Everywhere below V is identified with Rn+1 by means of the coordinates x j := Re z j , j = 0, . . . , n. Tube hypersurfaces arise, for instance, as the boundaries of tube domains, that is, domains of the form D + iRn+1, where D is a domain in Rn+1 . We refer to the hypersurface Γ and domain D as the bases of the above tubes. The study of tube domains is a classical subject in several complex variables and complex geometry, which goes back to the beginning of the 20th century. Indeed, already Siegel found it convenient to realize certain symmetric domains as tubes. For example (see Section 5.3 for details), the familiar unit ball in Cn+1 is biholomorphically equivalent to the tube domain with the base given by the inequality x0 >

n

∑ x2α .

(0.1)

α =1

Note that the boundary of the tube domain with base (0.1) is the tube hypersurface whose base is defined by the equation x0 =

n

∑ x2α .

(0.2)

α =1

This tube hypersurface is equivalent to the (2n + 1)-dimensional sphere in Cn+1 with one point removed.

v

vi

Preface

Although the definition of tube depends on the choice of the totally real subspace V , the structure of the direct product of a portion of V with all of iV is extremely useful. Indeed, the property that makes tube domains and hypersurfaces interesting from the complex-geometric point of view, is that they all possess an (n + 1)-dimensional commutative group of holomorphic symmetries, namely the group of translations {Z → Z + ib} with b ∈ V , Z ∈ Cn+1 . Furthermore, any affine automorphism of the base of a tube can be extended to a holomorphic affine automorphism of the whole tube (note, however, that in general – for example, for the tube domain with base (0.1) – there may be many more holomorphic automorphisms than affine ones). In the same way, any affine transformation between the bases of two tubes can be lifted to a holomorphic affine transformation between the tubes. This last observation, however simple, indicates an important link between complex and affine geometries. In this book we look at tube hypersurfaces from both the complex-geometric and affine-geometric points of view. One can endow a tube hypersurface (in fact any real hypersurface in complex space) with a so-called CR-structure, which is the remnant of the complex structure on the ambient space Cn+1 (see Section 1.1). We impose on the CR-structure the condition of sphericity (see Section 1.2). This is the condition for the hypersurface to be locally CR-equivalent (for example, locally biholomorphically equivalent – see Section 1.1) to the tube hypersurface with the base given by the equation x0 =

k

n

α =1

α =k+1

∑ x2α − ∑

x2α

for some 1 ≤ k ≤ n with n ≤ 2k (cf. equation (0.2)). For a given k the second fundamental form of the base of a locally closed spherical tube hypersurface is everywhere non-degenerate and has signature (k, n − k) up to sign. Interestingly, the sphericity condition coincides with the condition of the vanishing of the CR-curvature form (see Section 1.1), thus spherical hypersurfaces are exactly those that are flat in the CR-geometric context (the reader should not be alarmed by the apparent linguistic inconsistency between “sphericity” and “flatness”). In this book we offer a comprehensive exposition of the theory of spherical tube hypersurfaces starting with the idea proposed in the pioneering work by Yang (1982) and ending with a new approach due to Fels and Kaup (2009). Spherical tube hypersurfaces possess remarkable properties. For example, every such hypersurface is real-analytic (see Section 3.2) and extends to a real-analytic spherical (hence non-singular) tube hypersurface which is closed as a submanifold of Cn+1 (see Section 4.5). Thus, it suffices to consider only closed spherical tube hypersurfaces, and the main goal of this book is to explicitly classify such hypersurfaces whenever possible. Note that while for a fixed k all spherical tube hypersurfaces are CR-equivalent locally, they may not be CR-equivalent globally. We, however, aim at obtaining not just a classification up to CR-equivalence but a much finer classification up to affine equivalence (that is, a classification up to the affine equivalence of their bases). In 1982 Yang [108] proposed to approach this problem for k = n by means of utilizing the zero CR-curvature equations arising from the

Preface

vii

Cartan-Tanaka-Chern-Moser invariant theory, and we follow this approach throughout most of the book. We will now describe the book’s structure. In Chapter 1 we give a detailed exposition of Chern’s construction of a Cartan connection for a hypersurface satisfying a certain non-degeneracy condition (Levi non-degeneracy). For a locally closed tube hypersurface this condition is equivalent to the non-degeneracy of the second fundamental form of the base at every point. The curvature of the Cartan connection gives rise to the zero CR-curvature equations, which can be written in terms of any local defining function of the hypersurface (see Sections 1.3, 1.4). These equations involve partial derivatives of the defining function up to order 4 for n > 1 and up to order 6 for n = 1. In Chapter 3 we generalize the result of [108] from k = n to any value of k by showing that the zero CR-curvature equations significantly simplify for tube hypersurfaces and lead to systems of partial differential equations of order 2 of a very special form (we call them defining systems). As an application of this result, we show in Section 3.2 that every spherical tube hypersurface is real-analytic. Our exposition in Chapter 3 is based on results of [52], [56], [58], [64]. Further, in Chapter 4 we reduce every defining system to a system of one of three types by applying suitable linear transformations and give a certain representation of the solution for a system of each type. These representations imply the result already mentioned above: every spherical tube hypersurface extends to a real-analytic closed spherical tube hypersurface in Cn+1 (see Section 4.5). Our exposition in Chapter 4 is a refinement of that given in [56]. From Chapter 4 to the end of Chapter 8 we study only closed spherical tube hypersurfaces and concentrate on classifying such hypersurfaces up to affine equivalence. In Chapters 5–7 we consider the cases k = n, k = n − 1, k = n − 2. In each of these cases we use the representations of the solutions of defining systems found in Chapter 4. In Chapter 5 a complete classification for the case k = n is obtained. This classification is due to Dadok and Yang (see [27]), but our arguments are simpler than the original proof. In Chapter 6 we derive a complete classification for k = n − 1. This classification appeared in [64], but the present exposition is shorter and much more elegant. Finally, in Chapter 7 we give a complete classification for the case k = n − 2. This classification was found by the author in 1989 and announced in article [53], where a proof was also briefly sketched. Full details were given in a very long preprint (see [54]). Because of the prohibitive length of the preprint the complete proof was never published in a journal article. In this book it appears in print for the first time. One consequence of the results of Chapters 5–7 is the finiteness of the number of affine equivalence classes for every fixed n in each of the following cases: (a) k = n, (b) k = n − 1, and (c) k = n − 2 with n ≤ 6. In Chapter 8 we show that this number is infinite (in fact uncountable) in the cases: (i) k = n − 2 with n ≥ 7, (ii) k = n − 3 with n ≥ 7, and (iii) k ≤ n − 4. This result was announced in [53] but has only recently appeared with complete proofs (see [59]). Further, the question about the number of affine equivalence classes in the only remaining case k = 3, n = 6 had been open since 1989 until Fels and Kaup resolved it in 2009 by constructing an example of a family of spherical tube hypersurfaces in C7 for k = 3 that contains uncountably many pairwise affinely non-equivalent elements. In Chapter 8

viii

Preface

we present this family but deal with it by methods different from the original methods of Fels and Kaup. In particular, we use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces. The example mentioned above naturally arises from the new analytic-algebraic approach to studying spherical tube hypersurfaces developed by Fels and Kaup in [42]. It is based on their earlier work [41] concerned with the question of describing all (local) tube realizations of a real-analytic CR-manifold (cf. [4]). Fels and Kaup recover the real-analyticity result of Section 3.2 and the globalization results of Section 4.5 by their methods. Further, their approach yields the affine classifications of spherical tube hypersurfaces for k = n and k = n − 1 contained in Chapters 5, 6. We outline the main ideas of [41], [42] in Section 9.2 of Chapter 9. In Section 9.1 of Chapter 9 we consider tube hypersurfaces locally CR-equivalent to the tube hypersurface with the base given by the equation x0 =

k

∑

α =1

m

x2α − ∑ x2α ,

(0.3)

α =k+1

where 0 ≤ k ≤ m, m ≤ 2k, m < n. Such hypersurfaces are no longer Levi nondegenerate (in the locally closed case the second fundamental forms of their bases are everywhere degenerate), thus the standard Cartan-Tanaka-Chern-Moser theory does not apply to them. As we explain in Section 9.1, for m ≥ 1 every tube hypersurface of this kind is real-analytic and extends to a closed non-singular realanalytic tube hypersurface in Cn+1 represented as the direct sum of a complex (n − m)-dimensional linear subspace of Cn+1 and a closed spherical tube hypersurface lying in a complementary complex (m + 1)-dimensional subspace. For m = 0 such a hypersurface is an open subset of a real affine hyperplane in Cn+1 . Thus, the study of tube hypersurfaces locally CR-equivalent to the tube with base (0.3) reduces to the study of spherical tube hypersurfaces. Our exposition in Section 9.1 is based on results of [56]. In addition, the book includes a short chapter on spherical rigid hypersurfaces (see Chapter 2). A locally closed real hypersurface M in a complex (n + 1)dimensional manifold N is called rigid if near its every point in some local coordinates z0 = x0 + iy0 , z = (z1 , . . . , zn ) in N it can be given by an equation of the form x0 = F(z, z). Clearly, rigid hypersurfaces are much more general than tube ones, but it turns out that the zero CR-curvature equations significantly simplify already in the rigid case. One motivation for considering rigid hypersurfaces is that they naturally arise as a result of various scaling procedures (see references in Section 2.2). An application of the zero CR-curvature equations in the rigid case is given in Section 2.2. These equations serve as an intermediate step for obtaining defining systems in Chapter 3. Our exposition in Chapter 2 is an improvement of that given in [57]. I would like to thank Wilhelm Kaup for many valuable comments that helped improve the manuscript and Michael Eastwood for many inspiring conversations concerning the material included in Chapters 8 and 9. Special thanks go to Nikolay Kruzhilin for his help with obtaining a copy of preprint [54]. A significant portion

Preface

ix

of this book was written during my stay at the Max-Planck Institute in Bonn, which I thank for its hospitality and support. Canberra–Bonn, October 2010

Alexander Isaev

•

Contents

1

Invariants of CR-Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction to CR-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Chern’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Chern’s Invariants on Section of Bundle P 2 → M . . . . . . . . . . . . . . . 24 1.4 Umbilicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2

Rigid Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1 Zero CR-Curvature Equations for Rigid Hypersurfaces . . . . . . . . . . . 35 2.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3

Tube Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Zero CR-Curvature Equations for Tube Hypersurfaces. . . . . . . . . . . . 41 3.2 Analyticity of Spherical Tube Hypersurfaces . . . . . . . . . . . . . . . . . . . . 50

4

General Methods for Solving Defining Systems . . . . . . . . . . . . . . . . . . . . 4.1 Classification of Defining Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Defining Systems of Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Defining Systems of Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Defining Systems of Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Globalization of Spherical Tube Hypersurfaces . . . . . . . . . . . . . . . . . .

55 55 62 65 68 77

5

Strongly Pseudoconvex Spherical Tube Hypersurfaces . . . . . . . . . . . . . . 5.1 Defining Systems of Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Defining Systems of Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 93 93

6

(n − 1, 1)-Spherical Tube Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1 Real Canonical Forms of Pair of Matrices Q, X, where Q is Symmetric and X is Q-Symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Defining Systems of Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Defining Systems of Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xi

xii

Contents

6.4 Defining Systems of Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.5 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7

(n − 2, 2)-Spherical Tube Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Defining Systems of Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.3 Defining Systems of Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.4 Defining Systems of Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8

Number of Affine Equivalence Classes of (k, n − k)-Spherical Tube Hypersurfaces for k ≤ n − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.2 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 8.3 Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9

Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.1 Tube Hypersurfaces with Degenerate Levi Form . . . . . . . . . . . . . . . . . 195 9.1.1 Complex Foliations on CR-Manifolds . . . . . . . . . . . . . . . . . . . 195 9.1.2 Levi Foliation on Tube Manifold . . . . . . . . . . . . . . . . . . . . . . . 197 9.2 Approach of G. Fels and W. Kaup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

•

Chapter 1

Invariants of CR-Hypersurfaces

Abstract In this chapter we survey the invariant theory of Levi non-degenerate CR-hypersurfaces concentrating on Chern’s construction of Cartan connections.

1.1 Introduction to CR-Manifolds We start with a brief overview of necessary definitions and facts from CR-geometry (see [2], [25], [67], [105] for more detailed expositions). Unless stated otherwise, throughout the book manifolds are assumed to be connected, and differentialgeometric objects such as manifolds, distributions, fiber bundles, maps, differential forms, etc. are assumed to be C∞ -smooth. A CR-structure on a manifold M of dimension d is a distribution of linear subspaces of the tangent spaces Tpc (M) ⊂ Tp (M), p ∈ M, i.e. a subbundle of the tangent bundle T (M), endowed with  2 operators of complex structure JpM : Tpc (M) → Tpc (M), J pM = − id. For p ∈ M the subspace Tpc (M) is called the complex tangent space at p, and a manifold equipped with a CR-structure is called a CR-manifold. It follows that the number CRdim M := dimC Tpc (M) does not depend on p; it is called the CR-dimension of M. The number CRcodimM := d − 2 CRdim M is called the CR-codimension of M. Every complex (and even almost complex) manifold is a CR-manifold of zero CR-codimension. In this book we mostly consider CR-manifolds of CR-codimension one, or CR-hypersurfaces. Before constraining ourselves to this case, however, we will briefly discuss general CR-manifolds. CR-structures naturally arise on real submanifolds of complex manifolds. Indeed, if M is an immersed real submanifold of a complex manifold N, then one can consider the maximal complex subspaces of the tangent spaces to M Tp (M) := Tp (M) ∩ J pN Tp (M),

p ∈ M,

(1.1)

where J pN is the operator of complex structure on Tp (N). If dim Tp (M) is constant on M, then by setting

A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 1, 

1

2

1 Invariants of CR-Hypersurfaces

Tpc (M) := Tp (M),

  J pM := J pN 

Tpc (M)

for every p ∈ M, we obtain a CR-structure on M. The CR-structure defined above is called the CR-structure induced by N. We note that dim Tp (M) is constant if M is a real hypersurface in N (that is, an immersed real submanifold of N of codimension one), and therefore a real hypersurface in a complex manifold carries an induced CR-structure (which turns the hypersurface into a CR-hypersurface). For comparison, we remark that if the codimension of M in N is two, then dim Tp (M) need not be constant (see [48] for a study of generic compact codimension two submanifolds of CK ). Let M  be an immersed submanifold of a CR-manifold M, and suppose that  M is endowed with a CR-structure. Then M  is called a CR-submanifold of M    if for every p ∈ M  one has Tpc (M ) ⊂ Tpc (M) and J pM = J pM  c  . Clearly, if the Tp (M )

CR-structure of a CR-manifold M is induced by a complex manifold N, then M is a CR-submanifold of N. A map f : M1 → M2 between two CR-manifolds is called a CR-map if for every p ∈ M1 the following holds: (a) the differential d f (p) of f at p maps Tpc (M1 ) into T fc(p) (M2 ), and (b) d f (p) is complex-linear on Tpc (M1 ). Two CR-manifolds M1 , M2 of the same dimension and the same CR-dimension are called CR-equivalent if there is a diffeomorphism f from M1 onto M2 which is a CR-map (it follows that f −1 is a CR-map as well). Any such diffeomorphism is called a CR-isomorphism, or CR-equivalence. A CR-isomorphism from a CR-manifold M onto itself is called a CR-automorphism of M. CR-automorphisms of M form a group, which we denote by Aut(M). A CR-isomorphism between a pair of domains in M is called a local CR-automorphism of M. An infinitesimal CR-automorphism of M is a vector field on M whose local flow near every point consists of local CR-automorphisms of M. Infinitesimal CR-automorphisms form a (possibly infinite-dimensional) Lie algebra (see Theorem 12.4.2 in [2]). In the first instance, we are interested in the equivalence problem for CR-manifolds. This problem can be viewed as a special case of the equivalence problem for G-structures. Let G ⊂ GL(d, R) be a Lie subgroup. A G-structure on a d-dimensional manifold M is a subbundle S of the frame bundle F(M) over M which is a principal G-bundle. Two G-structures S1 , S2 on manifolds M1 , M2 , respectively, are called equivalent if there is a diffeomorphism f from M1 onto M2 such that the induced mapping f∗ : F(M1 ) → F(M2 ) maps S1 onto S2 . Any such diffeomorphism is called an isomorphism of G-structures. The CR-structure of a manifold M of CR-dimension n and CR-codimension k (here d = 2n + k) is a G-structure, where G is the group of all non-degenerate linear transformations of Cn ⊕ Rk that preserve the first component and are complex-linear on it. The notion of equivalence of such G-structures is then exactly that of CR-structures. For convenience, when speaking about G-structures below, we replace the frame bundle F(M) by the coframe bundle. ´ Cartan developed a general approach to the equivalence problem for GE. structures (see [18], [65], [67], [97]), which applies, for example, to Riemannian

1.1 Introduction to CR-Manifolds

3

and conformal structures. In Section 1.2 we outline a solution to the CR-equivalence problem for certain classes of CR-manifolds in the spirit of Cartan’s work focussing on the case of CR-hypersurfaces (for an alternative approach to the equivalence problem see, e.g. [71]). Namely, we describe some classes of CR-manifolds whose CR-structures reduce – in the sense defined below – to {e}-structures, or absolute parallelisms, where {e} is the one-element group. An absolute parallelism on an -dimensional manifold P is a 1-form σ on P with values in R such that for every x ∈ P the linear map σ (x) is an isomorphism from Tx (P) onto R . The equivalence problem for absolute parallelisms is reasonably well-understood (see [97]). Let C be a collection of manifolds equipped with G-structures. We say that the G-structures are s-reducible to absolute parallelisms if one can assign every M ∈ C some principal bundles πs

π s−1

π4

π3

π2

π1

P s → P s−1 → . . . → P 3 → P 2 → P 1 → M and an absolute parallelism σ on P s in such a way that the following holds: (i) any isomorphism of G-structures f : M1 → M2 for M1 , M2 ∈ C can be lifted to a diffeomorphism F : P1s → P2s satisfying F ∗ σ2 = σ1 , and (ii) any diffeomorphism F : P1s → P2s satisfying F ∗ σ2 = σ1 is a lift of an isomorphism of the corresponding G-structures f : M1 → M2 for M1 , M2 ∈ C. In the above definition we say that F is a lift of f if

π21 ◦ . . . ◦ π2s ◦ F = f ◦ π11 ◦ . . . ◦ π1s . Let M be a CR-manifold. For every p ∈ M consider the complexification Tpc (M) ⊗R C of the complex tangent space at p. Clearly, the complexification can be represented as the direct sum (1,0)

Tpc (M) ⊗R C = Tp where

(0,1)

(M) ⊕ Tp

(M),

(1,0)

(M) := {X − iJ pX : X ∈ Tpc (M)},

(0,1)

(M) := {X + iJ pX : X ∈ Tpc (M)}.

Tp Tp

(1.2)

The CR-structure on M is called integrable if for any pair of local sections z, z of the bundle T (1,0) (M) the commutator [z, z ] is also a local section of T (1,0) (M). It is not difficult to see that if M is a submanifold of a complex manifold N and the CR-structure on M is induced by N, then it is integrable. In this book we consider only integrable CR-structures. A C-valued function ϕ on a CR-manifold M is called a CR-function if for any local section z of T (0,1) (M) we have zϕ ≡ 0. If M is a real submanifold of a complex manifold N with induced CR-structure, then for any function ψ holomorphic on

4

1 Invariants of CR-Hypersurfaces

N its restriction ϕ := ψ |M is a CR-function on M. Let M1 , M2 be CR-manifolds, where M2 is a submanifold of CK with induced CR-structure. In this case any map f : M1 → M2 is given by K component functions. It is straightforward to verify that f is a CR-map if and only if all its component functions are CR-functions on M1 . An important characteristic of a CR-structure called the Levi form comes from taking commutators of local sections of T (1,0) (M) and T (0,1) (M). Let p ∈ M, (1,0) Z, Z ∈ Tp (M). Choose local sections z, z of T (1,0) (M) near p such that z(p) = Z, (1,0) z (p) = Z . The Levi form of M at p is the Hermitian form on Tp (M) with values c in (Tp (M)/Tp (M)) ⊗R C given by LM (p)(Z, Z ) := i[z, z ](p)(mod Tpc (M) ⊗R C).

(1.3)

For fixed Z and Z the right-hand side of the above formula is independent of the choice of z and z . We usually treat the Levi form as a Ck -valued Hermitian form (1,0) (i.e. a vector of k Hermitian forms) on Tp (M), where k is the CR-codimension of M. As a Ck -valued Hermitian form, the Levi form is defined uniquely up to the choice of coordinates in Tp (M)/Tpc (M). If M is a CR-hypersurface, we think of its (1,0)

Levi form at a given point p as a C-valued Hermitian form on Tp (M) defined up to a non-zero real multiple and speak of the signature of the Levi form up to sign. Let g, g˜ be two Ck -valued Hermitian forms on complex vector spaces V , V˜ , respectively. We say that g and g˜ are equivalent if there exists a complex-linear isomorphism A : V → V˜ and B ∈ GL(k, R) such that g(Az, ˜ Az) = Bg(z, z) for all z ∈ V . Clearly, the Levi form LM (p) defines an equivalence class of Ck -valued Hermitian forms. When we refer to LM (p) as a Ck -valued Hermitian form, we speak of a representative in this equivalence class. Let g = (g1 , . . . , gk ) be a Ck -valued Hermitian form on Cn . We say that g is nondegenerate if (i) the scalar Hermitian forms g1 , . . . , gk are linearly independent over R, and (ii) g(z, z ) = 0 for all z ∈ Cn implies z = 0. Observe that for a non-degenerate Hermitian form g one has 1 ≤ k ≤ n2 . If k = 1 and g is non-degenerate, we write the signature of g as (l1 , l2 ) with l1 + l2 = n, where l1 and l2 are the numbers of positive and negative eigenvalues of g, respectively. A CR-manifold M is called Levi non-degenerate if its Levi form at any p ∈ M is non-degenerate. Everywhere in this book, with the exception of Chapter 9, we consider only Levi non-degenerate CR-manifolds. Further, we call a CR-manifold M strongly uniform if LM (p) and LM (q) are equivalent for all p, q ∈ M. Every Levi non-degenerate CR-hypersurface is strongly uniform.

1.1 Introduction to CR-Manifolds

5

For any Ck -valued Hermitian form g on Cn we define a CR-manifold Qg ⊂ Cn+k of CR-dimension n and CR-codimension k as follows: Qg := {(z, w) ∈ Cn+k : Im w = g(z, z)},

(1.4)

where z = (z1 , . . . , zn ) is a point in Cn and w ∈ Ck . The manifold Qg is often called the quadric associated to g. The Levi form of Qg at every point is equivalent to g. If k = 1 and g(z, z) = ||z||2 (where || · || is the Euclidean norm on Cn ), the quadric Qg is CR-equivalent to the unit sphere in Cn+1 with one point removed. Indeed, the map   z w+1 (z, w) → (1.5) ,i 1−w 1−w transforms   Q||·||2 := (z, w) ∈ Cn+1 : ||z||2 + |w|2 = 1 \ {(0, 1)} into Q||·||2 . More generally, for k = 1 and an arbitrary Hermitian form g on Cn set   Sg := (z, w) ∈ Cn+1 : g(z, z) + |w|2 = 1 .

(1.6)

Map (1.5) transforms   Qg := Sg \ (z, 1) ∈ Cn+1 : g(z, z) = 0   into Qg \ (z, −i) ∈ Cn+1 : g(z, z) = −1 . Assume now that g is non-degenerate. In this case every local CR-automorphism of Qg extends to a birational map of Cn+k (see classical papers [1], [90], [99] for k = 1 and papers [6], [7], [44], [62], [67], [68], [70], [98], [106] for 1 < k ≤ n2 ). Let Bir(Qg ) denote the set of all such birational extensions. It turns out that Bir(Qg ) is a group (see [62]). For k = 1 every element of Bir(Qg ) is a linear-fractional transformation induced by an automorphism of CPn+1 (see [1], [90], [99]). For 1 < k ≤ n2 some formulas for the elements of Bir(Qg ) were given in [37]. It was shown in [62], [106] that the group Bir(Qg ) can be endowed with the structure of a Lie group with at most countably many connected components and the Lie algebra isomorphic to the Lie algebra gg of all infinitesimal CR-automorphisms of Qg . Every infinitesimal CR-automorphism of Qg is known to be polynomial (see [106]). We denote by Bir(Qg )◦ the connected component of Bir(Qg ) (with respect to the Lie group topology) that contains the identity.1 One can show that Bir(Qg )/ Bir(Qg )◦ is in fact finite. Note that Qg is a homogeneous manifold since the subgroup Hg ⊂ Bir(Qg ) of CR-automorphisms of the form (z, w) → (z + a, w + 2ig(z, a) + ig(a, a) + b), 1

In general, for a topological group G we denote its connected component containing the identity by G◦ .

6

1 Invariants of CR-Hypersurfaces

with a ∈ Cn , b ∈ Rk , acts transitively on Qg . Therefore, it is important to consider the subgroup of all elements of Bir(Qg ) that are defined and biholomorphic near a particular point in Qg , say the origin, and preserve it. This subgroup, which we denote by Bir0 (Qg ), is closed in Bir(Qg ), and Bir(Qg ) = Hg · Bir0 (Qg ) · Hg (this follows, for example, from results of [62]). Further, let Lin(Qg ) ⊂ Bir0 (Qg ) be the Lie subgroup of linear automorphisms of Qg . Every element of Lin(Qg ) has the form (z, w) → (Cz, ρ w), with C ∈ GL(n, C) and ρ ∈ GL(k, R) satisfying g(Cz,Cz) ≡ ρ g(z, z). It is shown in [37] that Bir0 (Qg ) = Lin(Qg ) · Bir0 (Qg )◦ . We call a Levi non-degenerate CR-manifold M weakly uniform if for any pair of points p, q ∈ M the Lie groups Lin(QLM (p) )◦ , Lin(QLM (q) )◦ are isomorphic by means of a map that extends to an isomorphism between Bir0 (QLM (p) )◦ and Bir0 (QLM (q) )◦ . Clearly, for a Levi nondegenerate CR-manifold strong uniformity implies weak uniformity. Existing results on the equivalence problem for CR-structures treat two classes of Levi non-degenerate manifolds: (i) the strongly uniform Levi non-degenerate manifolds, and (ii) the weakly uniform manifolds for which, in addition, the groups Bir0 (QLM (p) ) are “sufficiently small”, in particular Bir0 (QLM (p) ) = Lin(QLM (p) ). ´ Cartan solved the equivalence probIn [17] (see [67] for a detailed exposition) E. lem for all 3-dimensional Levi non-degenerate CR-hypersurfaces by reducing their CR-structures to absolute parallelisms (note that this reduction differs from Cartan’s approach to general G-structures mentioned earlier – cf. [9]). In 1967 Tanaka obtained a solution for all Levi non-degenerate strongly uniform manifolds (see [101]), but his result became widely known only after Chern-Moser’s work [24] was published in 1974 (see also [9], [10], [11], [23], [66]), where the problem was solved independently for all Levi non-degenerate CR-hypersurfaces. Although Tanaka’s pioneering construction is important and applies to very general situations (which include geometric structures other than CR-structures), his treatment of CR-hypersurfaces is less detailed and clear – and is certainly less useful in calculations – than that due to Chern (see [76] for a discussion of this matter). For example, Tanaka’s construction gives 3-reducibility to absolute parallelisms, whereas Chern’s construction gives 2-reducibility and in fact even 1-reducibility (see [9]). The structure group of the single bundle P 2 → M that arises in Chern’s construction is Bir0 (Qg ), where g is a Hermitian form equivalent to every LM (p), p ∈ M, and the absolute parallelism σ takes values in the Lie algebra gg (which is isomorphic to the Lie algebra of Bir(Qg )). The Lie algebra gg is well-understood for an arbitrary CR-codimension (see [7], [31], [34], [93]). In particular, gg is a graded Lie algebra: gg = ⊕2k=−2 gkg . In Tanaka’s construction, however, the absolute parallelism takes values in a certain prolongation g˜ g of ⊕0k=−2 gkg . The fact that g˜ g and gg coincide for an arbitrary CR-codimension is not obvious (see [31]). Further, the absolute parallelism σ from Chern’s construction is in fact a Cartan connection (to be defined in Section 1.2). In particular, it changes in a regular way under the action of the structure group of the bundle P 2 (see also

1.1 Introduction to CR-Manifolds

7

[9]). Namely, if for a ∈ Bir0 (Qg ) we denote by La the (left) action of a on P 2 , then L∗a σ = AdBir0 (Qg ),gg (a)σ , where AdBir0 (Qg ),gg is the adjoint representation of Bir0 (Qg ). It is not clear from [101] (even in the CR-hypersurface case) whether the sequence of bundles P˜ 3 → P˜ 2 → P˜ 1 → M constructed there can be reduced to a single bundle and whether the absolute parallelism defined on P˜ 3 behaves in any sense like a Cartan connection. [We note, however, that these points were clarified in Tanaka’s later work [102] (see also [103]), where complete proofs of the results announced in [100] were presented (see also Tanaka’s earlier work [99], where a special class of Levi non-degenerate CR-hypersurfaces was considered).] Being more detailed, Chern’s construction also allows one to investigate in detail the important curvature form of σ , i.e. the 2-form Σ := d σ − 1/2[σ , σ ] (this form is of particular importance to us throughout the book). It also can be used to introduce special invariant curves called chains, which have turned out to be important in the study of real hypersurfaces in complex manifolds (see, e.g. [107]). Due to these and other differences between Tanaka’s and Chern’s constructions, we prefer to use Chern’s approach in our treatment of Levi non-degenerate CR-hypersurfaces later in the chapter. We also remark here that in a certain more general situation (namely for Levi non-degenerate partially integrable CR-structures of CR-codimension one) Cartan connections were constructed in [14] as part of a general approach to producing Cartan connections for parabolic geometries (see also [13]). For more details on the parabolic geometry approach we refer the reader to recent monograph [16]. We finish this introduction with a brief survey of existing results for manifolds with CRcodim M ≥ 2. Certain Levi non-degenerate weakly uniform CR-structures of CR-codimension two were considered in [77], [85]. Conditions imposed on the Levi form in these papers are stronger than non-degeneracy and force the groups Bir0 (QLM (p) ) for all p ∈ M to be minimal possible. In particular, they contain only linear transformations of a special form (in this case gkLM (p) = 0 for k = 1, 2). Further, the situation where the groups Bir0 (QLM (p) ) are small and CRcodim M > 2, CRdim M > (CRcodim M)2 was treated in [47]. One motivation for considering manifolds with the Levi form satisfying conditions as in [85] (for CRdim M ≥ 7), [47], [77] is that these conditions are open, i.e. if they are satisfied at a point p, then they are also satisfied on a neighborhood of p in M. Moreover, the quadrics associated to Levi forms as in [85] (for CRdim M ≥ 7) and [77] are dense (in an appropriate sense) in the space of all Levi non-degenerate quadrics. Finally, the case CRdim M = CRcodim M = 2 has been studied very extensively in recent years. This is one of only two exceptional cases among all CRstructures with CRcodim M > 1 in the following sense: typically (in fact always except for the cases CRdim M = CRcodim M = 2 and (CRdim M)2 = CRcodim M) generic Levi non-degenerate quadrics have only linear automorphisms (see [36] and also [7], [85]). However, in the case CRdim M = CRcodim M = 2 Levi nondegenerate quadrics always have many non-linear automorphisms. Every nondegenerate C2 -valued Hermitian form g = (g1 , g2 ) on C2 is equivalent to one of the following:

8

1 Invariants of CR-Hypersurfaces

ghyp (z, z) := (|z1 |2 + |z2 |2 , z1 z2 + z2 z1 ), gell (z, z) := (|z1 |2 − |z2 |2 , z1 z2 + z2 z1 ), gpar (z, z) := (|z1 |2 , z1 z2 + z2 z1 ). These forms are called hyperbolic, elliptic, and parabolic, respectively. The groups Bir(Qg )◦ , Bir0 (Qg )◦ and the Lie algebra gg , where g is one of ghyp , gell , gpar , are quite large. They were explicitly found in [33] (see also [7], [35], [37]). A CR-manifold whose Levi form at every point is equivalent to ghyp or gell is called hyperbolic or elliptic, respectively. Clearly, the conditions of hyperbolicity and ellipticity are open. The equivalence problem for hyperbolic and elliptic CR-manifolds is of course covered by Tanaka’s construction in [101]. More explicit reductions of elliptic and hyperbolic CR-structures to absolute parallelisms, and even to Cartan connections, were obtained in [32], [94], [95]. The rich geometry of hyperbolic and elliptic CR-manifolds (and their partially integrable generalizations) was also studied in [12], [15], [38], [39].

1.2 Chern’s Construction From this moment to the end of Chapter 8 we only consider Levi non-degenerate CR-hypersurfaces with integrable CR-structure. In the present section we describe Chern’s construction from [24], which gives 2-reducibility of such CR-structures to absolute parallelisms. In fact, even 1-reducibility takes places for this construction (see [9]). Let M be a Levi non-degenerate CR-hypersurface with an integrable CR-structure of CR-dimension n. Locally on M the CR-structure is given by 1-forms μ , η α (here and below small Greek indices run from 1 to n unless specified otherwise), where μ is real-valued and vanishes exactly on the complex tangent spaces, η α are complexvalued and complex-linear on the complex tangent spaces. The integrability condition of the CR-structure is then equivalent to the Frobenius condition, which states that d μ , d η α belong to the differential ideal generated by μ , η α . Since μ is realvalued, this condition implies d μ ≡ ihαβ η α ∧ η β

(mod μ )

(1.7)

for some functions hαβ satisfying hαβ = hβ α . Here and below we use the convention

η β := η β , hβ α := hβ α , etc. as well as the usual summation convention for subscripts and superscripts. At every point the matrix (hαβ ) defines a Hermitian form on Cn equivalent to the Levi form of M, where α is the row index and β is the column index (see the footnote on the next two pages). For p ∈ M define E p as the collection of all covectors θ ∈ Tp∗ (M) such that c Tp (M) = {Y ∈ Tp (M) : θ (Y ) = 0}. Clearly, all elements in E p are real non-zero multiples of each other. Let E be the subbundle of the cotangent bundle of M whose

1.2 Chern’s Construction

9

fiber over p is E p . Define θ 0 to be the tautological 1-form on E, that is, for θ ∈ E and Y ∈ Tθ (E) set θ 0 (θ )(Y ) := θ (d πE (θ )(Y )), where πE : E → M is the natural projection. We now fix a non-degenerate Hermitian form on Cn with matrix g = (gαβ ) which is equivalent to every LM (p), p ∈ M. Identity (1.7) implies that for every θ ∈ E there exist a real-valued covector θ n+1 and complex-valued covectors θ α on Tθ (E) such that: (a) each θ α is a lift of a complex-valued covector on TπE (θ ) (M) which is complex-linear on TπcE (θ ) (M), (b) the covectors θ 0 (θ ), Re θ α , Im θ α , θ n+1 form a basis of the cotangent space Tθ∗ (E), and (c) the following identity holds: d θ 0 (θ ) = ±igαβ θ α ∧ θ β + θ 0 (θ ) ∧ θ n+1 .

(1.8)

For every p ∈ M the fiber E p has exactly two connected components, and if the numbers of positive and negative eigenvalues of (gαβ ) are distinct, the signs in the right-hand side of (1.8) coincide for all θ lying in the same connected component of E p and are opposite for θ1 and θ2 lying in different connected components irrespectively of the choice of θ α , θ n+1 . In this situation we define a bundle P 1 over M as follows: for every p ∈ M the fiber P p1 over p is connected and consists of all elements θ ∈ E p for which the plus sign occurs in the right-hand side of (1.8); we also set π 1 := πE P 1 . Next, if the numbers of positive and negative eigenvalues of (gαβ ) are equal, for every θ ∈ E and every choice of the sign in the right-hand side of (1.8) there are covectors θ α , θ n+1 on Tθ (E) satisfying (1.8). In this case we set P 1 := E and π 1 := πE . For θ ∈ P 1 we now only consider covectors θ α , θ n+1 on Tθ (P 1 ) satisfying conditions (a), (b) stated above and such that d θ 0 (θ ) = igαβ θ α ∧ θ β + θ 0 (θ ) ∧ θ n+1 .

(1.9)

The most general linear transformation of θ 0 (θ ), θ α , θ α , θ n+1 preserving equation (1.9) and the covector θ 0 (θ ) is given by the matrix (acting on the left) ⎛

1

0

0

0

⎞

⎜ α ⎟ uαβ 0 0⎟ ⎜v ⎜ ⎟ , ⎜ vα 0 uα 0⎟ ⎜ ⎟ β ⎝ ⎠ ρ s igρσ uβ vσ −igρσ uσ vρ 1

(1.10)

β

β

where s ∈ R, uαβ , vα ∈ C and gαβ uρα uσ = gρσ . In uαβ and vα the superscripts are used for indexing the rows and the subscript for indexing the columns.2 Let G1 be the 2

We follow this convention throughout the book whenever reasonable. However, the entries of the matrices of Hermitian and bilinear forms are indexed by subscripts or superscripts alone, e.g.

10

1 Invariants of CR-Hypersurfaces

group of matrices of the form (1.10). Clearly, P 1 is equipped with a G1 -structure (upon identification of G1 with a subgroup of GL(2n + 2, R)). Our immediate goal is to reduce this G1 -structure to an absolute parallelism. We define a principal G1 -bundle P 2 over P 1 as follows: for θ ∈ P 1 let the fiber Pθ2 over θ be the collection of all covectors (θ 0 (θ ), θ α , θ n+1 ) on Tθ (P 1 ) satisfying conditions (a), (b), (1.9), and let π 2 : P 2 → P 1 be the natural projection. Set ω := [π 2 ]∗ θ 0 and introduce a collection of tautological 1-forms on P 2 as follows:

ω α (Θ )(Y ) := θ α (d π 2 (Θ )(Y )), ϕ (Θ )(Y ) := θ n+1 (d π 2 (Θ )(Y )), where Θ = (θ 0 (θ ), θ α , θ n+1 ) is a point in Pθ2 and Y ∈ TΘ (P 2 ). It is clear from (1.9) that these forms satisfy d ω = igαβ ω α ∧ ω β + ω ∧ ϕ .

(1.11)

Further, the integrability of the CR-structure of M yields that locally on P 2 we have d ω α = ω β ∧ ϕβα + ω ∧ ϕ α

(1.12)

for some 1-forms ϕβα and ϕ α . In what follows we will study consequences of identities (1.11) and (1.12). Our calculations will be entirely local, and we will impose conditions that will determine the forms ϕβα and ϕ α (as well as another 1-form ψ introduced below) uniquely. This will allow us to patch the locally defined forms ϕβα , ϕ α , ψ into globally defined 1-forms on P 2 . Together with ω , ω α , ϕ these globally defined forms will be used to construct an absolute parallelism on P 2 with required properties. Let (gαβ ) be the matrix inverse to (gαβ ), that is, γ

gαβ gγβ = δα ,

γ β

gαβ gαγ = δ .

As is customary in tensor analysis, we use (gαβ ) and (gαβ ) to lower and raise indices, respectively. For quantities that have subscripts as well as a superscript it is important to know the location where the superscript can be lowered to, and this is indicated by a dot. Thus, we write ϕβα· for ϕβα and ϕβ γ for ϕβα· gαγ , etc. (gαβ ) and (gαβ ). For the matrix (gαβ ) the first subscript is the row index and the second one is the column index, whereas for the matrix (gαβ ) the first superscript is the column index and the second one is the row index. Further, coordinates are indexed by subscripts rather than superscripts everywhere in the book except Section 1.3. Accordingly, vectors are usually written as rows with the entries indexed by subscripts. When a matrix is applied to a row-vector on the left, it is meant that the vector needs to be transposed first.

1.2 Chern’s Construction

11

Above we assumed the matrix g to be constant, but for all calculations below we suppose that it is a matrix-valued map on P 1 . In this case the bundle P 2 must be replaced by a different bundle (see Section 1.3 for a precise construction). Allowing the matrix g to be variable makes our calculations more general than one needs just for the purposes of constructing an absolute parallelism on P 2 , but these more general calculations will have a further application in Section 1.3. Differentiation of (1.11) and (1.12) yields, respectively,   i dgαβ − ϕαβ − ϕβ α + gαβ ϕ ∧ ω α ∧ ω β +   −d ϕ + iωβ ∧ ϕ β + iϕβ ∧ ω β ∧ ω = 0

(1.13)

and 

 γ d ϕβα· − ϕβ · ∧ ϕγα· − iωβ ∧ ϕ α ∧ ω β +   d ϕ α − ϕ ∧ ϕ α − ϕ β ∧ ϕβα· ∧ ω = 0.

(1.14)

Lemma 1.1. There exist ϕβα· that satisfy (1.12) and the conditions dgαβ − ϕαβ − ϕβ α + gαβ ϕ = 0.

(1.15)

Such ϕβα· are unique up to an additive term in ω. Proof. It follows from (1.13) that dgαβ − ϕαβ − ϕβ α + gαβ ϕ = Aαβ γ ω γ + Bαβ γ ω γ + Cαβ ω for some functions Aαβ γ , Bαβ γ , Cαβ satisfying Aαβ γ = Aγβ α ,

Bαβ γ = Bαγβ .

(1.16)

Cαβ = Cβ α .

(1.17)

The Hermitian property of gαβ also yields Aαβ γ = Bβ αγ , Due to (1.16), (1.17) the forms 1 ϕ˜ αβ := ϕαβ + Aαβ γ ω γ + Cαβ 2 satisfy relations (1.15) and, upon raising indices, relations (1.12). Verification of the last statement of the lemma is straightforward.   From now on we suppose that (1.15) holds. Identity (1.13) then gives d ϕ = iωβ ∧ ϕ β + iϕβ ∧ ω β + ω ∧ ψ ,

(1.18)

12

1 Invariants of CR-Hypersurfaces

where ψ is a real 1-form. The forms ϕβα· , ϕ α , ψ satisfying (1.12), (1.15), (1.18) are defined up to transformations of the form

ϕβα· = ϕ˜ βα· + Dαβ · ω , ϕ α = ϕ˜ α + Dαβ · ω β + E α ω , ψ = ψ˜ + T ω + i(Eα

ωα

− Eα

(1.19)

ωα )

for some functions Dαβ · , E α , T , where T is real-valued and the following holds: Dαβ + Dβ α = 0.

(1.20)

  Observe also that one can choose a subset S of Re ϕβα· , Im ϕβα· such that for any  Θ ∈ P2  the values at Θ of the forms in the set S∪ ω , Re ω α , Im ω α , ϕ , Re ϕ α , Im ϕ α , ψ constitute a basis of TΘ∗ (P 2 ). Let γ Πβα· := d ϕβα· − ϕβ · ∧ ϕγα· . (1.21) Using (1.15) we obtain

Πβ α = gγα d ϕβγ · − ϕβγ · ∧ ϕγα = d ϕβ α − ϕβ α ∧ ϕ − ϕαγ ∧ ϕβγ · . Since

γ

γ

ϕβ γ ∧ ϕα · = ϕβ · ∧ ϕαγ ,

it then follows that

Πβ α + Παβ = d(ϕβ α + ϕαβ ) − (ϕβ α + ϕαβ ) ∧ ϕ . Differentiating (1.15) we obtain

Let

Πβ α + Παβ = gβ α d ϕ .

(1.22)

Γβα· := Πβα· − iωβ ∧ ϕ α + iϕβ ∧ ω α + iδβα (ϕσ ∧ ω σ ).

(1.23)

It follows from (1.14), (1.18), (1.22), (1.23) that

Γβα· ∧ ω β ≡ 0,

Γβ α + Γαβ ≡ 0 (mod ω ).

(1.24)

Lemma 1.2. We have

Γβ α ≡ Sβ γασ ω γ ∧ ω σ

(mod ω ),

where the functions Sβ γασ have the following symmetry properties: Sβ γασ = Sγβ ασ = Sγβ σ α = Sασ β γ .

(1.25)

1.2 Chern’s Construction

13

Proof. From the first set of equations in (1.24) we see

Γβ α ≡ χβ αγ ∧ ω γ

(mod ω ),

where χβ αγ are 1-forms. Hence, the second set of equations in (1.24) yields

χβ αγ ∧ ω γ + χαβ γ ∧ ω γ ≡ 0 (mod ω ), and therefore

χβ αγ ∧ ω γ ≡ Sβ γασ ω γ ∧ ω σ

(mod ω )

for some functions Sβ γασ . Symmetry properties (1.25) follow immediately from (1.24).   We will now impose conditions on the functions Sβ γασ from Lemma 1.2 to eliminate the remaining freedom in the choice of ϕβα· (see (1.19)). Lemma 1.3. The functions Dαβ · are uniquely determined by the conditions α Sρσ := Sαρ ·σ = 0.

(1.26) γ

Proof. We need to understand how the functions Sαρ ·σ change when a transformation of the form (1.19) is performed. Set S := Sαα · ,

D := Dαα · .

Since gαβ , Sαβ are Hermitian (see (1.25)) and Dαβ are skew-Hermitian (see (1.20)), it follows that S is real-valued and D is imaginary-valued. Indicating the new functions by tildas, we find   γ γ γ γ γ γ S˜αρ ·σ = Sαρ ·σ − i Dα · gρσ + Dρ ·gασ − δρ Dσα − δα Dσ ρ . Then we obtain   S˜ρσ = Sρσ − i gρσ D + Dρσ − (n + 1)Dσρ . To finish the proof of the lemma, we need to show that there exist uniquely defined Dαβ · satisfying (1.20) and such that gρσ D + (n + 2)Dρσ = −iSρσ .

(1.27)

Contracting (1.27) we get D=−

i S. 2(n + 1)

Substituting this back into (1.27) yields   1 i −iSρσ + Sgρσ . Dρσ = n+2 2(n + 1)

(1.28)

14

1 Invariants of CR-Hypersurfaces

It is immediately verified that the functions Dρσ given by formulas (1.28) satisfy (1.20) and (1.27).   From now on we assume that conditions (1.26) are satisfied, thus ϕβα· are uniquely defined. Further, Lemma 1.2 yields

Γβα· = Sβαρ ·σ ω ρ ∧ ω σ + λβα· ∧ ω ,

(1.29)

where λβα· are 1-forms. It follows from (1.14), (1.21), (1.23), (1.29) that d ϕ α − ϕ ∧ ϕ α − ϕ β ∧ ϕβα· − λβα· ∧ ω β = κ α ∧ ω ,

(1.30)

where κ α are also 1-forms. From (1.24), (1.25), (1.29) we get

λβ α + λαβ + gβ α ψ ≡ 0 (mod ω ).

(1.31)

We now differentiate (1.29) retaining only the terms that involve ω ρ ∧ ω σ . In doing so we use the following formulas, which are immediately obtained from (1.12), (1.15), (1.30): d ωα = −ω β ∧ ϕαβ + ωα ∧ ϕ + ω ∧ ϕα , (1.32) β β d ϕα = ϕαβ ∧ ϕ + λβ α ∧ ω + κα ∧ ω . Identities (1.11) (1.12), (1.21), (1.23), (1.29), (1.30), (1.32) then yield γ

γ

γ

γ

α ϕ − Sα α α dSβαρ ·σ − Sγρ ·σ β · β γ ·σ ϕρ · + Sβ ρ ·σ ϕγ · − Sβ ρ ·γ ϕσ ≡

i(λβα· gρσ + λρα· gβ σ − δβα λσ ρ − δρα λσ β ) (mod ω , ω γ , ω γ ), and by contraction we get γ

γ

β

dSρσ − Sγσ ϕρ · − Sργ ϕσ · ≡ i(λβ · gρσ + λρσ − (n + 1)λσρ ) (mod ω , ω γ , ω γ ). Now (1.26) and (1.31) imply 1 λρσ ≡ − gρσ ψ 2

(mod ω , ω γ , ω γ ).

Hence,

1 λρσ ≡ − gρσ ψ + Vρσβ ω β + Wρσβ ω β (mod ω ) (1.33) 2 for some functions Vρσβ , Wρσ β . Substituting this expression into (1.31) we obtain Vρσβ + Wσρβ = 0.

It now follows from (1.29) that

(1.34)

1.2 Chern’s Construction

15

1 Φβα· := Γβα· + δβα ψ ∧ ω = Sβαρ ·σ ω ρ ∧ ω σ + Vβα·ρ ω ρ ∧ ω − V·αβ σ ω σ ∧ ω . 2

(1.35)

Therefore, substitution of Γβα· − iϕβ ∧ ω α − iδβα (ϕσ ∧ ω σ ) into (1.14) implies 1 Φ α := d ϕ α − ϕ ∧ ϕ α − ϕ β ∧ ϕβα· + ψ ∧ ω α = 2

(1.36)

−Vβα·γ ω β ∧ ω γ + V·αβ σ ω β ∧ ω σ + ν α ∧ ω , where ν α are 1-forms. Formulas (1.35) yield that under transformation (1.19) with Dαβ · = 0 the functions Vβα·ρ change as follows:   1 V˜βα·ρ = Vβα·ρ + i δρα Eβ + δβα Eρ . 2 Contracting we obtain

  1 ρ ρ Eβ . V˜β ·ρ = Vβ ·ρ + i n + 2

This calculation leads to the following lemma. Lemma 1.4. The functions Eβ are uniquely determined by the conditions ρ

Vβ ·ρ = 0.

(1.37)

From now on we assume that conditions (1.37) are satisfied, thus ϕ α are uniquely defined. Next, we differentiate identity (1.18). Using (1.11), (1.12), (1.31), (1.32), (1.36), we get

ω ∧ (−d ψ + ϕ ∧ ψ + 2iϕ β ∧ ϕβ − iω β ∧ νβ − iν β ∧ ωβ ) = 0. Therefore, we have

Ψ := d ψ − ϕ ∧ ψ − 2iϕ β ∧ ϕβ = −iω β ∧ νβ − iν β ∧ ωβ + ξ ∧ ω ,

(1.38)

where ξ is a 1-form. We now differentiate (1.36) retaining only the terms that involve ω ρ ∧ ω σ . Using identities (1.11), (1.12), (1.18), (1.21), (1.23), (1.35), (1.36), (1.38), we obtain β

β

γ

dV·αρσ − V·αβ σ ϕρ · + V·ρσ ϕβα· − V·αργ ϕσ · − V·αρσ ϕ ≡ i Sβαρ ·σ ϕ β + igρσ ν α + δρα νσ 2 Conditions (1.37) are equivalent to

(mod ω , ω γ , ω γ ).

(1.39)

16

1 Invariants of CR-Hypersurfaces

V·αρσ gρσ = 0. Differentiating these identities and using (1.15), (1.25), (1.26), (1.37), (1.39), we obtain ν α ≡ 0 (mod ω , ω γ , ω γ ). Hence, we have

ν α ≡ Pβα· ω β + Qαβ· ω β

(mod ω )

(1.40)

for some functions Pβα· , Qα . Substitution of (1.40) into (1.36) now yields β·

Φ α = −Vβα·γ ω β ∧ ω γ + V·αβ σ ω β ∧ ω σ + Pβα·ω β ∧ ω + Qαβ· ω β ∧ ω .

(1.41)

Further, substituting (1.40) into (1.38) and absorbing into ξ the indeterminacy of ν α in ω , we obtain

Ψ = iQαβ ω α ∧ ω β − iQαβ ω α ∧ ω β − iPαβ ω α ∧ ω β + ξ ∧ ω , where

Eα

Pαβ := Pαβ + Pβ α .

(1.42) (1.43)

Formulas (1.36), (1.41) imply that under transformation (1.19) with Dαβ · = 0 and = 0 the functions Pβα· change as follows: 1 P˜βα· = Pβα· + δβα T, 2

which gives

n P˜αα· = Pαα· + T. 2 On the other hand, from (1.43) we see

(1.44)

Pαα· = 2 Re Pαα· , and therefore (1.44) yields

α

˜ = Pα + nT. P α· α·

This leads us to the following lemma. Lemma 1.5. The function T is uniquely determined by the condition Pαα· = 0.

(1.45)

With condition (1.45) satisfied, the form ψ is uniquely defined. Thus, the locally defined forms ϕβα· , ϕ α , ψ give rise to 1-forms (which we denote by the same respective symbols) defined on all of P 2 .

1.2 Chern’s Construction

17

We will now finalize our formula for Ψ . We differentiate (1.42) retaining only the terms that involve ω ρ ∧ ω σ . Using identities (1.11), (1.12), (1.18), (1.32), (1.33), (1.34), (1.36), (1.38), (1.42), we obtain β

γ

β σ ϕρ · − P ργ ϕ − P ρσ ϕ ≡ d Pρσ − P σ· β

2V·ρσ ϕβ + 2Vβ σρ ϕ β − gρσ ξ

(mod ω , ω γ , ω γ ).

(1.46)

Clearly, condition (1.45) can be written as follows: Pαβ gαβ = 0. Differentiating this identity and using (1.15), (1.37), (1.45), (1.46), we get

ξ ≡ 0 (mod ω , ω γ , ω γ ). Since Ψ is real-valued, we can write (1.42) in the form

Ψ = iQαβ ω α ∧ ω β − iQαβ ω α ∧ ω β − iPαβ ω α ∧ ω β + Rα ω α ∧ ω + Rα ω α ∧ ω

(1.47)

for some functions Rα . ´ Cartan Remark 1.1. For n = 1 all formulas derived above reduce to those given by E. in [17]. We now assume that the matrix g = (gαβ ) is constant and define a Hermitian   form H g on Cn+2 with matrix Hlmg l,m=0,...,n+1 by setting H g := gαβ , αβ

i H0gn+1 := − , 2

i g := , Hn+1 0 2

(1.48)

and letting the remaining matrix entries to be zero. Let SU± H g be the group of matrices A ∈ SL(n + 2, C) such that AH g A∗ = ±H g . The choice AH g A∗ = −H g is only possible if the numbers of positive and negative eigenvalues of the form g coincide, in which case the group SU± H g has exactly two connected components. If the numbers of positive and negative eigenvalues of g are distinct, SU± H g is connected. ± ± Let PSU± := SU /Z , where Z is the center of SU . We denote by H1 the Hg  Hg Hg ◦ ± subgroup of SUH g that consists of all matrices ⎛

t 0 0

⎞

⎜ tα tα 0 ⎟ ⎝ β· ⎠,

τ τβ t where |t| = 1 and the following holds:

(1.49)

18

1 Invariants of CR-Hypersurfaces

(i) t α = −2it ∑ tβα· gβ γ τγ , (ii) (iii)

βγ 2 α t det(tβ · ) = 1, β

∑ tρα·tσ · gρσ = gαβ ,

(1.50)

ρ ,σ

(iv)

i

∑ gρσ τρ τσ + 2 (τ t − τ t −1) = 0.

ρ ,σ

Let χ : H1 → G1 be the homomorphism that assigns matrix (1.10), with vα = it ∑ t β gαβ , β

−1  (uαβ · ) = t (tβα· )T , s = 4 Re(τ t −1 ), to matrix (1.49). The homomorphism χ is onto and its kernel coincides with Z . Hence, G1 is isomorphic to H1 /Z ⊂ PSU± H g , and we denote by χ1 the isomorphism between H1 /Z and G1 induced by χ . The Lie algebra suH g of SU± H g consists of all matrices A ∈ sl(n + 2, C) such that AH g + H g A∗ = 0. We now define an suH g -valued absolute parallelism σ = (σlm )l,m=0,...,n+1 on P 2 by the formulas

σ00 := −

1 (ϕ α + ϕ ) , σα0 := ω α , n + 2 α·

0 σn+1 := 2ω ,

σ0α := −iϕα ,

α := 2iω , σβα := ϕαβ · + δαβ σ00 , σn+1 α

1 σ0n+1 := − ψ , 4

1 σαn+1 := ϕ α , 2

(1.51)

n+1 σn+1 := −σ00 .

It is easy to observe that σ defines an isomorphism between TΘ (P 2 ) and suH g for every Θ ∈ P 2 (see (1.15)). Consider the following form called the curvature form of σ : 1 Σ := d σ − [σ , σ ] = d σ − σ ∧ σ . 2

(1.52)

This is an suH g -valued 2-form with

Σ = (Σlm )l,m=0,...,n+1 ,

Σlm := d σlm − σkm ∧ σlk .

0 , It is often referred to as the CR-curvature form of M. The components Σα0 , Σn+1 α Σn+1 are called the torsion of σ . Conditions (1.11), (1.12), (1.18) yield that the torsion of σ in fact vanishes. Further, a straightforward calculation shows

1.2 Chern’s Construction

Σ00 = −

19

1 Φα , n + 2 α· 1 δαβ Φγγ· , n+2

Σ0α = −iΦα ,

Σβα = Φαβ · −

1 Σ0n+1 = − Ψ , 4

1 Σαn+1 = Φ α , 2

(1.53) n+1 Σn+1 = −Σ00 .

For any 2-form Ω on P 2 in ω α , ω α , ω such that

Ω ≡ aαβ ω α ∧ ω β + terms quadratic in ω γ , ω γ set

(mod ω )

Tr Ω := aαα · .

Then conditions (1.26), (1.37), (1.45) can be restated, respectively, as follows: (i) Tr Σβα = 0,

Tr Σ00 = 0,

(ii) Tr Σ0α = 0,

Tr Σαn+1 = 0,

(iii) Tr Σ0n+1 = 0, and their totality can be summarized by the equation Tr Σ = 0.

(1.54)

It follows from Chern’s construction described above that the absolute parallelism σ defined in (1.51) is uniquely determined by the vanishing of its torsion and by condition (1.54). To describe further properties of σ , we need a general definition. Let R be a Lie group with Lie algebra r and S a closed subgroup of R with Lie algebra s ⊂ r acting by diffeomorphisms on a manifold P such that dim P = dim R. For every element s ∈ s denote by Xs the fundamental vector field arising from the one-parameter subgroup {exp(ts), t ∈ R} of S, i.e. Xs (x) :=

 d   exp (−ts) x  , dt t=0

x ∈ P.

A Cartan connection of type R/S on the manifold P is an r-valued absolute parallelism ρ on P such that (i) ρ (x)(Xs (x)) = s for all s ∈ s and x ∈ P, and (ii) L∗a ρ = AdS,r (a)ρ for all a ∈ S, where La denotes the action of a on P and AdS,r is the adjoint representation of S. A straightforward calculation shows that, upon identification of the group G1 with the group H1 /Z ⊂ PSU± H g by means of the isomorphism χ1 , the absolute parallelism σ is in fact a Cartan connection of type PSU± H g /G1 on the bundle P 2 → P 1 . Thus, we have proved the following theorem.

20

1 Invariants of CR-Hypersurfaces

Theorem 1.1. [24] If g is a non-degenerate Hermitian form on Cn and Cg the collection of CR-hypersurfaces of CR-dimension n whose Levi form at every point is equivalent to g, then the CR-structures of the manifolds in Cg are 2-reducible to absolute parallelisms. For M ∈ Cg the absolute parallelism σ on P 2 → P 1 → M establishes an isomorphisms between TΘ (P 2 ) and the Lie algebra suH g at every point Θ ∈ P 2 . Furthermore, σ is a Cartan connection on P 2 → P 1 and is determined by the vanishing of the torsion and curvature condition (1.54). As was noted by S. Webster (see the Appendix to [24]), there are further symmetries for the functions occurring in formulas (1.35), (1.41), (1.47), which give expansions of the components of the CR-curvature form Σ with respect to ω α , ω α , ω . The additional symmetries follow from the Bianchi identities, which one obtains by differentiating equation (1.52). Namely, differentiation of (1.52) yields dΣ = σ ∧ Σ − Σ ∧ σ , which in terms of components is written as follows: d Σlm = σkm ∧ Σlk − Σkm ∧ σlk .

(1.55)

Webster shows that the Bianchi identities imply Vαα·β = 0,

Vαβ γ = Vγβ α ,

Qαβ = Qβ α ,

Pαβ = Pβ α .

(1.56)

Hence, (1.41) and (1.47) become, respectively,

Φ α = V·αβ σ ω β ∧ ω σ + Pβα·ω β ∧ ω + Qαβ· ω β ∧ ω , Ψ = −2iPαβ ω α ∧ ω β + Rα ω α ∧ ω + Rα ω α ∧ ω .

(1.57)

Identities (1.26), (1.35), (1.37), (1.56) yield Φαα· = 0. Thus, (1.53) implies

Σ00 = 0,

n+1 Σn+1 = 0,

Σβα = Φαβ · .

(1.58)

In addition, from (1.56) we see Pαα· = 2 Re Pαα· = 2Pαα· , and therefore condition (1.45) is equivalent to Pαα· = 0.

(1.59)

For a non-degenerate C-valued Hermitian form g on Cn consider the quadric Qg associated to g (see (1.4)). We will now give an explicit description of the group Bir(Qg ). Consider CPn+1 with homogeneous coordinates Z = (ζ0 : ζ1 : . . . : ζn+1 ) and realize Cn+1 in CPn+1 as the set of points (1 : z1 : . . . : zn : w). Let Qg be the closure of Qg in CPn+1 . Clearly, we have

1.2 Chern’s Construction

21

  Qg = Z ∈ CPn+1 : H g (Z, Z) = 0 ,

(1.60)

where Z := (ζ0 , ζ1 , . . . , ζn+1 ) and H g is the Hermitian form defined in (1.48), that is, H g (Z, Z) = g(ζ , ζ ) + i/2(ζn+1 ζ 0 − ζ0 ζ n+1 ) with ζ := (ζ1 , . . . , ζn ). We consider Qg with the CR-structure induced by CPn+1 . If g is sign-definite, Qg is CR-equivalent to the unit sphere in Cn+1 . In general, Qg is CR-equivalent to the closure Sg in CPn+1 of the hypersurface Sg defined in (1.6). Indeed, we have   Sg = Z ∈ CPn+1 : g(ζ , ζ ) + |ζn+1 |2 − |ζ0 |2 = 0 , and the map

Z → (ζ0 − ζn+1 : ζ1 : . . . : ζn : i(ζ0 + ζn+1 ))

(1.61)

transforms Sg into Qg (observe that the restriction of map (1.61) to Cn+1 \ {w = 1} coincides with map (1.5)). n+1 by assigning a matrix We define an action of the group SU± H g on CP   n+1 given by Z → AT −1 Z. A ∈ SU± H g the holomorphic automorphism of CP Clearly, every such automorphism preserves Qg , thus its restriction to Qg is a CR-automorphism of Qg . The kernel of this action is the center Z of SU± H g , hence ± the group PSUH g acts on Qg effectively and transitively by CR-automorphisms. One can show that every local automorphism of Qg extends to a CR-automorphism of Qg induced by this action. This continuation result goes back to Poincar´e for the case n = 1 (see [90]). It was obtained by Tanaka in [99] for arbitrary n ≥ 1 and g for all local CR-automorphisms of Qg that can be holomorphically continued to a neighborhood in Cn+1 of a domain in Qg (see also [1]). In fact, every local CR-automorphism of Qg admits a local holomorphic continuation required by Tanaka’s result. Indeed, let f : V → V  be a CR-isomorphism between domains V and V  in Qg . If the form g is indefinite, the existence of a holomorphic continuation of f to a neighborhood of V in Cn+1 follows from a well-known fact that appears as Theorem 3.3.2 in [22] (see references therein for details). If the form g is signdefinite, a continuation of f to a neighborhood of V is provided by [88]. [Note that the existence of a local holomorphic continuation also follows from Theorem 3.1 of [3].] Thus, the group Bir(Qg ) endowed with the compact-open topology arising from its action on Qg admits the structure of a Lie group isomorphic to PSU± H g. It can be shown that the Lie algebra of Bir(Qg ) with respect to this structure is isomorphic to the Lie algebra of infinitesimal CR-automorphisms of Qg . As a Lie group, Bir(Qg ) acts on Qg transitively by CR-automorphisms. Clearly, Bir(Qg ) is connected if the numbers of positive and negative eigenvalues of g are distinct and has exactly two connected components otherwise. From now on we identify the g group Bir(Qg ) with PSU± H g and its Lie algebra with suH . [We note in passing that the effect of continuation of local CR-automorphisms and, more generally, locally defined CR-isomorphisms to globally defined maps for manifolds other than Qg has been observed by many authors (see, e.g. [62], [69], [78], [86], [89], [107]). A related continuation result for global CR-automorphisms in the case where the Hermitian form g is degenerate was obtained in [63].]

22

1 Invariants of CR-Hypersurfaces

Let H := Bir0 (Qg ) and H0 be the subgroup of SU± H g that consists of all matrices of the form ⎞ ⎛ t 0 0 ⎜ tα tα 0 ⎟ ⎠, ⎝ β·

τ τβ ±t −1 where conditions (1.50) are replaced by the conditions (i) t α = ∓2it ∑ tβα· gβ γ τγ , βγ

(ii) ±tt −1 det(tβα· ) = 1, (iii)

β

∑ tρα·tσ · gρσ = ±gαβ ,

ρ ,σ

(iv)

i

∑ gρσ τρ τσ ± 2 (τ t −1 − τ t −1) = 0,

ρ ,σ

with the bottom choice of the sign only possible if the numbers of positive and negative eigenvalues of the form g are equal. Clearly, H1 is the codimension one subgroup of H0 given by the top choice of the sign and the condition |t| = 1 (see (1.49), (1.50)). It is straightforward to check that the isomorphism PSU± H g → Bir(Qg ) identifies the subgroup H0 /Z with H, thus the group G1  H1 /Z can be viewed as a codimension one subgroup of H. It was shown in [9] that the manifold P 2 constructed above is in fact a principal H-bundle over M with the projection π := π 1 ◦ π 2 and, upon identification of H and H0 /Z , the parallelism σ is a Cartan connection of type PSU± H g /H on the bundle P 2 → M. Thus, the following variant of Theorem 1.1 holds. Theorem 1.2. [9], [24] The CR-structures of the manifolds from Cg are 1-reducible to absolute parallelisms. For M ∈ Cg the absolute parallelism σ on P 2 → M establishes an isomorphisms between TΘ (P 2 ) and the Lie algebra suH g at every point Θ ∈ P 2 . Furthermore, σ is a Cartan connection on P 2 → M and is determined by the vanishing of the torsion and curvature condition (1.54). Inspection of Chern’s construction yields that for the manifold Qg the bundle πg

P 2 → Qg is the bundle Bir(Qg ) → Bir(Qg )/H, where the quotient Bir(Qg )/H is identified with the Bir(Qg )-homogeneous manifold Qg in the usual way and πg is the quotient map. In this case the Cartan connection σ is the Maurer-Cartan form σBir(Qg ) on the group Bir(Qg ). Recall that the Maurer-Cartan form σR on a Lie group R is the right-invariant 1-form with values in the Lie algebra r of R such that σR (e) : r → r is the identity map. The Maurer-Cartan form satisfies the Maurer-Cartan equation 1 d σR − [σR , σR ] = 0 2 and under the left multiplication La by a ∈ R transforms as follows:

1.2 Chern’s Construction

23

L∗a σR = AdR,r (a)σR . The Maurer-Cartan equation implies that the CR-curvature form of Qg vanishes. Conversely, suppose that the CR-curvature form of a manifold M ∈ Cg is zero. Then for every point Θ ∈ P 2 there is a neighborhood U of Θ , a neighborhood identity in Bir(Qg ), and a diffeomorphism F : U → V such that  V of the    ∗  = σ  . By Theorem 1.2 the diffeomorphism F is a lift of a CRF σBir(Q ) g

V

U

isomorphism f : π (U) → πQg (V ). Therefore, every point of M has a neighborhood CR-equivalent to an open subset of Qg . A CR-hypersurface M ∈ Cg is called spherical if it is locally CR-equivalent to Qg , i.e. if every point in M has a neighborhood CR-equivalent to an open subset of Qg . If the signature of the non-degenerate Hermitian form g is (k, n − k) for some 0 ≤ k ≤ n, and M is locally CR-equivalent to Qg , we also say that M is (k, n − k)-spherical. It is usually assumed, without loss of generality, that n ≤ 2k. [We will generalize the above definition of sphericity to the Levi degenerate case in Section 9.1. Until then we only consider Levi non-degenerate CRhypersurfaces.] Further, a CR-hypersurface with vanishing CR-curvature form is called CR-flat. We summarize the content of the preceding paragraph as follows. Corollary 1.1. A CR-hypersurface is spherical if and only if it is CR-flat. In this book we study spherical CR-hypersurfaces. Corollary 1.1 and formulas (1.53), (1.58) yield that such CR-hypersurfaces are characterized by the conditions

Φαβ · = 0,

Φ α = 0,

Ψ = 0,

or, equivalently, by the conditions β

Sαρ ·σ = 0,

β

Vα ·ρ = 0,

Pβα· = 0,

Qαβ · = 0,

Rβ = 0.

(1.62)

Due to the transformation law L∗a σ = AdH,suH g (a)σ ,

a ∈ H,

where La is the (left) action of a on the bundle P 2 → M, the CR-curvature form Σ transforms in a similar way L∗a Σ = AdH,suH g (a)Σ .

(1.63)

Transformation law (1.63) implies that conditions (1.62) hold everywhere on P 2 if for every p ∈ M there is a local section ΓW of P 2 over a neighborhood W of p in M such that these conditions hold on the submanifold ΓW (W ) of P 2 . Throughout the book we only consider real hypersurfaces in complex manifolds with induced CR-structure, and our next step is to write sphericity conditions (1.62) on a certain local section of P 2 defined in terms of a local defining function of the hypersurface (cf. [76], Section 5).

24

1 Invariants of CR-Hypersurfaces

1.3 Chern’s Invariants on Section of Bundle P 2 → M Let M be a Levi non-degenerate CR-hypersurface with an integrable CR-structure of CR-dimension n. Fix a Hermitian form on Cn with matrix g which is equivalent to every LM (p), p ∈ M, and consider the fiber bundle P 1 over M and the tautological 1-form θ 0 on P 1 as constructed in Section 1.2. Let W be an open subset of M and U := [π 1 ]−1 (W ). Further, let G = (Gαβ ) be a matrix-valued map on U such that for every θ ∈ U the value G (θ ) is the matrix of a Hermitian form whose signature coincides with that of the Hermitian form defined by g. Then for every θ ∈ U there exist a real-valued covector θ n+1 and complex-valued covectors θ α on Tθ (P 1 ) such that: (a) each θ α is a lift of a complex-valued covector on Tπ 1 (θ ) (M) which is complex-linear on Tπc1 (θ ) (M), (b) the covectors θ 0 (θ ), Re θ α , Im θ α , θ n+1 form a basis of the cotangent space Tθ∗ (P 1 ), and (c) the following identity holds: d θ 0 (θ ) = iGαβ (θ )θ α ∧ θ β + θ 0 (θ ) ∧ θ n+1 .

(1.64)

The most general linear transformation of θ 0 (θ ), θ α , θ α , θ n+1 preserving equation (1.64) and the covector θ 0 (θ ) is given by the matrix (acting on the left) ⎛ ⎞ 1 0 0 0 ⎜ α ⎟ uαβ 0 0⎟ ⎜v ⎜ ⎟ , ⎜ vα 0 uα 0⎟ ⎜ ⎟ β ⎝ ⎠ ρ s iGρσ (θ )uβ vσ −iGρσ (θ )uσ vρ 1 β

β

where s ∈ R, uαβ , vα ∈ C and Gαβ (θ )uρα uσ = Gρσ (θ ). For θ ∈ U let Pθ2,G be the collection of all covectors (θ 0 (θ ), θ α , θ n+1 ) on Tθ (P 1 ) satisfying conditions (a), (b), (c) above. The sets Pθ2,G , θ ∈ U, form a fiber bundle over U, which we denote by PG2 . Let πG2 : PG2 → U be the projection (θ 0 (θ ), θ α , θ n+1 ) → θ . For every point θ0 ∈ U there is a neighborhood U0 of θ0 in U such that the open sets [πG2 ]−1 (U0 ) and [π 2 ]−1 (U0 ) are diffeomorphic, with the fiber Pθ2,G mapped onto the fiber Pθ2 for every θ ∈ U0 as follows: F : (θ 0 (θ ), θ α , θ n+1 ) → (θ 0 (θ ), Cβα (θ )θ β , θ n+1 ), where Cβα are complex-valued functions on U0 and the matrix (Cβα ) is everywhere non-degenerate. Next, set ωG := [πG2 ]∗ θ 0 and introduce a collection of tautological 1-forms on PG2 as follows:

ωGα (Θ )(Y ) := θ α (d πG2 (Θ )(Y )),

ϕG (Θ )(Y ) := θ n+1 (d πG2 (Θ )(Y )),

1.3 Chern’s Invariants on Section of Bundle P 2 → M

25

where Θ = (θ 0 (θ ), θ α , θ n+1 ) is a point in Pθ2,G and Y ∈ TΘ (PG2 ). Identity (1.64) implies   β

d ωG = i [πG2 ]∗ Gαβ ωGα ∧ ωG + ωG ∧ ϕG .

As in Section 1.2, starting with the forms ωG , ωGα , ϕG we can construct 1-forms ϕβα·,G , ϕGα , ψG and 2-forms Φβα·,G , ΦGα , ΨG on PG2 (recall that in our calculations in Section 1.2 we allowed (gαβ ) to be a matrix-valued map). A straightforward calculation yields that on [πG2 ]−1 (U0 ) we have

ωG = F ∗ ω , ωGα = Dβα F ∗ ω β , ϕG = F ∗ ϕ ,

γ

γ

ϕβα·,G = −dDγα · Cβ + Dγα Cβν F ∗ ϕν · , ϕGα = Dβα F ∗ ϕ β , ψG = F ∗ ψ and

μ

γ

Sβαρ ·σ ,G = Dγα Cβν Cρ Cση F ∗ Sν μ ·η , μ

γ

Vβα·ρ ,G = Dγα Cβν Cρ F ∗Vν ·μ , γ

Pβα·,G = Dγα Cβν F ∗ Pν · ,

(1.65)

γ

Qα

β ·,G

= Dγα C η F ∗ Qη · ,

β γ Rα ,G = Cα F ∗ Rγ ,

where (Dβα ) is the matrix inverse to (Cβα ) and Sβαρ ·σ ,G , Vβα·ρ ,G , Pβα·,G , Qαβ·,G , Rα ,G

are the corresponding functions in the expansions of the forms Φβα·,G , ΦGα , ΨG with ρ

ρ

respect to the forms ωG , ωG , ωG . Let γU,G : U → PG2 be a section of PG2 and γW a local section of P 1 over W . Formulas (1.65) imply that if the functions Sβαρ ·σ ,G , Vβα·ρ ,G , Pβα·,G , Qα , Rα ,G β ·,G

vanish on the submanifold (γU,G ◦ γW )(W ) of PG2 , then conditions (1.62) hold on the submanifold ΓW (W ) of P 2 , where ΓW := F ◦ γU,G ◦ γW is a section of the bundle P 2 → M over the set W . [Here we assume for simplicity that F is defined on all of [πG2 ]−1 (U). To be absolutely precise, one must consider for every θ0 ∈ U a neighborhood U0 as above.] Suppose now that M is an immersed Levi non-degenerate real hypersurface in a complex manifold N of dimension n + 1 with n ≥ 1. Fix p ∈ M and consider a neighborhood M  of p in M which is locally closed in N.3 Then there exist a neighborhood W of p in N, holomorphic coordinates z0 , z = (z1 , . . . , zn ) in W , and 3

We say that an immersed submanifold S of a manifold R is locally closed if the immersion ι : S → R, ι (x) := x, is a locally proper map, or, equivalently, if S is a closed submanifold of an open submanifold of R. We say that S is closed in R if ι is a proper map.

26

1 Invariants of CR-Hypersurfaces

a real-valued function r(z0 , z0 , z, z) on W such that the set W := M  ∩ W coincides with the set {r = 0} and r0 := ∂ r/∂ z0 = 0 on W (note that r0 := ∂ r/∂ z0 = 0 on W as well since r0 = r0 ).4 The CR-structure of M, being induced by N, is given on W by setting    ∂r β  0  μ = i∂ r := i dz + r0 dz  , W W ∂ zβ (1.66)   α α η = dz  W

(cf. the beginning of Section 1.2). Then on W we have d μ = ihαβ dzα ∧ dzβ + μ ∧ φ , with

(1.67)

hαβ = −rαβ + r0−1 rα r0β + r0−1 rβ r0α − |r0 |−2 r00 rα rβ ,

  φ := −r0−1 r0γ dzγ − r0−1 r0γ dzγ + |r0 |−2 r00 rγ dzγ + rγ dzγ

(1.68)

(cf. (1.7)), where we use the following notation: rα :=

∂r , ∂ zα

rβ :=

∂r ∂ zβ

,

rαβ :=

∂ 2r ∂ zα ∂ zβ

,

etc.

Clearly, for every q ∈ W the Levi form of M at q is equivalent to the Hermitian form with the matrix h(q) := (hαβ (q)). We now choose a matrix g such that the Hermitian form defined by g has the same signature as the Hermitian form defined by h(q) for every q ∈ W . Then the fiber of the bundle P 1 over q is {uμ (q) : u > 0} in the case where the numbers of positive and negative eigenvalues of g are distinct and {u μ (q) : u ∈ R∗ } otherwise. For the form θ 0 on U = [π 1 ]−1 (W ) we have     du d θ 0 = iu [π 1 ]∗ hαβ [π 1 ]∗ dzα ∧ [π 1 ]∗ dzβ + θ 0 ∧ − + [π 1 ]∗ φ . u We now let Gαβ = u[π 1 ]∗ hαβ on U and choose the section γU,G as follows:    1 ∗ α du 0 1 ∗ γU,G (uμ (q)) = θ (uμ (q)), [π ] dz (u μ (q)), − + ([π ] φ )(u μ (q)) . u Next, choose the section γW by setting u = 1, i.e. γW (q) = μ (q). Our goal is to compute the forms ωG , ωGα , ϕG , ϕβα·,G , ϕGα , ψG and the functions Sβαρ ·σ ,G , Vβα·ρ ,G , Pβα·,G , Qα , Rα ,G on the submanifold W := (γU,G ◦ γW )(W ) of PG2 . In fact, we β ·,G

4

For notational convenience, in this section we index coordinates by superscripts rather than subscripts. We will return to indexing coordinates by subscripts in Chapter 2.

1.3 Chern’s Invariants on Section of Bundle P 2 → M

27

compute the push-forwards of these quantities to W under the diffeomorphism π 1 ◦ πG2 W : W → W . Clearly, on W we have   ωG = [πG2 ]∗ [π 1 ]∗ μ ,   ωGα = [πG2 ]∗ [π 1 ]∗ dzα ,   ϕG = [πG2 ]∗ [π 1 ]∗ φ ,    thus the push-forwards of ωG W , ωGα W , ϕG W from W to W are μ , dzα , φ , respectively. Differentiating (1.67) we obtain i(dhαβ + hαβ φ ) ∧ dzα ∧ dzβ − μ ∧ d φ = 0. Hence,

dhαβ + hαβ φ = aαβ γ dzγ + aβ αγ dzγ + cαβ μ ,

(1.69)

d φ = icαβ dzα ∧ dzβ + μ ∧ ζ (1) for some 1-form ζ (1) and functions aαβ γ , cαβ satisfying aαβ γ = aγβ α ,

cαβ = cβ α .

With μ given in (1.66) and hαβ , φ given in (1.68), the functions aαβ γ , cαβ and the

form ζ (1) are completely determined by formulas (1.69) if we assume that ζ (1) is a linear combination of dzα and dzα . These quantities involve partial derivatives of the function r up to order 3. Everywhere below indices are lowered by means of the matrix h = (hαβ ) and γ

γ β

raised by means of its inverse (hαβ ), where hαβ hγβ = δα , hαβ hαγ = δ . Set α (1)

φβ ·

1 := aαβ ·γ dzγ + cαβ · μ , 2

1 φ α (1) := cαβ · dzβ . 2

(1.70)

Identities (1.69) imply α (1)

dzβ ∧ φβ ·

+ μ ∧ φ α (1) = 0, (1) (1) −φ αβ βα

dhαβ + hαβ φ − φ

d φ = idzβ ∧ φ β (1) + iφ

= 0,

(1) ∧ dzβ β

(1.71) + μ ∧ ζ (1) .

 On the other hand, let 1-forms φβα· , φ α , ζ be the push-forwards of ϕβα·,G W ,   ϕGα W , ψG W from W to W , respectively. It follows from identities (1.12), (1.15), (1.18) applied to the forms ωG , ωGα , ϕG , ϕβα·,G , ϕGα , ψG that φβα· , φ α , ζ satisfy

28

1 Invariants of CR-Hypersurfaces

dzβ ∧ φβα· + μ ∧ φ α = 0, dhαβ + hαβ φ − φαβ − φβ α = 0,

(1.72)

d φ = idzβ ∧ φ β + iφβ ∧ dzβ + μ ∧ ζ . α (1)

It is straightforward to see from (1.71), (1.72) that φβα· , φ α , ζ are related to φβ ·

φ α (1) ,

ζ (1)

,

as follows: α (1)

φβ ·

= φβα· + dβα· μ ,

φ α (1) = φ α + dβα·dzβ + eα μ ,

(1.73)

ζ (1) = ζ + t μ + i(eα dzα − eα dzα ), where dβα· , eα , t are functions on W , t is real-valued and the following holds: dαβ + dβ α = 0.

(1.74)

We will now find dβα· , eα , t from conditions (1.26), (1.37), (1.59). Identities (1.35) imply γ

d φβα· − φβ · ∧ φγα· − idzβ ∧ φ α + iφβ ∧ dzα + iδβα (φσ ∧ dzσ ) ≡ Sβαγ ·σ dzγ ∧ dzσ

(mod μ ),

(1.75)

 where Sβαγ ·σ are the push-forwards of the functions Sβαγ ·σ ,G W from W to W . It follows from (1.67), (1.73), (1.75) that α (1)

d φβ ·

where

γ (1)

α (1)

− idzβ ∧ φ α (1) +   (1) (1) α (1) iφβ ∧ dzα + iδβα φσ ∧ dzσ ≡ Sβ γ ·σ dzγ ∧ dzσ − φβ · ∧ φγ ·

α (1)

(mod μ ),

Sβ γ ·σ := Sβαγ ·σ + i(dβα·hγσ + dγα hβ σ − δγα dσβ − δβα dσγ ). α (1)

Note that with φβ ·

α (1) functions Sβ γ ·σ

(1.76)

(1.77)

, φ α (1) given by (1.70), where aαβ γ , cαβ are found from (1.69),

are completely determined by formulas (1.76) and involve parthe tial derivatives of r up to order 4. Define (1) α (1) γ (1) (1.78) Sγσ := Sαγ ·σ , S (1) := Sγ · . Contracting (1.77) and using conditions (1.26) we obtain (1)

hγσ d + dγσ − (n + 1)dσγ = −iSγσ , where d := dαα· . Identities (1.74) and (1.79) imply

(1.79)

1.3 Chern’s Invariants on Section of Bundle P 2 → M

29 (1)

hγσ d + (n + 2)dγσ = −iSγσ .

(1.80)

Contracting (1.80) we get d=−

i S (1) . 2(n + 1)

Substituting this back into (1.80) yields   i 1 (1) dγσ = −Sγσ + S (1) hγσ . n+2 2(n + 1)

(1.81)

Formulas (1.81) determine the functions dβα· in terms of partial derivatives of r up to order 4, and we set φ α (2) := φ α (1) − dβα·dzβ . (1.82) Next, identities (1.35) imply γ

d φβα· − φβ · ∧ φγα· − idzβ ∧ φ α + iφβ ∧ dzα + iδβα (φσ ∧ dzσ ) + (1.83) 1 α δβ ζ ∧ μ = Sβαγ ·σ dzγ ∧ dzσ + Vβα·γ dzγ ∧ μ − V·βασ dzσ ∧ μ , 2  where Vβα·γ are the push-forwards of the functions Vβα·γ ,G W from W to W . It follows from (1.73), (1.82), (1.83) that   (2) (2) γ d φβα· − φβ · ∧ φγα· − idzβ ∧ φ α (2) + iφβ ∧ dzα + iδβα φσ ∧ dzσ + (1.84) 1 α (1) α (1) α (1) δβ ζ ∧ μ = Sβαγ ·σ dzγ ∧ dzσ + Vβ ·γ dzγ ∧ μ − V·β σ dzσ ∧ μ , 2 where α (1)

Vβ · γ

  1 := Vβα·γ − i δγα eβ + δβα eγ . 2

(1.85)

Note that with ζ (1) found from (1.69), φβα· given by

φβα· = aαβ ·γ dzγ +



 1 α cβ · − dβα· μ , 2

(1.86)

φ α (2) given by (1.82), φ α (1) given by (1.70), where aαβ γ , cαβ are found from (1.69) α (1)

and dβα· are found from (1.81), the functions Vβ ·γ are completely determined by formulas (1.84) and involve partial derivatives of r up to order 5. Contracting (1.85) and using conditions (1.37) we obtain eβ =

2i α (1) V . 2n + 1 β ·α

(1.87)

Formulas (1.87) determine the functions eα in terms of partial derivatives of r up to order 5, and we set

30

1 Invariants of CR-Hypersurfaces

  ζ (2) := ζ (1) − i eα dzα − eα dzα .

(1.88)

Further, identities (1.36), (1.57) imply 1 d φ α − φ ∧ φ α − φ β ∧ φβα· + ζ ∧ dzα = 2 V·βασ dzβ

∧ dzσ

(1.89)

+ Pβα· dzβ

∧ μ + Q α dzβ β·

∧ μ,

  where Pβα· and Qα are the push-forwards of the functions Pβα·,G W and Qα W β· β ·,G from W to W , respectively. It follows from (1.73), (1.88), (1.89) that 1 d φ α − φ ∧ φ α − φ β ∧ φβα· + ζ (2) ∧ dzα = 2 V·βασ dzβ where

∧ dzσ

(1.90)

α (1) + Pβ · dzβ

∧ μ + Q α dzβ β·

1 := Pβα· − δβα t. 2 Note that with φ given in (1.68), φβα· given by (1.86), φ α given by α (1)

Pβ ·

α

φ =



 1 α α c − dβ · dzβ − eα μ , 2 β·

∧ μ,

(1.91)

(1.92)

ζ (2) given by (1.88), where aαβ γ , cαβ , ζ (1) are found from (1.69), dβα· are found from α (1)

(1.81), and eα are found from (1.87), the functions Pβ · and Q α are completely β· determined by formulas (1.90) and involve partial derivatives of r up to order 6. Contracting (1.91) and using condition (1.59) we obtain 2 α (1) t = − Pα · . n

(1.93)

Formula (1.93) determines the function t in terms of partial derivatives of r up to order 6, and ζ is given by   ζ = ζ (1) − i eα dzα − eα dzα − t μ . (1.94) Finally, identities (1.38), (1.57) imply d ζ − φ ∧ ζ − 2iφ β ∧ φβ = −2iPαβ dzα ∧ dzβ + Rα dzα ∧ μ + Rα dzα ∧ μ , (1.95)  where Rα are the push-forwards of the functions Rα ,G W from W to W . Since the forms φβα· , φ α , ζ have now been determined, identities (1.83), (1.89), (1.95) can be used to find the functions Sβαγ ·σ , Vβα·γ , Pβα· , Q α , Rα in terms of partial derivatives β·

1.4 Umbilicity

31

of r up to order 7. More precisely, Sβαγ ·σ are determined by the partial derivatives of order 4, Vβα·γ by the partial derivatives of order 5, Pβα· and Q α by the partial β· derivatives of order 6, and Rα by the partial derivatives of order 7. The discussion at the end of Section 1.2 and transformation law (1.65) now yield that the system of equations Sβαγ ·σ = 0,

Vβα·γ = 0,

Pβα· = 0,

Qβα· = 0,

Rα = 0

(1.96)

is equivalent to the sphericity of the locally closed portion W of the real hypersurface M. System (1.96) involves partial derivatives of r up to order 7 and is hard to deal with in general. However, for special classes of hypersurfaces it can be simplified and becomes a rather useful tool for identifying spherical hypersurfaces. In this book we consider hypersurfaces of such a kind.

1.4 Umbilicity Before we turn to special classes of hypersurfaces, we will show that system (1.96) can be simplified to some extent in general. To describe this simplification, we introduce the notion of umbilic point in a Levi non-degenerate CR-hypersurface M of CR-dimension n. For n ≥ 2 a point p ∈ M is called umbilic if all functions Sβαγ ·σ vanish on the fiber π −1 (p) of the bundle P 2 → M. For n = 1 conditions (1.26), (1.37), (1.59) become 1 = 0, S11·1

1 V1·1 = 0,

P1·1 = 0,

(1.97)

respectively, and for n = 1 we call a point p ∈ M umbilic if Q11· vanishes on the fiber π −1 (p). Due to transformation law (1.63), it is sufficient to require in the definition of umbilicity that Sβαγ ·σ and Q11· vanish only at some point of the fiber π −1 (p) for n ≥ 2 and n = 1, respectively. We will now prove the following useful proposition. Proposition 1.1. [9] A Levi non-degenerate CR-hypersurface M is spherical if and only if every point of M is umbilic. Proof. If M is spherical, then its every point is umbilic due to conditions (1.62). Conversely, assume that every point of M is umbilic. To show that conditions (1.62) hold on P 2 , we use the Bianchi identities (see (1.55)). First, suppose n = 1. Due to (1.35), (1.53), (1.57), (1.58), (1.97), all components of the curvature form Σ are equal to zero except possibly for  1 1 Σ02 = − Ψ = − R1 ω 1 ∧ ω + R1 ω 1 ∧ ω . 4 4 From identities (1.55) for m = 1, l = 0 and (1.51) we see

32

1 Invariants of CR-Hypersurfaces

ω1 ∧ Ψ = 0, which implies

R1 ω1 ∧ ω 1 ∧ ω = 0.

Hence R1 = 0, and therefore Σ = 0 as required. Now, suppose n ≥ 2. In this case due to (1.35) we have

Φαβ · = Vαβ·ρ ω ρ ∧ ω − V·βασ ω σ ∧ ω .

(1.98)

Further, from identities (1.55) for m = α , l = β and (1.51), (1.53), (1.57), (1.58) we obtain   β β β β β d Φα = ∑ σγα ∧ Φγ + iωα ∧ V·γσ ω γ ∧ ω σ + Pγ · ω γ ∧ ω + Qγ · ω γ ∧ ω + γ

  γ γ i Vαγρ ω γ ∧ ω ρ + Pγα ω γ ∧ ω + Qγα ω γ ∧ ω ∧ ω β − ∑ Φα ∧ σβ .

(1.99)

γ

Considering in identities (1.99) the terms not involving ω and using (1.11), (1.98), β we get Vα ·ρ = 0. Hence, (1.99) yields     ωα ∧ Pγβ· ω γ ∧ ω + Qβγ· ω γ ∧ ω + Pγα ω γ ∧ ω + Qγα ω γ ∧ ω ∧ ω β = 0, β

β

which implies Pγ · = 0 and Qγ · = 0. Thus, all components of the curvature form Σ are equal to zero except possibly for  1 1 Σ0n+1 = − Ψ = − Rα ω α ∧ ω + Rα ω α ∧ ω . 4 4 From identities (1.55) for m = β , l = 0 and (1.51) we see

ωβ ∧ Ψ = 0. Hence Rα = 0, and therefore Σ = 0 as required.   Due to Proposition 1.1 and transformation laws (1.63), (1.65), system of equations (1.96), which characterizes the sphericity of a locally closed portion of an immersed real hypersurface in a complex (n + 1)-dimensional manifold, can be replaced by the system of equations Sβαγ ·σ = 0

(1.100)

1 Q1· =0

(1.101)

for n ≥ 2 and by the single equation

for n = 1. System (1.100) involves partial derivatives of r up to order 4, whereas equation (1.101) involves partial derivatives of r up to order 6.

1.4 Umbilicity

33

We also remark that in the real-analytic case the sphericity condition can be expressed in terms of a so-called complex defining function (see [83], [84]). In this case, analogously to (1.100), (1.101), sphericity is equivalent to a system of equations involving partial derivatives up to order 4 for n ≥ 2 and to a single equation involving partial derivatives up to order 6 for n = 1 .

•

Chapter 2

Rigid Hypersurfaces

Abstract In this short chapter we consider a class of real hypersurfaces in complex manifolds for which zero CR-curvature equations (1.96) (or, equivalently, (1.100) for n ≥ 2 and (1.101) for n = 1) substantially simplify.

2.1 Zero CR-Curvature Equations for Rigid Hypersurfaces Let M be a locally closed real hypersurface in a complex manifold N of dimension n + 1 with n ≥ 1. For every point p ∈ M there exist a coordinate chart W in N containing p and holomorphic coordinates z0 = x0 + iy0 , z = (z1 , . . . , zn )1 in W such that: (i) W = {x0 ∈ I, y0 ∈ J, z ∈ U } for some intervals I, J in R and a domain U in Cn , and (ii) the set W := M ∩ W is given as a graph x0 = F(y0 , z, z), where F is a function on U := {y0 ∈ J, z ∈ U }. The hypersurface M is called rigid if for every p ∈ M the chart W and coordinates z0 , z in W can be chosen so that the function F is independent of y0 , that is, W is given by x0 = F(z, z).

(2.1)

In this case, the value of F at a point in U is equal to the value of F at its projection to U . With W and the coordinates z0 , z so chosen, equation (2.1) is called a rigid representation of M in W . For M given in a rigid representation in W we now set r(z0 , z0 , z, z) :=

1

z0 + z0 − F(z, z) 2

In this section we return to indexing coordinates by subscripts.

A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 2, 

35

36

2 Rigid Hypersurfaces

and find the functions Sβαγ ·σ , Vβα·γ , Q α by going through the calculations of Section β· 1.3. The remaining functions Pβα· and Rα can of course be determined by this process as well, but they are not required for our purposes. Everywhere in this chapter we use the notation Fα :=

∂F , ∂ zα

Fβ :=

∂F , ∂ zβ

Fαβ :=

∂ 2F , ∂ zα ∂ zβ

etc.

(as before, small Greek indices run from 1 to n). In (1.68) we clearly have

φ = 0.

hαβ = Fαβ , From (1.69) we then find aαβ γ = Fαβ γ ,

cαβ = 0,

ζ (1) = 0.

(2.2)

Hence, from (1.70), (1.76) we obtain α (1)

Sβ γ ·σ = −

∂ aαβ ·γ ∂ zσ

= F αν F ρη Fρνσ Fβ γη − F αν Fβ γνσ ,

where we set F αβ := hαβ . Formulas (1.81) then give   i 1 (1) (1) −Sγσ + S Fγσ . dγσ = n+2 2(n + 1)

(2.3)

(2.4)

From (1.77) we now obtain   α (1) Sβαγ ·σ = Sβ γ ·σ − i dβα· Fγσ + dγα· Fβ σ − δγα dσβ − δβα dσ γ .

(2.5)

Next, formulas (1.70), (1.82), (1.84), (1.86) yield α (1)

Vβ ·γ

=−

∂ dβα· ∂ zγ

ρ

ρ

+ aβ ·γ dρα· − aαρ·γ dβ · .

(2.6)

Further, with eβ found from (1.87), identities (1.83), (1.86), (1.92), (1.94) imply V·βασ =

∂ dβα·

i − ieα Fβ σ − δβα eσ . ∂ zσ 2

(2.7)

Finally, from identities (1.86), (1.88), (1.90), (1.92) we get Qβα· = −

∂ eα . ∂ zβ

(2.8)

2.1 Zero CR-Curvature Equations for Rigid Hypersurfaces

37

Note that since the function F depends only on z and z, so do all quantities found above. We will now obtain the main result of this chapter. Theorem 2.1. [57], [58] Let N be a complex manifold of dimension n + 1 with n ≥ 1, and W ⊂ W a real Levi non-degenerate hypersurface given in a rigid representation in a suitable coordinate chart W ⊂ N . Then W is spherical if and only if the corresponding function F satisfies on U a system of partial differential equations of the following form:   γ γ γ Fαβ = Fγ Eγ Fα Fβ + Dα Fβ + Dβ Fα + Cαβ + Hαβ , (2.9) γ

γ

where Eγ , Dα , Cαβ , Hαβ are functions holomorphic on U . Proof. 2 Assume first that M is spherical. In this part of the proof all functions are assumed to be restricted to U . By (1.96) we have Sβαγ ·σ = 0,

V·βασ = 0,

Qβα· = 0.

(2.10)

From the third set of identities in (2.10) and formulas (2.8) we see that eα are holomorphic. Therefore, it follows from the second set of identities in (2.10) and formulas (2.7) that the following holds: i dβα· = ieα Fβ + δβα eρ Fρ + iDαβ , 2

(2.11)

where Dαβ are holomorphic. Next, symmetry conditions (1.74), the first set of identities in (2.10), and formulas (2.3), (2.5), (2.11) yield

∂ aαβ ·γ ∂ zσ

 = ∂ /∂ zσ eα Fβ Fγ + (δβα Fγ + δγα Fβ )(eρ Fρ )+

 ρ ρ Dγα Fβ + Dαβ Fγ + δβα Dγ Fρ + δγα Dβ Fρ .

Integrating these identities, lowering indices, and using (2.2), we obtain   ∂ Fβ γ = ∂ /∂ zσ eα Fα Fβ Fγ + Dαβ Fα Fγ + Dαγ Fα Fβ + Cαβ γ Fα , ∂ zσ where Cαβ γ are holomorphic. Integration of these identities yields a system of partial differential equations of the form (2.9) with Eα = eα and some holomorphic functions Hαβ . Conversely, suppose that the function F satisfies on U system (2.9). Hence, F γ γ satisfies this system everywhere on U, with Eγ , Dα , Cαβ , Hαβ extended to U as 2

This proof corrects the proof of Proposition 2.1 in [57] from formulas (2.12) onwards.

38

2 Rigid Hypersurfaces

functions independent of the variable y0 . Due to Proposition 1.1, to show that W is spherical it is sufficient to prove that identities (1.100) hold for n ≥ 2 and identity (1.101) holds for n = 1. α (1) First, suppose n ≥ 2. To find Sβ γ ·σ from formulas (2.3), we determine the required third-order partial derivatives of F by differentiating the equations of system (2.9) as follows: Fβ γη =

Fρνσ =

  ∂ Fβ γ = Fκη Eκ Fβ Fγ + Dκβ Fγ + Dκγ Fβ + Cκβ γ + ∂ zη   Fκ Eκ Fβ η Fγ + Eκ Fβ Fγη + Dκβ Fγη + Dγκ Fβ η ,

(2.12)

  ∂ Fνσ = Fρκ Eκ Fν Fσ + Dκν Fσ + Dσκ Fν + Cκνσ + ∂ zρ   Fκ Eκ Fρν Fσ + Eκ Fν Fρσ + Dκν Fρσ + Dκσ Fρν .

(2.13)

Next, to find the required fourth-order partial derivatives of F, we further differentiate (2.12) and replace the resulting third-order partial derivatives by the corresponding expressions from (2.13). Plugging the obtained formulas together with (2.12), (2.13) into (2.3) and cancelling terms yields  α (1) Sβ γ ·σ = − δγα Eρ Fρ Fβ σ + δγα Eρ Fρσ Fβ + δβα Eρ Fρ Fγσ + δβα Eρ Fρσ Fγ +  (2.14) ρ ρ Eα Fβ σ Fγ + Eα Fγσ Fβ + Dαβ Fγσ + Dγα Fβ σ + δγα Dβ Fρσ + δβα Dγ Fρσ . It then follows by contraction (see (1.78)) that   (1) ρ Sγσ = − (n + 2)Eρ Fρ Fγσ + (n + 2)Eρ Fρσ Fγ + (n + 2)Dγ Fρσ + DFγσ ,   S (1) = −(n + 1) (n + 2)Eρ Fρ + 2D , where D := Dαα . Hence, formulas (2.4) give i dβα· = iEα Fβ + δβα Eρ Fρ + iDαβ . 2

(2.15)

Identities (2.5), (2.14), (2.15) yield   Sβαγ ·σ = i δγα (dβ σ + dσβ ) + δβα (dγσ + dσγ ) . Now symmetry conditions (1.74) imply Sβαγ ·σ = 0, which completes the proof of the theorem for n ≥ 2. 1 = 0, which due to (2.8) Next, suppose n = 1. In this case we need to obtain Q1· is equivalent to showing that ∂ e1 = 0. (2.16) ∂ z1

2.2 Application

39

We will prove that e1 = E1 , and (2.16) will then follow since E1 is holomorphic on U . Identities (1.87) yield 2i 1 (1) (2.17) e1 = − V1·1 F 11 . 3 From (2.6) we find ∂ d1 1 (1) V1·1 = − 1· . (2.18) ∂ z1 From formulas (2.15), which remain valid for n = 1, we obtain 1 = d1·

3i 1 E F1 + iD11 . 2

Then symmetry conditions (1.74) imply 1 d1· =

3i 1 E F1 + iD11 . 2

Plugging this expression into (2.18) and taking into account that E1 and D11 are holomorphic on U , we obtain 1 (1)

V1·1 = −

3i 1 E F11 . 2

 Together with (2.17) this gives e1 = E1 as required.  We remark that an alternative characterization of spherical rigid hypersurfaces in 2-dimensional complex manifolds was obtained in [80] by a method that uses the Chern-Moser normal form rather than Chern’s construction (for a connection between these see [10], [24]).

2.2 Application As an application of Theorem 2.1, in this section we characterize spherical rigid polynomial hypersurfaces in C2 . A real hypersurface in C2 is called rigid polynomial if in some global holomorphic coordinates z0 = x0 + iy0 , z in C2 it is defined by an equation of the form x0 = P(z, z), (2.19) where P is a real-valued polynomial. Clearly, by means of a holomorphic automorphism of C2 hypersurface (2.19) is equivalent to any hypersurface of the form x0 = P(z, z) + ReQ(z), where Q is a holomorphic polynomial, and therefore we can assume that P does not contain harmonic terms. Spherical hypersurfaces of this kind are described in the following proposition.

40

2 Rigid Hypersurfaces

Proposition 2.1. [26], [49] Let M be a rigid polynomial hypersurface in C2 given by equation (2.19), where P does not contain harmonic terms. Assume further that there exists an open subset of M which is spherical. Then P(z, z) = ± |R(z)|2 , where R(z) is a holomorphic polynomial. Proof. Let W be a spherical subset of M and U its projection to the z-coordinate. By Theorem 2.1 the polynomial P satisfies on U a differential equation of the form Pzz = E(Pz )3 + D(Pz)2 + CPz + H,

(2.20)

where Pz and Pzz denote, respectively, the first- and second-order partial derivatives of P with respect to z, and E, D, C, H are functions holomorphic on U . Since P is not harmonic, Pz is not holomorphic on U . Therefore, considering in both parts of equation (2.20) the terms of the highest degree with respect to z, we see E = D = 0. Furthermore, since P does not contain harmonic terms, it follows that H = 0, thus equation (2.20) takes the form Pzz = CPz . Integrating this equation, we obtain that on U (and hence everywhere on C) the polynomial P has the form P(z) = S(z)T (z), where S and T are holomorphic polynomials. Since P is real-valued, T = aS, with a ∈ R, and the proposition follows.   We quoted the above proof of Proposition 2.1 from [49]. The proof given in [26] ´ Cartan’s invariants constructed in [17] (recall that Chern’s construcdirectly uses E. tion in Section 1.2, on which Theorem 2.1 is based, essentially reduces to that due ´ Cartan for n = 1). Polynomial rigid hypersurfaces arose in [26], [49] as a result to E. of scaling procedures, and in order to obtain the main theorems of these papers the spherical and non-spherical cases had to be considered separately. Proposition 2.1 was used for treating the former case. We also note that in [20] certain spherical rigid a priori not necessarily polynomial hypersurfaces in C2 were classified. The original proof does not use Theorem 2.1, although one can also obtain the result of [20] by an argument based on this theorem.

Chapter 3

Tube Hypersurfaces

Abstract In this chapter we consider tube hypersurfaces in complex vector spaces. They form a natural subclass of the class of rigid hypersurfaces for which the zero CR-curvature equations admit further simplification.

3.1 Zero CR-Curvature Equations for Tube Hypersurfaces Everywhere below, with the exception of Chapter 9, the ambient complex manifold N is assumed to be the complex space Cn+1 with n ≥ 1, in which we fix a totally real (n + 1)-dimensional linear subspace V . A tube hypersurface in Cn+1 is a real hypersurface in Cn+1 of the form M = MR + iV, where MR is a hypersurface in V (i.e. an immersed submanifold of V of codimension one) called the base of M. Clearly, the geometry of a tube hypersurface is fully (z1 , . . . , zn ) in Cn+1 determined bythat of its base. Choosing  coordinates z0 , z = n+1 such that V = Im z j = 0, j = 0, . . . , n , we identify V with R by means of the coordinates x j := Re z j , j = 0, . . . , n. Thus, we always regard MR as a hypersurface in Rn+1 and represent the tube hypersurface M as M = MR + iRn+1 . Note that for Z := (z0 , z) any transformation of Cn+1 of the form Z → Z + ib,

b ∈ Rn+1

is a CR-automorphism of M. We let x := (x1 , . . . , xn ), X := (x0 , x) and consider the map Π : Cn+1 → Rn+1 , Z → X. Clearly, if M is a tube hypersurface, we have Π −1 (MR ) = M. For p ∈ Rn+1 any set of the form Π −1 (V ), where V is a neighborhood of p in Rn+1 , A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 3, 

41

42

3 Tube Hypersurfaces

is called a tube neighborhood of p in Cn+1 . In most cases, for a locally closed tube hypersurface all local considerations take place in a tube neighborhood of a point in its base. There is a natural equivalence relation for tube hypersurfaces. Namely, two tube hypersurfaces M1 , M2 are called affinely equivalent if their bases are affinely equivalent in Rn+1 , i.e. if there exists an affine transformation of Cn+1 of the form Z → CZ + b,

C ∈ GL(n + 1, R), b ∈ Rn+1

(3.1)

that maps M1 onto M2 . Clearly, affine maps of the form (3.1) transform tube hypersurfaces into tube hypersurfaces, bases into bases, and tube neighborhoods into tube neighborhoods. If M is a locally closed tube hypersurface and p ∈ MR , then there exist a tube neighborhood W of p and an affine transformation A of Cn+1 of the form (3.1) such that: (i) A (p) = 0, (ii) A (W) = Π −1 (V ) for a neighborhood V of the origin in Rn+1 of the form V = Ω + I, where Ω is a domain in Rn and I is an interval in the line {x = 0}, with Rn identified with the linear subspace of Rn+1 given by x0 = 0, and (iii) for W := M ∩ W the base of the tube hypersurface A (W ) is represented in V as a graph x0 = F(x) with F(0) = 0, Fα (0) = 0,

(3.2)

where F is a function on Ω . Note that from this point on we use the notation Fα :=

∂F ∂ 2F , Fαβ := , etc., ∂ xα ∂ xα ∂ xβ

which should not be confused with the analogous notation for the partial derivatives with respect to holomorphic and anti-holomorphic variables used in the previous two chapters (here, as before, small Greek indices run from 1 to n). Observe that the signature of the Levi form of A (W ) at a point  Z coincides up to sign with that of the bilinear form defined by the matrix Fαβ (x) , which gives the second fundamental form of the base of A (W ) at the point X = Π (Z). Extending F to Π −1 (Ω ) identically along the fibers of the projection Π (let us denote this extension by F), we obtain a rigid representation of the hypersurface A (M) in W = A (W) with U = Π −1 (Ω ) = Ω + iRn+1 and U = {y0 = 0} ∩ U = Ω + iRn (see Section 2.1). Hence, every locally closed tube hypersurface in Cn+1 is rigid. Throughout the book we only consider tube hypersurfaces up to affine equivalence, and therefore for the purposes of local analysis we usually represent a locally closed portion of a tube hypersurface in the form (3.2), in which case we say that the portion is given in a standard representation. To every standard representation we associate the sets Ω , I, V = Ω + I, W = Π −1 (V ), U = Ω + iRn+1 , U = Ω + iRn and the functions F, F as above. We will now obtain a characterization of spherical tube hypersurfaces by refining Theorem 2.1. Theorem 3.1. [52], [58], [64] Let W be a tube hypersurface in a standard representation. Then the following holds:

3.1 Zero CR-Curvature Equations for Tube Hypersurfaces

43

(i) If W is (k, n − k)-spherical for some 1 ≤ k ≤ n with n ≤ 2k, then there exist a neighborhood W  of the origin in W and a transformation A of the form (3.1) with b = 0 that maps W  onto a tube hypersurface in a standard representation for which the corresponding function F satisfies on the corresponding set Ω a system of partial differential equations of the form   γ γ γ Fαβ = Fγ Dα Fβ + Dβ Fα + Cαβ + Hαβ , F(0) = 0, Fα (0) = 0, (3.3) γ

γ

where Dα , Cαβ , Hαβ are real constants. Moreover, A can be chosen so that the following holds: ⎧ ⎪ 1, α = β , α = 1, . . . , k, ⎪ ⎪ ⎨ Hαβ = −1, α = β , α = k + 1, . . ., n, (3.4) ⎪ ⎪ ⎪ ⎩ 0, α = β . (ii) If the function F corresponding to W satisfies on the set Ω a system of the form (3.3) with Hαβ given by (3.4), then there exists a neighborhood Ω  ⊂ Ω of the origin in Rn such that W  := W ∩ Π −1 (Ω  + I) is (k, n − k)-spherical. Remark 3.1. For the case where W is strongly pseudoconvex, i.e. the Levi form at every point of W is sign-definite, part (i) of Theorem 3.1 was obtained in [108] (in this case k = n). Proof. We will first prove part (ii). Since Fαβ (0) = Hαβ , there exists a neighbor  hood Ω  ⊂ Ω of the origin in Rn in which the bilinear form defined by Fαβ (x) is non-degenerate  and  therefore has the same signature as the bilinear form defined by the matrix Hαβ . Let V  := Ω  + I and W  := Π −1 (V  ). Then the tube hypersurface W  := W ∩ W  is Levi non-degenerate. Since the function F satisfies on Ω  system (3.3), its extension F to the set U satisfies on U  := Ω  + iRn the following system:

∂ 2F ∂F 1 1 γ γ ∂F γ ∂F D + Hαβ . C = + Dβ + α ∂ zα ∂ zβ ∂ zγ ∂ zβ ∂ zα 2 αβ 4 Theorem 2.1 now implies that W  is (k, n − k)-spherical. We will now prove part (i). By Theorem 2.1 the extension F of F to the set U satisfies on U some system

∂ 2F ∂F γ ∂F γ ∂F γ γ ∂F ∂F E = + Dα + Dβ + Cαβ + Hαβ , ∂ zα ∂ zβ ∂ zγ ∂ zα ∂ zβ ∂ zβ ∂ zα γ

γ

where Eγ , Dα , Cαβ , Hαβ are holomorphic on U . Since F is independent of Im z, the proof of Theorem 2.1 and the formulas that precede it in Section 2.1 yield that Eγ , Dαγ , Cγαβ , Hαβ do not depend on Imz either and are real-valued. Hence, these functions are in fact real constants. Thus, the function F satisfies on Ω the system   γ γ γ Fαβ = Fγ Eγ Fα Fβ + Dα Fβ + Dβ Fα + Cαβ + Hαβ , (3.5)

44

3 Tube Hypersurfaces γ

γ

γ

γ

where Eγ := Eγ /2, Dα := Dα , Cαβ := 2Cαβ , Hαβ := 4Hαβ . Since W is in a standard representation, we have F(0)= 0,Fγ (0) = 0. Therefore, (3.5) implies Hαβ = Fαβ (0), hence the matrix H := Hαβ is symmetric. Performing the transformation (x0 , x) → (−x0 , x) if necessary, we can assume that the number of negative eigenvalues of H does not exceed the number of positive ones. Further, performing a transformation of the form (x0 , x) → (x0 , C x),

C ∈ GL(n, R),

(3.6)

we can assume that the entries of H are given by formulas (3.4). Indeed, under transformation (3.6) system (3.5) changes into a system of the same kind with H replaced by (C −1 )T HC −1 (see Proposition 4.1). We will now show that by applying an appropriate linear transformation one can eliminate all terms of order 3 with respect to Fα in the right-hand side of (3.5). Fix a vector E = (E1 , . . . , En ) ∈ Rn and apply to WR the following linear transformation AE of Rn+1 (x0 , x) → (x0 , x + E x0 ). (3.7) Clearly, AE transforms WR into the hypersurface given in AE (V ) by the equation x0 = F (x − E x0 ) . In a suitable neighborhood Vˇ = Ωˇ + Iˇ ⊂ AE (V ) of the origin in Rn+1 this equation ˇ can be resolved with respect to x0 as x0 = F(x) and gives rise to a tube hypersurface in a standard representation in the tube neighborhood Π −1 (Vˇ ). A long but straightforward calculation (which we omit) leads to the following lemma. Lemma 3.1. The function Fˇ satisfies on Ωˇ a system of the form (3.5). Furthermore, ˇ =H if we indicate the respective system parameters by check marks, we have H and γ γ (3.8) Eˇ γ = Eγ − 2 ∑ Dρ Eρ + ∑ Cρσ Eρ Eσ − Eγ E , E , ρ

ρ ,σ

1 γ γ ˇ αγ = Dαγ − ∑ Cαρ Eρ + Hαα Eα Eγ + δα E , E , D 2 ρ ˇ γ = Cγ − Hαβ Eγ − δ γ Hαα Eα − δαγ Hβ β Eβ , C αβ αβ β

(3.9)

where ·, · is the bilinear form on Rn defined by the matrix H. Thus, part (i) of the theorem will follow if we show that there exists a vector E ∈ Rn for which the right-hand side of (3.8) (we denote it by Fγ (E )) vanishes for all γ . Let F be the self-map of Rn defined as F := (F1 , . . . , Fn ). Assume first that the bilinear form ·, · is positive-definite, i.e. k = n. Suppose that F is nowhere zero and consider the following self-map of the unit sphere Sn−1 : G(E ) :=

F(E ) , ||F(E )||

||E || = 1,

3.1 Zero CR-Curvature Equations for Tube Hypersurfaces

45

where, as before, || · || denotes the Euclidean norm in Cn . Considering the homotopy of self-maps of Sn−1 Gt (E ) :=

F(tE ) , ||F(tE )||

||E || = 1,

0 ≤ t ≤ 1,

we see that G = G1 is homotopic to the constant map G0 and therefore has degree 0. On the other hand, consider the family of self-maps of Sn−1 ˜ t (E ) := F(E /t) , G ||F(E /t)|| ˜ 0 by and define G

||E || = 1,

0
˜ 0 (E ) := lim G ˜ t (E ) = −E . G t→0,t>0

˜ 1 is homotopic ˜ t , with 0 ≤ t ≤ 1, is a homotopy, hence G = G We see that the family G ˜ 0 and therefore has a non-zero degree. This contradiction shows that to the map G the map F vanishes at some point of Rn as required.1 In fact, we have just proved that if ·, · is positive-definite, then any polynomial self-map of Rn of the form E → −E E , E + lower-order terms in Eα vanishes at some point. The following simple example shows that this is no longer the case if ·, · is indefinite, i.e. k < n. Example 3.1. Let n = 2 and E , E := E12 − E22 . Then it is easy to see that the map E → −E E , E + (1, 1) does not vanish anywhere in R2 . Assume now that ·, · is indefinite. First of all, we make a useful remark. Remark 3.2. For any v = (v1 , . . . , vn ) ∈ Rn set v := (v1 , . . . , vk ), v := (vk+1 , . . . , vn ). Then, restricting F to the linear subspace {v = 0} and arguing as in the case k = n above, one can show that the map F := (Fk+1 , . . . , Fn ) has a zero lying in this subspace. Thus, we can assume Eγ = 0 for γ = k + 1, . . ., n. In view of Example 3.1, in order to show that the map F vanishes at some point of Rn , we need to perform a detailed analysis of the terms of order less than 3 with respect to Eα in the right-hand side of (3.8). We will now derive certain relations for γ γ the constants Eγ , Dα , Cαβ , Hαβ occurring in system (3.5). These relations are of utmost importance not only for the proof of Theorem 3.1 but also for a majority of results in the rest of the book. 1

The above proof is due to M. A. Mishchenko. The original proof given in [108] for the strongly pseudoconvex case seems to be incomplete.

46

3 Tube Hypersurfaces

We differentiate the (α , β )th equation in system (3.5) with respect to xν and the (α , ν )th equation with respect to xβ . Clearly, we have

∂ Fαβ ∂ Fαν ≡ . ∂ xν ∂ xβ

(3.10)

Both sides of (3.10) involve some first- and second-order partial derivatives of the function F. Using system (3.5), we now replace in (3.10) the second-order partial derivatives of F by the corresponding expressions in terms of its first-order partial derivatives. This turns (3.10) into a collection of identities of the form 5

∑ Pαβj ν (F1 , . . . , Fn) = 0,

(3.11)

j=0 j

where Pαβ ν is a homogeneous polynomial of order j on Rn . Each of the polynomials in identities (3.11) can be explicitly computed, and it turns out, in particular, that j Pαβ ν = 0 for j = 4, 5. Since the matrix (Fαβ (0)) = H is non-degenerate, the map x → (F1 (x), . . . , Fn (x)) is a diffeomorphism between some neighborhoods of the origin in Rn . Therefore, Fα can be treated as independent variables in (3.5), (3.11). γ γ This observation yields certain relations for the constants Eγ , Dα , Cαβ , Hαβ , which we list in the lemma below. γ In the lemma we denote the (γ , β )th entry of an n × n-matrix C by (C)β , where, as before, γ and β are the row and column indices, respectively. Also, if H is a symmetric matrix of size n × n with complex entries, we say that a matrix C with complex entries is H-symmetric if CT H = HC. In particular, if In is the n × n identity matrix, then every In -symmetric matrix is symmetric. If [·, ·] is the bilinear form defined by H, then the above definition is equivalent to the identity [Cz, z ] = [z, Cz ] being satisfied for all z, z ∈ Cn . Lemma 3.2. If a system of the form (3.5) with zero initial conditions and matrix H of the form (3.4) for some 0 ≤ k ≤ n has a C∞ -smooth solution, then the following relations hold: γ

γ

(i) Cαβ = Cβ α , γ

γ

(ii) each matrix Cα , where (Cα )β := Cαβ , is H-symmetric, γ

γ

(iii) the matrix D, where (D)β := Dβ , is H-symmetric, β

γ

γ

(iv) ([Cα , Cβ ])γα = Dγ Hβ β , ([Cα , Cβ ])α = −Dβ Hαα , β ([Cα , Cβ ])γ

= −Dγα Hαα ,

γ ([Cα , Cβ ])β

(3.12)

γ = Dα Hβ β ,

β

β

β

([Cα , Cβ ])αβ = (Dαα + Dβ )Hβ β , ([Cα , Cβ ])α = −(Dαα + Dβ )Hαα

for α = β , γ = α , β , and all other entries of the commutator [Cα , Cβ ] are zero,

3.1 Zero CR-Curvature Equations for Tube Hypersurfaces

47

γ

(v) ([Cα , D])α = −Eγ Hαα , ([Cα , D])γα = Eγ Hγγ for α = γ,

and all other entries of the commutator [Cα , D] are zero, γ

γ

β

γ

β

γ

(vi) (D2 )β = Eα Cαβ , (D2 )β − (D2 )γ = Eα Cαβ − Eα Cαγ for β = γ. We note that relations (i) follow directly from system (3.5), relations (ii) from 0 1 the identities Pαβ ν = 0, relations (iii) and (iv) from the identities Pαβ ν ≡ 0, relations 2 3 (v) from the identities Pαβ ν ≡ 0, and relations (vi) from the identities Pαβ ν ≡ 0. The calculations required to prove Lemma 3.2 are quite lengthy but straightforward, and we omit them. We also remark that relations (i), (ii), (iii), (vi) hold for any system of the form (3.5), where H is not necessarily diagonal with entries ±1 but an arbitrary non-degenerate symmetric matrix. Our proof of the claim that the map F has a zero in Rn relies on relations (3.12). The proof is rather technical, and here we only present its special case for n = 2, k = 1, that is, for the situation where the bilinear form ·, · is as in Example 3.1. By Remark 3.2 we assume E2 = 0. Relations (3.12) then imply

a −b −β − δ α −β , C2 = , , C1 = D= b d β δ δ γ where the real numbers a, b, d, α , β , γ , δ satisfy the conditions b(β + γ ) + δ (a − d) = E1 , b(α − δ ) − β (a − d) = 0, δ (α − δ ) + β (β + γ ) = a + d.

(3.13)

Perform the transformation AE with E1 = (α − δ )/2, E2 = β (see (3.7)). Transformation law (3.8) and conditions (3.13) imply that for the resulting system we have ˇ 1 is a scalar matrix. Thus, Eˇ 2 = 0. In addition, transformation law (3.9) yields that C dropping check marks, we assume δ = α and β = 0. In this case the last identity in (3.13) implies d = −a, hence the first identity yields E1 = 2α a + γ b. Therefore, the two components of the map F are given by the following formulas:   F1 (E ) = 2α a + γ b − 2aE1 + 2bE2 + α (E12 − E22 ) − E1 E12 − E22 , (3.14)   F2 (E ) = −2bE1 + 2aE2 + 2α E1E2 + γ E22 − E2 E12 − E22 . We need to show that for any choice of a, b, α , γ ∈ R the functions F1 , F2 simultaneously vanish at some point (E1 , E2 ) ∈ R2 . If b = 0, then F vanishes at the point (α , 0), thus we suppose b = 0. In this case, performing if necessary the transformation (x0 , x1 , x2 ) → (x0 , −x1 , x2 ), we can assume b < 0. Further, we say that Condition (S) is satisfied for the parameters a, b, α , γ if the following holds:

48

3 Tube Hypersurfaces

16α b + 8γ a + 4α 2γ − γ 3 = 0. In this case F vanishes at the point (α , −γ /2), thus everywhere below we assume that Condition (S) is not satisfied. For E1 = α we consider the equation F1 (E ) = 0 as a quadratic equation with respect to E2 (see (3.14)) and find √ −2b + D , (3.15) E2 = 2(E1 − α ) where D is the equation’s discriminant D = 4E14 − 8α E13 + 4(α 2 + 2a)E12 − 4(4α a + γ b)E1 + 4(2α 2 a + αγ b + b2 ) (3.16) √ and D denotes any of the two square roots of D. Plugging expression (3.15) into the equation F2 (E ) = 0 (see (3.14)), we obtain the identity √ DQ1 + Q2 = 0, where Q1 := 4α E13 + 8(a − α 2)E12 + 2(2α 3 − 3γ b − 8α a)E1 + 2(4α 2 a + 3αγ b + 4b2), Q2 := (γ E1 − αγ − 4b)D. Set P := Q22 − DQ21 . Observe that if e is a real root of P, then D(e) ≥ 0. Indeed, D(e) < 0 yields that e is a root of each of Q1 , Q2 . The condition Q2 (e) = 0 then implies

γ e − αγ − 4b = 0. Since b = 0, we have γ = 0, and therefore e=

4b + α. γ

Plugging this expression into the equation Q1 (e) = 0, we see after a short calculation that Condition (S) is satisfied. This contradiction shows that we have D(e) ≥ 0 for any real root e of P. Thus, part (i) of the theorem will follow from formula (3.15) if we prove that the polynomial P has a real root distinct from α . Further, the polynomial P can be written as P = DR, where

(3.17)

3.1 Zero CR-Curvature Equations for Tube Hypersurfaces

49

R := D(γ E1 − αγ − 4b)2 − Q21 . Direct calculation now shows ˜ R = (E1 − α )3 R, where the polynomial R˜ has degree at most 3 and can vanish at α only if Condition (S) is satisfied. Hence, any real root of R˜ would be a suitable root of P. The leading term of R˜ is 4(γ 2 − 4α 2 )E13 . Hence, for γ = ±2α the polynomial R˜ has a real root, and we only need to consider the case γ = ±2α . In this case α = 0 (for otherwise ˆ where Condition (S) would be satisfied), and we have R˜ = −32(a ± b)R, Rˆ := 2α E12 + (2(a ∓ b) − 3α 2)E1 + α (α 2 − 2a). The discriminant D of the quadratic equation Rˆ = 0 is computed as follows: 2  D = 2(a ∓ b) + α 2 ± 16α 2b. Since b < 0, we have D > 0 for γ = −2α . Furthermore, D ≥ 0 if α 2 − 2a ≤ 0. Hence, in each of these cases Rˆ has a real root, thus we assume γ = 2α and α 2 − 2a > 0. In this situation we have D < 0 if and only if b lies in the interval

  3 2 3 2 2 2 Iα ,a := a − α − |α | 2(α − 2a), a − α + |α | 2(α − 2a) . 2 2

(3.18)

Since Rˆ does not have a real root for b ∈ Iα ,a , in order to show that P does have a real root distinct from α for such values of b we will prove that this holds for the polynomial D (see (3.16), (3.17)). We write D for γ = 2α as follows:    2 D = 4(E1 − α )2 E12 − (α 2 − 2a) + 4 α (E1 − α ) − b . Since b < 0, the polynomial D does not vanish at α . Furthermore, any real root of D lies in the interval     − α 2 − 2a, α 2 − 2a . Clearly, if e is a real root of D, then we have

 b = (e − α ) α ± (α 2 − 2a) − e2 . Parametrizing all possible roots of D for fixed α and a as  e(s) = s α 2 − 2a, −1 ≤ s ≤ 1, we obtain the following parametrization of the values of b for which D has a real root:

50

3 Tube Hypersurfaces

     b± (s) = s α 2 − 2a − α α ± α 2 − 2a · 1 − s2 . We will now show that the union of the ranges of the functions b+ and b− contains the interval Iα ,a defined in (3.18). Indeed, we have √ b± (1) = −α 2 + α α 2 − 2a =: b1 ,    3 b+ − √12 = a − α 2 − α 2(α 2 − 2a) =: b2 , 2    3 b− √12 = a − α 2 + α 2(α 2 − 2a) =: b3 . 2 Therefore, the ranges of b+ and b− contain the following intervals: [min{b1 , b2 }, max{b1 , b2 }],

[min{b1 , b3 }, max{b1 , b3 }],

respectively. Thus, the union of these ranges contains the interval I α ,a = [min{b2 , b3 }, max{b2 , b3 }]. Hence, for every b ∈ Iα ,a there exists −1 ≤ s0 ≤ 1 such that either b = b+ (s0 ) or b = b− (s0 ). Then e(s0 ) is a real root of D as required. Thus, we have shown that P always has a real root which is distinct from α .   A system of partial differential equations of the form (3.3) having a C∞ -smooth solution is called a defining system if the matrix H := (Hαβ ) is non-degenerate and the number of negative eigenvalues of H does not exceed the number of positive ones. As specified in (3.3), a defining system is always assumed to have zero initial conditions. The parameters of a defining system satisfy (i), (ii), (iii), (vi) of (3.12) with Eα = 0. In particular, D2 is a scalar matrix. Furthermore, if the matrix H is given in the form (3.4) for some 1 ≤ k ≤ n with n ≤ 2k, the parameters also satisfy (iv) and (v) of (3.12) with Eα = 0. In particular, D commutes with every Cα (in fact, as explained at the beginning of Section 4.1, this last statement holds irrespectively of the form of H). Defining systems are our main tool for studying spherical tube hypersurfaces. In Chapter 4 a general theory of such systems will be developed.

3.2 Analyticity of Spherical Tube Hypersurfaces Our first application of the characterization of spherical tube hypersurfaces by means of defining systems is the following proposition. Proposition 3.1. [56] A spherical tube hypersurface is real-analytic. Remark 3.3. For a non-tube spherical hypersurface the assertion of Proposition 3.1 may not hold. An example can be constructed as follows. Let ψ be a function holomorphic on the unit ball Bn+1 ⊂ Cn+1 and smooth up to the boundary ∂ Bn+1 = S2n+1

3.2 Analyticity of Spherical Tube Hypersurfaces

51

that does not extend to a function holomorphic on any larger domain (for the existence of such functions see, e.g. [19]). Then ϕ := ψ |S2n+1 is a CR-function on S2n+1. Fix a point p0 ∈ S2n+1 . Adding to ψ a complex-linear function if necessary, we can (1,0)  2n+1 S such that Z0 ϕ (p0 ) = 0 (see assume that there exists an element Z0 ∈ Tp0 (1.2)). We can then find n complex-linear functions ϕ1 , . . . , ϕn on Cn+1 such that the map f := (ϕ , ϕ1 |S2n+1 , . . . , ϕn |S2n+1 ) from S2n+1 to Cn+1 has rank 2n + 1 at the point p0 . Choose a neighborhood V of p0 in S2n+1 on which f is one-to-one and has maximal rank and let M := f (V ). Clearly, M is a hypersurface in Cn+1 and f is a CR-isomorphism between V and M. The hypersurface M is C∞ -smooth, strongly pseudoconvex, and spherical, but is not real-analytic. Indeed, if M were real-analytic, then by Theorem 3.1 of [3] the map f would be real-analytic and hence holomorphically extendable to a neighborhood of the point p0 , which contradicts the non-extendability of f past S2n+1 . As we will see below, examples of spherical smooth non–real-analytic hypersurfaces exist only in the strongly pseudoconvex case. We will now give two proofs of Proposition 3.1. Both proofs refer to defining systems but do so in different ways. Yet another proof can be found in recent paper [41] (see Section 9.2 below for details). Proof 1. Let M be any (not necessarily tube) spherical hypersurface. Assume first that the Levi form of M is indefinite. Fix p ∈ M and let V be a neighborhood of p in M which is CR-equivalent to an open subset V  of the corresponding quadric Qg and which is locally closed in Cn+1 . Let f : V → V  be a CR-isomorphism. By Theorem 3.3.2 in [22] (see also references therein) the map f holomorphically extends to a map f˜ : V˜ → Cn+1 , where V˜ is an open subset of Cn+1 containing V . The map f is a diffeomorphism on V , and therefore one can find a neighborhood Vˆ ⊂ V˜ of p in Cn+1 on which f˜ is biholomorphic. Since Qg is real-analytic, this implies that M is real-analytic near p. Thus, M is real-analytic. Assume now that M is a strongly pseudoconvex spherical tube hypersurface. In this case the analyticity of M follows from [27], where all such hypersurfaces were explicitly determined up to affine equivalence (a precise formulation of this result will be given in Chapter 5). This was achieved by solving defining systems for k = n up to linear equivalence. All tube hypersurfaces in the classification of [27] are realanalytic, and the proposition follows.   In fact, one could avoid using the holomorphic extendability of a local CR-equivalence between M and Qg in the case of indefinite Levi form as well as the exact forms of the solutions of defining systems in the strongly pseudoconvex case. Below we will give a proof based solely on certain elementary estimates of the partial derivatives of the function F implied by defining systems. This proof does not require separate arguments depending on the signature of the Levi form. Proof 2. Let F be a C∞ -smooth solution of a defining system, where the matrix H is assumed to be of the form (3.4). We will show that F is in fact real-analytic on some neighborhood of the origin. By Theorem 3.1 this will imply that for every p ∈ M a locally closed portion of M near p is real-analytic, hence M is real-analytic.

52

3 Tube Hypersurfaces

Fix μ ≥ 1 satisfying

 γ Dα  ≤ μ ,

   γ  Cαβ  ≤ μ ,

(3.19)

and choose ε > 0 such that one has |Fα (x)| ≤ 1 for x ∈ Bε , where Bε is the ball of radius ε in Rn centered at the origin. We will prove that for every m ∈ N and j1 , . . . , jn ∈ Z+ , with j1 + . . . + jn = m, the following estimate holds:2     ∂ mF   sup  j (x) (3.20)  ≤ (4n μ )m−1 (m − 1)!. jn 1  x∈Bε  ∂ x1 . . . ∂ xn Estimate (3.20) implies that F is real-analytic near the origin. Indeed, (3.20) yields that the remainder rm (x) of order m in Taylor’s formula for F(x), with x ∈ Bε , is estimated as |rm (x)| ≤ (4n2 με )m . Thus, for ε satisfying 4n2 με < 1 the remainder rm (x) tends to zero in Bε as m → ∞, hence F is real-analytic on Bε . We will now prove estimate (3.20) by induction on m. Clearly, (3.20) holds for m = 1, 2. Let m ≥ 3 and write ∂ m F/∂ x1j1 . . . ∂ xnjn as ∂ m−2 Fα0 β0 /∂ x11 . . . ∂ xnn for some indices 1 ≤ α0 , β0 ≤ n and non-negative integers 1 , . . . , n satisfying 1 + . . . + n = m − 2. System (3.3) then yields

∂ mF j

j

∂ x11 . . . ∂ xnn =

∑

=

lν ≤ν , l1 +...+ln <m−2



1 l1

 ... n ln

m−2−(l1 +...+ln ) F α0 1 −l1 n −ln ∂ x1 . . . ∂ xn

γ ∂ Dβ 0



 +

∂ l1 +...+ln Fγ

 γ

Dα0

∂ xl11 . . . ∂ xlnn ∂ m−2 Fγ ∂ x11 . . . ∂ xnn



∂ m−2−(l1 +...+ln ) Fβ0 ∂ x11 −l1 . . . ∂ xnn −ln

γ

γ

γ

Dα0 Fβ0 + Dβ Fα0 + Cα 0

+ 

0 β0

.

Now estimates (3.19), the induction hypothesis, and elementary combinatorics imply that for x ∈ Bε we have      ∂ mF 1    m−2 2n ... n × ≤ (4n (x) μ ) μ   j1 ∑ l ln  ∂ x . . . ∂ xnjn  1 lν ≤ν , l1 +...+ln <m−2 1  (l1 + . . . + ln )!(m − 2 − (l1 + . . . + ln ))! + 3n μ (m − 2)! ≤ (4n μ )m−1 × 2

Here and below N and Z+ are the sets of positive and non-negative integers, respectively.

3.2 Analyticity of Spherical Tube Hypersurfaces

∑

lν ≤ν



1 l1

1 ! · ... · n!



...

n ln

∑

(l1 + . . . + ln )! (m − 2 − (l1 + . . . + ln ))! = (4n μ )m−1 (m − 1)!. l1 ! · ... · ln! (1 − l1 )! · ... · (n − ln )!

lν ≤ν

(l1 + . . . + ln )! (m − 2 − (l1 + . . . + ln ))! = (4nμ )m−1 ×

Thus (3.20) holds, and our second proof of the proposition is complete.  

53

•

Chapter 4

General Methods for Solving Defining Systems

Abstract In this chapter we describe a general approach to finding the solutions of defining systems. Since we are interested in classifying spherical tube hypersurfaces up to affine equivalence, we attempt to solve defining systems up to linear equivalence. More precisely, if F(x) is the solution of a defining system near the origin, we are interested in determining the hypersurface x0 = F(x) up to linear transformations in the variables x0 , x. Thus, we begin the chapter by simplifying defining systems by means of such transformations. One of the consequences of our approach is the following globalization result: every spherical tube hypersurface in Cn+1 extends to a spherical real-analytic hypersurface which is closed as a submanifold of Cn+1 .

4.1 Classification of Defining Systems First of all, we need to understand how a defining system changes when a linear transformation of the form (3.6) is performed. In the following proposition we consider more general systems, namely systems of the form (3.5). Proposition 4.1. [64] If a C∞ -smooth function F defined on a neighborhood of the origin in Rn satisfies a system of the form (3.5) with not necessarily zero initial conditions and the matrix H = (Hαβ ) not necessarily non-degenerate, then for ˆ C ∈ GL(n, R) the function F(x) := F(Cx) satisfies near the origin a system of the same form. If we indicate the parameters of the new system by hats, then we have ˆ α = C−1 Cβ C(C)βα , H ˆ = C−1 DC, C ˆ = CT HC, Eˆ = C−1 E, D

(4.1)

where E is the vector (E1 , . . . , En ). Proposition 4.1 is proved by a straightforward argument, which we omit. This proposition yields, in particular, that D changes as the matrix of a linear operator and H changes as the matrix of a bilinear form (observe that H is symmetric for a system of the form (3.5) with zero initial conditions). If H is symmetric and nondegenerate, it can always be reduced to the form (3.4) for some 0 ≤ k ≤ n. Thus, for A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 4, 

55

56

4 General Methods for Solving Defining Systems

a defining system Proposition 4.1 and relations (v) of Lemma 3.2 imply [Cα , D] = 0.

(4.2)

We also recall that for a defining system relations (vi) of Lemma 3.2 yield D2 = λ · In , where, as before, In is the n × n identity matrix. For an arbitrary defining system we will now attempt to find a linear transformation of Rn that simultaneously reduces D and H to simple forms. We need the following general fact from linear algebra (cf. Proposition 6.1). Proposition 4.2. Let Q be a real non-degenerate symmetric matrix of size n × n and X a real Q-symmetric matrix. Then there exists a linear transformation of Rn that takes Q and X into ⎛  ⎞ ⎛  ⎞ Q1 X1 ⎜ ⎟ ⎜ ⎟ .. .. ⎜ ⎜ . . 0 ⎟ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Qm Xm ⎜ ⎟ ⎜ ⎟, and (4.3)   ⎜ ⎟ ⎜ ⎟ Q1 X1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ .. .. ⎝ ⎠ ⎝ 0 ⎠ . . 0 Ql

Xl

respectively, where the Jordan normal form of each matrix Xi consists of a single cell having a real eigenvalue, the Jordan normal form of each X j consists of two cells of the same size corresponding to two non-real mutually conjugate eigenvalues, Xi is Qi -symmetric and X j is Qj -symmetric for i = 1, . . . , m, j = 1, . . . , l .1 Proof. Let v1 , v2 ∈ Cn be two elements of a Jordan basis for X corresponding to distinct cells in the Jordan normal form, and let λ1 and λ2 be the eigenvalues of X corresponding to these cells, respectively. Suppose λ1 = λ2 . It is easy to show by an inductive argument that v1 , v2 are orthogonal with respect to the bilinear form on Cn defined by Q. It then follows that there exists a linear transformation of Rn that takes X and Q, respectively, into forms (4.3), where Xi has a single eigenvalue and this eigenvalue is real, X j has exactly two eigenvalues and these eigenvalues are non-real, mutually conjugate and have equal multiplicities. Thus, to prove the proposition it is sufficient to consider the following two cases: (i) X has a single eigenvalue and this eigenvalue is real, and (ii) X has exactly two eigenvalues and these eigenvalues are non-real, mutually conjugate and have equal multiplicities. Case (i). Without loss of generality we assume that the eigenvalue of X is zero. Let v11 , . . . , v1n1 , . . . , v1 , . . . , vn , with n1 ≤ n2 ≤ . . . ≤ n , n1 + . . . + n = n, be a Jordan basis for X, where j

j

Xvk j = vk j −1 , 1

j

v0 := 0,

k j = 1, . . . , n j ,

j = 1, . . . , .

Here Q transforms as the matrix of a bilinear form, whereas X transforms as the matrix of a linear operator, that is, Q → CT QC, X → C−1 XC for C ∈ GL(n, R).

4.1 Classification of Defining Systems

57

Here  is the number of cells in the Jordan normal form of X and n1 , . . . , n are the cell sizes. Let [·, ·] denote the bilinear form on Rn corresponding to Q. Every entry in the matrix of this bilinear form in the above Jordan basis either is equal to zero or coincides with one of the following numbers (some of which may also be zero):  i, j (4.4) aki := viki , vnj j , ki = 1, . . . , ni , i ≤ j, i, j = 1, . . . , . j Our aim is to choose a Jordan basis for which a1, k1 = 0 for all k1 = 1, . . . , n1 , j = 2, . . . , .

Suppose a1,1 1 = 0. Since Q is non-degenerate, there exists j0 ∈ {2, . . . , } such 1, j that a1 0 = 0 and n j0 = n1 . For α ∈ R consider the Jordan basis for X defined as follows: j v˜1k1 := v1k1 + α vk01 , k1 = 1, . . . , n1 , (4.5) j j k j = 1, . . . , n j , j = 2, . . . , . v˜k j := vk j , 

1 1  1, j0 j0 j0 2 a˜1,1 1 := v˜1 , v˜n1 = 2α a1 + α v˜1 , v˜n1 .

Then we have

(4.6)

Choosing α ∈ R such that a˜1,1 1 = 0, without loss of generality we thus can assume 1,1 a1 = 0. 1(1)

(1)

1(1)

(1)

Let v1 , . . . , vn1 , . . . , v1 , . . . , vn be the following Jordan basis for X: 1(1)

:= v1k1 ,

j(1) vk j

j vk j ,

vk1

j(1)

vk j

:=

k1 = 1, . . . , n1 , k j = 1, . . . , n j − n1, j = 2, . . . , ,

(4.7)

:= vkj j + α1j v1k j −n j +n1 , k j = n j − n1 + 1, . . . , n j , j = 2, . . . , ,

where α1j are the real numbers found from the conditions  v1n1 , vnj j −n1 +1 + α1j a1,1 j = 2, . . . , . 1 = 0, 1(q)

Next, for every 2 ≤ q ≤ n1 define a Jordan basis v1 for X inductively as follows: 1(q)

:= vk1

j(q)

:= vk j

vk1 vk j

j(q)

vk j

1(q−1) j(q−1)

j(q−1)

:= vk j

1(q)

(4.8) (q)

(q)

, . . . , vn1 , . . . , v1 , . . . , vn

,

k1 = 1, . . . , n1 ,

,

k j = 1, . . . , n j − n1 + q − 1, j = 2, . . . , , j 1(q−1)

+ αq vk j −n j +n1 −q+1 , k j = n j − n1 + q, . . ., n j , j = 2, . . . , ,

where αqj are the real numbers determined from the conditions

(4.9)

58

4 General Methods for Solving Defining Systems

 1(q−1) j(q−1) , vn j −n1 +q + αqj a1,1 v n1 1 = 0, 1(n1 )

For the basis v1

1(n )

(n )

j = 2, . . . , .

(4.10)

(n )

, . . . , vn1 1 , . . . , v1 1 , . . . , vn 1 we then have  1(n ) j(n ) vk1 1 , vn j 1 = 0, k1 = 1, . . . , n1 , j = 2, . . . , 

as required. Therefore, there exists a linear transformation of Rn that takes X and Q into the forms



Q1 0 X1 0 , , (4.11) 0 X2 0 Q2 respectively, where Xi is Qi -symmetric for i = 1, 2 and the Jordan normal form of X1 consists of a single cell. Case (ii). Let μ and μ , with μ ∈ R, be the eigenvalues of X. Let further v11 , . . . , v1n1 , . . . , v1 , . . . , vn , v11 , . . . , v1n1 , v1 , . . . , vn , with n1 ≤ n2 ≤ . . . ≤ n , n1 + . . . + n = n/2, be a Jordan basis for X, where Xvk j = μ vk j + vk j −1 , j

j

j

j

v0 := 0,

k j = 1, . . . , n j ,

j = 1, . . . , .

Here 2 is the number of cells in the Jordan normal form of X and n1 , . . . , n are the cell sizes. Let [·, ·] denote the bilinear form on Cn corresponding to Q. Every entry in the matrix of this bilinear form in the above Jordan basis either is equal to zero or i, j coincides with either one of the numbers aki given by formulas (4.4) or one of their

j conjugates. As in case (i), our aim is to choose a Jordan basis for which a1, k1 = 0 for all k1 = 1, . . . , n1 , j = 2, . . . , . Suppose a1,1 1 = 0. Since Q is non-degenerate, there exists j0 ∈ {2, . . . , } such j0 =  0 and n j0 = n1 . For α ∈ C we define a Jordan basis v˜11 , . . . , v˜1n1 , . . ., that a1, 1 1

1





v˜1 , . . . , v˜n , v˜1 , . . . , v˜n1 , v˜1 , . . . , v˜n for X by formulas (4.5). Then, as before, (4.6) holds. Choosing α ∈ C such that a˜1,1 1 = 0, without loss of generality we thus can =  0. assume a1,1 1 1(1)

1(1)

(1)

(1)

1(1)

1(1)

(1)

(1)

Let v1 , . . . , vn1 , . . . , v1 , . . . , vn , v1 , . . . , vn1 , v1 , . . . , vn be the Jordan j basis for X defined by (4.7), where α1 are the complex numbers found from conditions (4.8). Next, for every integer 2 ≤ q ≤ n1 we define a Jordan basis 1(q) 1(q) (q) (q) 1(q) 1(q) (q) (q) v1 , . . . , vn1 , . . . , v1 , . . . , vn , v1 , . . . , vn1 , v1 , . . . , vn for X inductively by formulas (4.9), where αqj are the complex numbers determined from conditions 1(n ) 1(n ) (n ) (n ) 1(n ) 1(n ) (4.10). For the basis v1 1 , . . . , vn1 1 , . . . , v1 1 , . . . , vn 1 , v1 1 , . . . , vn1 1 , (n1 ) (n1 ) v1 , . . . , vn we then have  1(n ) j(n ) vk1 1 , vn j 1 = 0, k1 = 1, . . . , n1 , j = 2, . . . ,  as required.

4.1 Classification of Defining Systems

59

We have shown that there exists a linear transformation of Rn that takes X and Q into forms (4.11), respectively, where Xi is Qi -symmetric for i = 1, 2 and the Jordan normal form of X1 consists of two cells of the same size, one corresponding to the eigenvalue μ , the other to the eigenvalue μ . Arguing by induction we obtain the statement of the proposition.  Remark 4.1. For a real non-degenerate symmetric matrix Q denote by χQ the minimum of the numbers of positive and negative eigenvalues of Q. Further, denote by ni and nj the sizes of the cells in the Jordan normal forms of the matrices Xi and X j in formulas (4.3), respectively. It is then not hard to observe (see, e.g. [96]) that   n χQi = i , i = 1, . . . , m, 2 χQj = nj , j = 1, . . . , l. Remark 4.2. For Xi and X j in formulas (4.3) let λi be the eigenvalue of Xi and μ j , μ j , with μ j ∈ R, the eigenvalues of X j . Applying Proposition 4.2 and Remark 4.1 to X = D, Q = H and recalling that D2 = λ · In with λ ∈ R, we see that in this case ni ≤ 2, nj ≤ 1 for all i, j and the following holds: √ • If λ > 0, then l = 0, m = n, ni = 1, λi = ± λ , i = 1, . . . , n, • If λ < 0, then m = 0, n = 2l, nj = 1, μ j = ±i |λ |, and χQj = 1, j = 1, . . . , l, • If λ = 0, then l = 0 and λi = 0, i = 1, . . . , m. As we will see below, the three possibilities listed in Remark 4.2 give rise to three types of defining systems. Set

1 0  H := . (4.12) 0 −1 Proposition 4.3. [56] Let X be a real H -symmetric matrix. (i) If the Jordan normal form of X is non-diagonal with eigenvalue λ ∈ R, then by a linear transformation of R2 preserving H the matrix X can be transformed into λ I2 + T (τ ) with τ = ±1, where T (τ ) :=

τ −τ τ −τ

.

(4.13)

(ii) If X has eigenvalues η ± iδ with η , δ ∈ R, δ = 0, then by a linear transformation of R2 preserving H the matrix X can be transformed into η I2 + S(δ ), where S(δ ) :=

0 −δ δ 0

.

(4.14)

60

4 General Methods for Solving Defining Systems

Proof. (i). We can assume λ = 0. Under any linear transformation of R2 taking X into its Jordan normal form the matrix H (regarded as the matrix of a bilinear form) is transformed into a matrix

0 a1 , a1 a2 where a1 , a2 ∈ R, a1 = 0. For τ = ±1, b1 , b2 ∈ R, b1 = 0 define a basis e1 , e2 in R2 as follows: e1 := (b1 , b1 ), e2 := (τ b1 + b2 , b2 ). By passing to the basis e1 , e2 one takes the matrix T (τ ) into its Jordan normal form (identical to that of X) and transforms H into

0 c1 , c1 c2 with c1 := τ b21 , c2 := b21 + 2τ b1 b2 . Setting τ := sgn a1 and choosing b1 , b2 such that ci = ai , i = 1, 2, we obtain statement (i). (ii). In this case we have

X=

a −b b c

for some a, b, c ∈ R satisfying the inequality (a − c)2 < 4b2 .

(4.15)

Observe that for any q ∈ R the linear transformation given by the matrix ⎛ ⎞ 1 + q2 q ⎝ ⎠  1 + q2 q preserves H . Then statement (ii) follows by choosing q to be a real solution of the equation  4bq 1 + q2 = (a − c)(1 + 2q2), which exists due to inequality (4.15). 

Remark 4.3. It follows from Propositions 4.1, 4.3 and Remark 4.2 that by a transformation of the form (x0 , x) → (ρ x0 , C x),

C ∈ GL(n, R), ρ ∈ R∗

(4.16)

every defining system can be reduced to a defining system of one of three types described below. Each of the types is given by the following specific forms of the matrices D and H:

4.1 Classification of Defining Systems

TYPE I.

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

61

⎞

0

1/2 ..

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. N times

..

. 1/2 −1/2 ..

.

0

−1/2

where 0 ≤ N ≤ n. Here H is diagonal with entries ±1 such that the number of negative eigenvalues does not exceed the number of positive ones.

TYPE II. ⎛ S(−1/2) ⎜ .. ⎜ . ⎜ n/2 times D=⎜ ⎜ ⎜ .. ⎝ .

0

0

⎞

⎛

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

⎜ ⎜ ⎜ H=⎜ ⎜ ⎜ ⎝

H

0 ..

. n/2 times

..

. H

0

S(−1/2)

where n is even.

TYPE III. ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

T (1) ..

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0

. N times

..

. T (1)

T (−1) ..

. K times

..

. T (−1) 0

0

..

. 0

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

62

4 General Methods for Solving Defining Systems

⎛ ⎜ ⎜ ⎜ ⎜ H=⎜ ⎜ ⎜ ⎜ ⎝

H

0 ..

. N + K + L times

..

. H

0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

In−2(N+K+L)

where 0 ≤ N + K + L ≤ [n/2]. We refer to a defining system for which the pair of matrices D, H has one of the above three forms as a defining system of type I, II, III, respectively. In the next three sections we give general methods for finding the solutions of systems of each of the three types. For the case where the signature of the Levi form up to sign is equal to either (n, 0) (the strongly pseudoconvex case) or (n − 1, 1) with n ≥ 2, or (n − 2, 2) with n ≥ 4 these methods will be refined in Chapters 5, 6, 7 to yield explicit affine classifications of spherical tube hypersurfaces.

4.2 Defining Systems of Type I Since the matrix D commutes with each of the matrices Cα (see (4.2)), every defining system of type I splits into two subsystems as follows: N

γ

Fαβ = Fα Fβ + ∑ Cαβ Fγ + Hαβ , γ =1 n

α , β = 1, . . . , N,

γ

Fαβ = −Fα Fβ + ∑ Cαβ Fγ + Hαβ , α , β = N + 1, . . . , n, γ =N+1

α = 1, . . . , N, β = N + 1, . . . , n,

Fαβ = 0, F(0) = 0,

α = 1, . . . , n.

Fα (0) = 0,

Then F = F 1 + F 2 , where F 1 (x1 , . . . , xN ) and F 2 (xN+1 , . . . , xn ) are the solutions of the first and second subsystems, respectively, where we set F 1 := 0 for N = 0 and F 2 := 0 for N = n. Define G1 := exp(−F 1 ),

G2 := exp(F 2 ).

The functions G1 , G2 satisfy the following systems: G1αβ =

N

γ

∑ Cαβ G2γ − Hαβ G1 ,

γ =1

G1 (0) = 1,

G1α (0) = 0,

α , β = 1, . . . , N,

4.2 Defining Systems of Type I

and G2αβ =

63

n

γ

∑

γ =N+1

G2 (0) = 1,

Cαβ G2γ + Hαβ G2 , G2α (0) = 0,

α , β = N + 1, . . . , n.

Let N ≥ 1. It follows that the vector V 1 := (G1 , G11 , . . . , G1N ) satisfies the linear system

∂V 1 = A1α V 1 , ∂ xα

V 1 (0) = (1, 0, . . . , 0),

α = 1, . . . , N,

where A1α is the following (N + 1) × (N + 1)-matrix: ⎞ ⎛ 0 ... 0 1 0 ... 0 ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 1 1 T ⎟. ⎜ Aα := ⎜ −Hαα (Cα ) ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎠ ⎝ ..

(4.17)

(4.18)

0 Here the rows and columns are enumerated from 0 to N, the non-zero entries γ γ in the 0th row and column occur at the α th positions, and (C1α )β := Cαβ for

α , β , γ = 1, . . . , N. Relations (ii) of Lemma 3.2 imply that each of the matrices A1α ˜ 1 -symmetric, where is H ⎞ ⎛ −1 0 . . . 0 ⎟ ⎜ 0 ⎟ ˜ 1 := ⎜ H (4.19) ⎟ ⎜ .. 1 ⎠ ⎝ . H 0

with (H1 )β γ := Hβ γ for β , γ = 1, . . . , N. Further, relations (iv) of Lemma 3.2 imply that the matrices A1α pairwise commute for α = 1, . . . , N. Hence, the solution of (4.17) is   V 1 = exp

N

∑ A1α xα

α =1

v1 ,

(4.20)

where v1 := (1, 0, . . . , 0) ∈ RN+1 . Similarly, for N < n the vector V 2 := (G2 , G2N+1 , . . . , G2n ) satisfies the linear system ∂V 2 = A2α V 2 , V 2 (0) = (1, 0, . . . , 0), α = N + 1, . . ., n, (4.21) ∂ xα

64

4 General Methods for Solving Defining Systems

where A2α is the following (n − N + 1) × (n − N + 1)-matrix: ⎞ 0 ... 0 1 0 ... 0 ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 2 )T ⎟. H A2α := ⎜ (C α ⎟ ⎜ αα ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎠ ⎝ .. ⎛

(4.22)

0 In the above matrix the rows and columns are enumerated by the elements of the set Σ := {0, N + 1, . . . , n}, the non-zero entries in the 0th row and column occur at the γ γ α th positions, and (C2α )β := Cαβ for α , β , γ = N + 1, . . . , n. As before, relations (ii) ˜ 2 -symmetric, where of Lemma 3.2 imply that each of the matrices A2α is H ⎞ ⎛ 1 0 ... 0 ⎟ ⎜0 ⎟ ˜ 2 := ⎜ H (4.23) ⎟ ⎜ .. ⎝ . H2 ⎠ 0 ˜ 2 are with (H2 )β γ := Hβ γ for β , γ = N + 1, . . . , n (here the rows and columns of H also enumerated by the elements of the set Σ ). Further, relations (iv) of Lemma 3.2 imply that the matrices A2α pairwise commute for α = N + 1, . . . , n, and therefore the solution of (4.21) is   V 2 = exp

n

∑

α =N+1

A2α xα

v2 ,

(4.24)

where v2 := (1, 0, . . . , 0) ∈ Rn−N+1 . Next, near the origin the hypersurface x0 = F(x) can be written as

ϕ := exp(x0 )G1 − G2 = 0,

(4.25)

where the functions G1 and G2 are determined from formulas (4.20), (4.24). In particular, it follows that G1 extends to a function real-analytic on RN and G2 extends to a function real-analytic on Rn−N . Observe next that grad ϕ does not vanish anywhere on Rn+1 . Indeed, assuming otherwise, for N ≥ 1 we see that G1 and G1α , α = 1, . . . , N, simultaneously vanish at some point in RN , hence system (4.17) implies G1 ≡ 0, which is impossible. If N = 0, we have ϕ = exp(x0 ) − G2 , thus grad ϕ does not vanish anywhere on Rn+1 in this case either. Considering the connected component of the real-analytic subset of Rn+1 given by equation (4.25) that contains the hypersurface x0 = F(x), we arrive at the following proposition.

4.3 Defining Systems of Type II

65

Proposition 4.4. [56] If F is the solution of a defining system of type I near the origin, then its graph x0 = F(x) extends to a non-singular real-analytic hypersurface which is closed as a submanifold of Rn+1 . In the next two sections we observe a similar extension phenomenon for the graphs of the solutions of defining systems of types II and III as well.

4.3 Defining Systems of Type II Let F be the solution of a defining system of type II on a neighborhood Ω of the origin in Rn . By Proposition 3.1 the function F is real-analytic on Ω if Ω is sufficiently small, and therefore F can be extended to a function F holomorphic on a neighborhood ΩC of the origin in Cn , where Cn is identified with the subspace {z0 = 0} of Cn+1 . Observe that F satisfies on ΩC the system

∂ 2F ∂F γ ∂F γ ∂F γ = + Dβ + Cαβ + Hαβ , Dα ∂ zα ∂ zβ ∂ zγ ∂ zβ ∂ zα (4.26) ∂F (0) = 0. F (0) = 0, ∂ zα For α = 1, . . . , n/2 set

w2α −1 := iz2α −1 + z2α , w2α := −iz2α −1 + z2α ,

and let C0 be the matrix obtained by placing n/2 blocks   −i/2 i/2 1/2 1/2 on the diagonal. Expressing F in the variables w1 , . . . , wn , we obtain the function Fˆ (w) := F (C0 w), with w := (w1 , . . . , wn ). It can be shown analogously to Proposition 4.1 that the function Fˆ also satisfies a system of the form (4.26) on a neighborhood of the origin in Cn . Furthermore, the respective parameters of the new system (which we indicate by hats) are related to those of system (4.26) by formulas (4.1) with C replaced by C0 . In particular, we have ⎛

⎞ i/2 0 0 ⎜ 0 −i/2 ⎟ ⎜ ⎟ ⎜ ⎟ . .. ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ˆ n/2 times D=⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎝ i/2 0 ⎠ 0 0 −i/2

66

4 General Methods for Solving Defining Systems

ˆ = − 1 · In . and H 2 We split the index set {1, . . . , n} into the subsets σ1 and σ2 of odd and even indices, respectively, and say for every μ = 1, . . . , n/2 that the element 2μ − 1 of σ1 ˆ commutes with each and the element 2μ of σ2 are neighbors. Since the matrix D ˆ ˆ of the matrices Cα , the system for the function F splits into two subsystems as follows: ˆ ∂ 2 Fˆ ∂ Fˆ ∂ Fˆ ˆ γ ∂ F − 1 δαβ , =i +∑ C ∂ wα ∂ wβ ∂ wα ∂ wβ γ ∈σ1 αβ ∂ wγ 2

α , β ∈ σ1 ,

ˆ ∂ 2 Fˆ ∂ Fˆ ∂ Fˆ ˆ γ ∂ F − 1 δαβ , α , β ∈ σ2 , = −i +∑ C αβ ∂ wα ∂ wβ ∂ wα ∂ wβ γ ∈σ2 ∂ wγ 2

∂ 2 Fˆ = 0, ∂ wα ∂ wβ Fˆ (0) = 0,

α ∈ σ1 , β ∈ σ2 ,

∂ Fˆ (0) = 0, ∂ wα

α = 1, . . . , n.

Then Fˆ = Fˆ 1 + Fˆ 2 , where Fˆ 1 and Fˆ 2 are the solutions of the first and second subsystems, respectively. Therefore, we have F = Fˆ 1 (iz1 + z2 , . . . , izn−1 + zn ) + Fˆ 2 (−iz1 + z2 , . . . , −izn−1 + zn ) . Hence, on the neighborhood Ω of the origin in Rn we have F = Fˆ 1 (ix1 + x2 , . . . , ixn−1 + xn ) + Fˆ 2 (−ix1 + x2 , . . . , −ixn−1 + xn ) . It is straightforward to observe that the two terms in the right-hand side of the above formula are conjugate to each other. This follows from the fact that, due to  ˆ γ   , where ˆ γ , with α , β , γ ∈ σ2 , is conjugate to the number C (4.1), each number C αβ αβ α  , β  , γ  ∈ σ1 are the neighbors of α , β , γ , respectively. Thus, we have obtained F = 2 Re Fˆ 1 (ix1 + x2 , . . . , ixn−1 + xn ) , where Fˆ 1 is a function holomorphic near the origin in Cn/2 and satisfying the system ˆ1 ∂ 2 Fˆ 1 ∂ Fˆ 1 ∂ Fˆ 1 ˆ γ ∂ F − 1 δαβ , α , β ∈ σ1 , =i +∑ C αβ ∂ wα ∂ wβ ∂ wα ∂ wβ γ ∈σ1 ∂ wγ 2 Fˆ 1 (0) = 0,

∂ Fˆ 1 (0) = 0, ∂ wα

α ∈ σ1 .

4.3 Defining Systems of Type II

67

  Setting G := exp −iFˆ 1 we get

∂ 2G ˆ γ ∂ G + i δαβ G, α , β ∈ σ1 , =∑ C ∂ wα ∂ wβ γ ∈σ1 αβ ∂ wγ 2 ∂G G(0) = 1, (0) = 0, α ∈ σ1 . ∂ wα It follows that the vector



∂G ∂G ∂G , ,..., V := G, ∂ w1 ∂ w3 ∂ wn−1

satisfies the linear system

∂V = Aα V, ∂ wα

V (0) = (1, 0, . . . , 0),

α ∈ σ1 ,

(4.27)

where Aα is the following (n/2 + 1) × (n/2 + 1)-matrix: ⎛ ⎞ 0 ... 0 1 0 ... 0 ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟  ⎜ ⎟. ˆ Aα := ⎜ i/2 Cα ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎝ .. ⎠ 0 Here the rows and columns are enumerated by the elements of the set {0} ∪ σ1 , the non-zero entries in the 0th row and column occur at the α th positions, and ˆ α )γ := C ˆ γ for α , β , γ ∈ σ1 . Relations (ii) of Lemma 3.2 imply that each of (C β αβ the matrices Aα is symmetric with respect to   i/2 0

0

In/2

ˆ α is symmetric). Further, relations (iv) of Lemma 3.2 (in particular, each matrix C imply that the matrices Aα pairwise commute for α ∈ σ1 , and therefore the solution of (4.27) is   V = exp

∑

α ∈σ1

Aα wα

v,

where v := (1, 0, . . . , 0) ∈ Cn/2+1. Next, near the origin the hypersurface x0 = F(x) can be written as   ψ := Im R(x) · exp(ix0 /2) = 0,

(4.28)

(4.29)

68

4 General Methods for Solving Defining Systems

where R := G (ix1 + x2, . . . , ixn−1 + xn ) .

(4.30)

The function G can be explicitly determined from formula (4.28). In particular, it follows that G extends to a function holomorphic on Cn/2 , and therefore R(x) extends to a function real-analytic on Rn . Observe next that grad ψ does not vanish at the points in Rn+1 where ψ vanishes. Indeed, we have  1  ∂ψ = Re R(x) · exp(ix0 /2) , ∂ x0 2

∂ψ ∂G = Re · exp(ix0 /2) , α = 1, . . . , n/2, ∂ x2α −1 ∂ w2α −1

∂ψ ∂G = Im · exp(ix0 /2) , ∂ x 2α ∂ w2α −1

α = 1, . . . , n/2,

where the partial derivatives of G are calculated at the point (ix1 +x2 , . . . , ixn−1 +xn ). It then follows that if ψ and grad ψ simultaneously vanish at some point in Rn+1 , then G and grad G simultaneously vanish at some point in Cn/2 . Hence, system (4.27) implies G ≡ 0, which is impossible. Considering the connected component of the real-analytic subset of Rn+1 given by equation (4.29) that contains the hypersurface x0 = F(x), we obtain the following proposition. Proposition 4.5. [56] If F is the solution of a defining system of type II near the origin, then its graph x0 = F(x) extends to a non-singular real-analytic hypersurface which is closed as a submanifold of Rn+1 .

4.4 Defining Systems of Type III Since the matrices Cα commute with D and are H-symmetric, for a defining system of type III they have the form ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ Cα = ⎜ ⎜ ⎜ ⎜ ⎝

ωα1

Ωα

πα1 −πα1 . . . παN+K −παN+K

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ωαN+K ⎟ ⎟ ωαN+K ⎟ ⎠ Λα

ωα1 .. .

(4.31)

4.4 Defining Systems of Type III

69

where Ωα is a 2(N + K) × 2(N + K)-matrix symmetric with respect to ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

H

0 ..

. N + K times

..

.

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

H

0

Λα is an (n − 2(N + K)) × (n − 2(N + K))-matrix symmetric with respect to ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

H

0 ..

. L times

..

. H

0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

In−2(N+K+L)

πα1 , . . . , παN+K are (n − 2(N + K))-column-vectors, and ωα1 , . . . , ωαN+K are (n − 2(N + K))-row-vectors. In addition, for each α the following relations hold: 2γ −1

2γ

2γ −1

2γ −1

2γ

2γ

(Ωα )2β −1 − (Ωα )2β −1 = (Ωα )2β −1 + (Ωα )2β

=

(Ωα )2β −1 + (Ω α )2β ,

γ , β = 1, . . . , N and γ , β = N + 1, . . . , N + K, 2γ −1

2γ

2γ −1

2γ −1

2γ

2γ

(Ωα )2β −1 − (Ωα )2β −1 = −(Ωα )2β −1 − (Ωα )2β

=

−(Ωα )2β −1 − (Ω α )2β ,

γ = 1, . . . , N, β = N + 1, . . . , N + K and γ = N + 1, . . ., N + K, β = 1, . . . , N, (πασ )2γ −1 = (ωασ )2γ −1 ,

γ = N + K + 1, . . ., N + K + L, σ = 1, . . . , N + K, (πασ )2γ = −(ωασ )2γ ,

γ = N + K + 1, . . ., N + K + L, σ = 1, . . . , N + K, (πασ )γ = (ωασ )γ ,

γ = 2(N + K + L) + 1, . . ., n, σ = 1, . . . , N + K,

(4.32)

70

4 General Methods for Solving Defining Systems

where the entries of πασ and ωασ are enumerated from 2(N + K) + 1 to n. In addition, relations (iv) of Lemma 3.2 imply that the matrices Λα pairwise commute. We first assume N + K ≥ 1 and for α = 1, . . . , N + K set uα := x2α −1 + x2α , vα := x2α −1 − x2α .

(4.33)

Expressing F in the variables u1 , . . . , uN+K , v1 , . . . , vN+K , x2(N+K)+1 , . . . , xn , we obtain the function  Fˆ := F (u1 + v1 ) /2, (u1 − v1 ) /2, . . . , (uN+K + vN+K ) /2,  (4.34) (uN+K − vN+K ) /2, x2(N+K)+1 , . . . , xn .

∂ Fˆ . ∂ uα Proposition 4.6. [56] For every α = 1, . . . , N + K the function Ψα is independent of u1 , . . . , uN+K and x2(N+K)+1 , . . . , xn .

Further, for α = 1, . . . , N + K define Ψα := 2

Proof. The defining system yields  1  γ ∂Ψα γ γ γ = Fγ C2α −1 2β −1 + C2α 2β −1 + C2α −1 2β + C2α 2β , ∂ uβ 2

β = 1, . . . , N + K, 



∂Ψα γ γ = Fγ C2α −1 β + C2α β , ∂ xβ

β = 2(N + K) + 1, . . ., n,

where the variables x2γ −1 and x2γ for γ = 1, . . . , N + K are expressed in terms of uγ and vγ (see (4.33)). It follows from (4.31) and (4.32) that the right-hand sides of the above identities vanish.  Thus, the functions Ψα depend only on v1 , . . . , vN+K and are defined and realanalytic on some ball Bε centered at the origin in RN+K . On Bε these functions satisfy the following equations:  N+K  ∂Ψα 2γ −1 2γ −1 = ΨαΨβ + ∑ C2α −1 2β −1 + C2α 2β −1 Ψγ + H2α −1 2β −1, ∂ vβ γ =1

α , β = 1, . . . , N,  N+K  ∂Ψα 2γ −1 2γ −1 = −ΨαΨβ + ∑ C2α −1 2β −1 + C2α 2β −1 Ψγ + H2α −1 2β −1, ∂ vβ γ =1

α , β = N + 1, . . ., N + K,  N+K  ∂Ψβ ∂Ψα 2γ −1 2γ −1 =− = −ΨαΨβ + ∑ C2α −1 2β −1 + C2α 2β −1 Ψγ , ∂ vβ ∂ vα γ =1

α = 1, . . . , N, β = N + 1, . . . , N + K,

(4.35)

4.4 Defining Systems of Type III

Ψα (0) = 0,

71

α = 1, . . . , N + K.

Consider the following map from Bε into RN+K : P:

(v1 , . . . , vN+K ) → (Ψ1 , . . . , ΨN , −ΨN+1 , . . . , −ΨN+K ) .

(4.36)

Relations (4.31), (4.32) and equations (4.35) imply that the Jacobian matrix of P is symmetric, and therefore there exists a function P(v1 , . . . , vN+M ) real-analytic on Bε and vanishing at the origin such that the following holds:

∂P = Ψα , α = 1, . . . , N, ∂ vα

∂P = −Ψα , α = N + 1, . . ., N + K. ∂ vα

Furthermore, equations (4.35) yield that P satisfies the following system:

∂ 2P ∂ P ∂ P N+K ˜ γ ∂ P ˜ αβ , α , β = 1, . . . , N + K, α ≤ β , = + +H Cαβ ∂ vα ∂ vβ ∂ vα ∂ vβ γ∑ ∂ vγ =1 P(0) = 0,

∂P (0) = 0, ∂ vα

(4.37)

α = 1, . . . , N + K,

where for α ≤ β we set ⎧ 2γ −1 2γ −1 ⎪ C2α −1 2β −1 + C2α 2β −1, α , β , γ = N + 1, . . ., N + K and ⎪ ⎪ ⎨ α , γ = 1, . . . , N, β = 1, . . . , N + K, ˜ γ := C αβ ⎪   ⎪ ⎪ 2γ −1 ⎩ − C2γ −1 2α −1 2β −1 + C2α 2β −1 , otherwise (4.38) and ⎧ 1, α = β = 1, . . . , N, ⎪ ⎪ ⎪ ⎨ ˜ αβ := −1, α = β = N + 1, . . . , N + K, H ⎪ ⎪ ⎪ ⎩ 0, otherwise. Observe that the form of system (4.37) is similar to that of defining systems of ˜ may type I (the only difference being that the number of negative eigenvalues of H exceed the number of positive ones). This system can be solved as explained in Section 4.2. Namely, setting Q := exp(−P) (4.39) we obtain the system N+K ∂ 2Q ˜ γ ∂Q −H ˜ αβ Q, α , β = 1, . . . , N + K, α ≤ β , = ∑ C αβ ∂ v ∂ vα ∂ vβ γ γ =1

Q(0) = 1,

∂Q (0) = 0, ∂ vα

α = 1, . . . , N + K.

(4.40)

72

4 General Methods for Solving Defining Systems

Considering the vector

∂Q ∂Q V := Q, ,..., ∂ v1 ∂ vN+K we get the system

∂V = Bα V , ∂ vα

V (0) = (1, 0, . . . , 0),

α = 1, . . . , N + K,

(4.41)

where Bα for α = 1, . . . , N + K are the following pairwise commuting (N + K + 1) × (N + K + 1)-matrices: ⎞ ⎛ 0 ... 0 1 0 ... 0 ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ T ⎟. ⎜ ˜ ˜ Bα := ⎜ −Hαα (4.42) (Cα ) ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎠ ⎝ .. 0 Here the rows and columns are enumerated from 0 to N + K, the non-zero entries in the 0th row and column occur at the α th positions, and we define ˜ α )γ := (C ˜ β )γα := C ˜ γ for α , β , γ = 1, . . . , N + K, α ≤ β . Hence, (C β αβ  V = exp

N+K

∑ Bα vα

α =1



ϑ,

(4.43)

where ϑ := (1, 0, . . . , 0) ∈ RN+K+1 . It then follows that Q extends to a function real-analytic on RN+K . Furthermore, Q and its gradient do not vanish simultaneously anywhere in RN+K since otherwise system (4.41) would imply Q ≡ 0, which clearly does not hold on Bε . Next, on Bε , we have 1 ∂Q Ψα = − · , α = 1, . . . , N, Q ∂ vα (4.44) 1 ∂Q Ψα = · , α = N + 1, . . . , N + K. Q ∂ vα Identities (4.44) imply that each product QΨα extends to a function real-analytic on RN+K . Further, for the function Fˆ defined in (4.34), from (4.31) we obtain n N+K ∂ 2 Fˆ γ ∂ Fˆ γ = ∑ (Λα )β + ∑ (ωα )β Ψγ + Hαβ , ∂ xα ∂ xβ γ =2(N+K)+1 ∂ xγ γ =1

α , β = 2(N + K) + 1, . . ., n,

(4.45)

4.4 Defining Systems of Type III

73

where the rows and columns of Λα are enumerated from 2(N + K) + 1 to n. Therefore, the vector   ∂ Fˆ ∂ Fˆ W := ,..., ∂ x2(N+K)+1 ∂ xn satisfies the non-homogeneous linear system

∂W = ΛαT W + μα , ∂ xα

W (0) = 0,

α = 2(N + K) + 1, . . ., n,

(4.46)

where μα is the (n − 2(N + K))-vector depending only on v1 , . . . , vN+K whose β th component is equal to N+K

γ

∑ (ωα )β Ψγ + Hαβ ,

β = 2(N + K) + 1, . . ., n.

γ =1

(4.47)

Since the matrices Λα pairwise commute and system (4.46) has a solution, the following holds:

ΛαT μβ = ΛβT μα , Hence, W has the form  W = exp



n

∑

α =2(N+K)+1



α , β = 2(N + K) + 1, . . ., n.

ΛαT xα

×

n



∑

β =2(N+K)+1

xβ 0

 exp

−ΛβT t





 dt ×

n

exp − ∑

γ =β +1

(4.48) 

ΛγT xγ



μβ + Φ ,

where Φ is an (n − 2(N + K))-vector whose components Φα depend only on u1 , . . . , uN+K , v1 , . . . , vN+K and vanish at the origin. It follows from (4.47), (4.48) that for every α = 2(N + K) + 1, . . ., n we have N+K n ∂ Fˆ = ∑ σαβ Ψβ + ∑ λαγ Φγ + ρα , ∂ xα β =1 γ =2(N+K)+1

(4.49)

where σαβ , λαγ , ρα are functions of the variables x2(N+K)+1 , . . . , xn that extend to functions real-analytic on Rn−2(N+K) ; furthermore, σαβ and ρα vanish at the origin. ˆ ∂ xα for α = 2(N + K) + 1, . . . , n is independent of Observe now that ∂ F/ u1 , . . . , uN+K . Indeed, by Proposition 4.6 we have



∂ ∂ Fˆ ∂ 1 Ψ = 0, α = 2(N + K) + 1, . . ., n, β = 1, . . . , N + K. = ∂ u β ∂ xα ∂ xα 2 β

74

4 General Methods for Solving Defining Systems

Observe next that the functions λαγ form an everywhere invertible matrix (which we denote by Λ ). In fact, one has α  

λαγ =

n

exp

∑

β =2(N+K)+1

ΛβT xβ

. γ

The last two observations together with (4.49) yield that the functions Φα for α = 2(N + K) + 1, . . ., n in fact depend only on the variables v1 , . . . , vN+K . Next, (4.31) yields   ∂ Fˆ n 1 ∂ 2 Fˆ ∂ Fˆ γ γ = Ψα + Λ ) − ( Λ ) ( 2 α −1 2 α ∑ β β ∂x + ∂ v α ∂ xβ ∂ xβ 2 γ =2(N+K)+1 γ  1 N+K  2γ −1 2γ −1 C − C 2α −1 β 2α β Ψγ , 2 γ∑ =1

α = 1, . . . , N, β = 2(N + K) + 1, . . ., n,   ∂ Fˆ n 1 ∂ 2 Fˆ ∂ Fˆ γ γ = −Ψα + Λ ) − ( Λ ) + ( 2 α −1 2 α ∑ β β ∂x ∂ v α ∂ xβ ∂ xβ 2 γ =2(N+K)+1 γ

(4.50)

 1 N+K  2γ −1 2γ −1 − C C 2α −1 β 2α β Ψγ , 2 γ∑ =1

α = N + 1, . . ., N + K, β = 2(N + K) + 1, . . ., n. Plugging identities (4.49) into both sides of (4.50) and using (4.35), we obtain the equations

Λ

∂Φ = (Ψα In−2(N+K) + Λ˜ αT )Λ Φ + πα , ∂ vα

α = 1, . . . , N,

∂Φ Λ = (−Ψα In−2(N+K) + Λ˜ αT )Λ Φ + πα , α = N + 1, . . ., N + K. ∂ vα

(4.51)

Here

1 Λ˜ α := (Λ2α −1 − Λ2α ) 2 and πα is an (n − 2(N + K))-vector independent of the variables u1 , . . . , uN+K and having the form πα = sα Ψ + tα , α = 1, . . . , N + K, (4.52) where Ψ := (Ψ1 , . . . , ΨN+K ), sα is an (n − 2(N + K)) × (N + K)-matrix, and tα is an (n − 2(N + K))-vector. Here sα , tα depend only on x2(N+K)+1 , . . . , xn , with their components extending to functions real-analytic on Rn−2(N+K) , and tα vanishes at the origin for every α . Since Λ is everywhere invertible and commutes with each of the matrices Λ˜ αT , system (4.51) reduces to the system

4.4 Defining Systems of Type III

75

∂Φ = (Ψα In−2(N+K) + Λ˜ αT )Φ + Λ −1 πα , ∂ vα ∂Φ = (−Ψα In−2(N+K) + Λ˜ αT )Φ + Λ −1 πα , ∂ vα

α = 1, . . . , N, (4.53)

α = N + 1, . . ., N + K.

It then follows that each vector Λ −1 πα depends only on the variables v1 , . . . , vN+K . Since πα has the form (4.52) and the Jacobian matrix of the map P is non-degenerate at the origin (see (4.35), (4.36)), we see that Λ −1 sα is a constant matrix and tα ≡ 0. Thus, we have Λ −1 πα = s˜α Ψ , α = 1, . . . , N + K (4.54) for a constant (n − 2(N + K)) × (N + K)-matrix s˜α . Now (4.44) and (4.53) imply that the vector W := QΦ satisfies the following linear system:

∂W = Λ˜ αT W + ηα , ∂ vα

W (0) = 0,

α = 1, . . . , N + K,

(4.55)

where the components of each vector ηα := QΛ −1 πα extend to functions realanalytic on RN+K (see (4.44), (4.54)). Since the matrices Λ˜ α pairwise commute and system (4.55) has a solution, the following holds:

∂ ηβ ∂ ηα Λ˜ αT ηβ + = Λ˜ βT ηα + , ∂ vβ ∂ vα Therefore, we have  W = exp

N+K

∑

α =1

α , β = 1, . . . , N + K.



Λ˜ αT vα

×

N+K  vβ



∑

β =1 0

    N+K T T exp −Λ˜ β t exp − ∑ Λ˜ γ vγ ×

(4.56)

γ =β +1

   ηβ 0, . . . , 0,t, vβ +1, . . . , vN+K dt.

The above formula implies that QΦα extends to a function real-analytic on RN+K for α = 2(N + K) + 1, . . ., n. Further, (4.49) yields Fˆ =

N+K

n

β =1

γ =2(N+K)+1

∑ σβ Ψβ + ∑

λγ Φγ + ρ + r,

(4.57)

where σβ , λγ , ρ are functions of the variables x2(N+K)+1 , . . . , xn that extend to functions real-analytic on Rn−2(N+K) , and r is independent of x2(N+K)+1 , . . . , xn . Clearly,

76

4 General Methods for Solving Defining Systems

σβ , λγ , ρ may be chosen to vanish at the origin for all β , γ , in which case r vanishes at the origin with all its first-order partial derivatives. We will now deal with the function r. We have ∂r ∂ Fˆ 1 = = Ψα , ∂ uα ∂ uα 2

α = 1, . . . , N + K,

which implies r=

Ψα uα + R, α =1 2

N+K

∑

(4.58)

where R is a function of v1 , . . . , vN+M satisfying R(0) = 0 and ∂ R/∂ vα (0) = 0, α = 1, . . . , N + K. Plugging (4.57) and (4.58) into the equations for the function Fˆ that arise from the original defining system, collecting the terms independent of u1 , . . . , uN+K , and using (4.31), (4.32), (4.35), (4.44), (4.53), (4.54), (4.56), we get

∂ 2R ∂R ∂ R N+K ˜ γ ∂ R = Ψα + Ψβ + + ξαβ , Cαβ ∂ vα ∂ vβ ∂ vβ ∂ vα γ∑ ∂ vγ =1 α , β = 1, . . . , N, α ≤ β , ∂ 2R ∂R ∂ R N+K ˜ γ ∂ R = Ψα − Ψβ + + ξαβ , Cαβ ∂ vα ∂ vβ ∂ vβ ∂ vα γ∑ ∂ vγ =1 α = 1, . . . , N, β = N + 1, . . ., N + K,

(4.59)

∂ 2R ∂R ∂ R N+K ˜ γ ∂ R Cαβ = −Ψα − Ψβ + + ξαβ , ∂ vα ∂ vβ ∂ vβ ∂ vα γ∑ ∂ vγ =1 α , β = N + 1, . . . , N + K, α ≤ β , R(0) = 0,

∂R (0) = 0, ∂ vα

α = 1, . . . , N + K,

γ

˜ are defined in (4.38) and ξαβ are functions such that each product Qξαβ where C αβ extends to a function real-analytic on RN+K . Further, (4.40), (4.44), (4.59) yield that the vector

∂ (QR) ∂ (QR) W := QR, ,..., ∂ v1 ∂ vN+K satisfies the linear system

∂W = Bα W + ζα , ∂ vα

W(0) = 0,

α = 1, . . . , N + K,

(4.60)

where Bα are the matrices defined in (4.42) and ζα are (N + K + 1)-vectors whose components extend to functions real-analytic on RN+K . Since the matrices Bα

4.5 Globalization of Spherical Tube Hypersurfaces

77

pairwise commute and system (4.60) has a solution, the following holds: Bα ζ β +

∂ ζβ ∂ ζα = Bβ ζα + , ∂ vβ ∂ vα

α , β = 1, . . . , N + K.

Therefore, W has the form   W = exp

N+K

∑ Bα vα

α =1

×

N+K  vβ

∑

β =1 0

  N+K   exp −Bβ t exp − ∑ Bγ vγ ×



(4.61)

γ =β +1

   ζβ 0, . . . , 0,t, vβ +1 , . . . , vN+K dt.

The above formula implies that QR extends to a function real-analytic on RN+K . It now follows from (4.34), (4.44), (4.56), (4.57), (4.58), (4.61) that near the origin the hypersurface x0 = F(x) can be written as

χ := x0 Q +

 uβ  ∂ Q N+K  uβ  ∂ Q σβ + − ∑ σβ + − 2 ∂ vβ β =N+1 2 ∂ vβ β =1 N

∑

n

∑

γ =2(N+K)+1

(4.62)

λγ (QΦγ ) − Qρ − QR = 0,

where the variables u1 , . . . , uN+K and v1 , . . . , vN+K are expressed in terms of the variables x1 , . . . , x2(N+K) (see (4.33)) and each summand extends to a function realanalytic on Rn+1 . Observe that χ and its gradient do not vanish simultaneously at any point of Rn+1 since otherwise Q and its gradient would vanish simultaneously at some point in Rn , which is impossible (see (4.41)). If N = K = 0, then identity (4.57) holds with r ≡ 0, and (4.62) becomes x0 − ρ = 0, where ρ extends to a function real-analytic on Rn . Considering the connected component of the real-analytic subset of Rn+1 given by this equation for N = K = 0 and by equation (4.62) for N + K ≥ 1, we arrive at the following proposition. Proposition 4.7. [56] If F is the solution of a defining system of type III near the origin, then its graph x0 = F(x) extends to a non-singular real-analytic hypersurface which is closed as a submanifold of Rn+1 .

4.5 Globalization of Spherical Tube Hypersurfaces Theorem 3.1, Remark 4.3, and Propositions 4.4, 4.5, 4.7 yield the following result.

78

4 General Methods for Solving Defining Systems

Theorem 4.1. [56] If M is a spherical tube hypersurface in Cn+1 , then M extends to a non-singular real-analytic hypersurface Mext which is closed as a submanifold of Cn+1 . In this section we show that M ext inherits the sphericity property of M as stated below. Theorem 4.2. [56] The hypersurface M ext from Theorem 4.1 is spherical. Note that Theorems 4.1 and 4.2 do not hold for general (not necessarily tube) real-analytic spherical hypersurfaces. Indeed, the real-analytic hypersurface     1 2 2 ,k∈Z M1 := (z, w) ∈ C2 : Im w = sin  , z = 0, z = z π (2k + 1) can be mapped into the quadric " ! Q| · |2 = (z, w) ∈ C2 : Im w = |z|2 by means of the locally biholomorphic map

1 (z, w) → sin , w z and hence is spherical. However, M1 clearly does not extend to a closed non-singular hypersurface in C2 . Theorem 4.1 thus shows that M1 cannot be mapped onto a tube 2 hypersurface by any holomorphicautomorphism  of C . Observe also that M1 ex2 tends to every point of the form π (2k+1) , u + i , k ∈ Z, u ∈ R, as a non-singular real-analytic hypersurface, but the extension has a vanishing Levi form at these points and therefore is not spherical near any such point. Even if a real-analytic spherical hypersurface admits an extension to a closed non-singular real-analytic hypersurface in Cn+1 , the extension may be non-spherical. Indeed, the hypersurface ! " M2 := (z, w) ∈ C2 : Im w = |z|4 , z = 0 can be mapped into Q| · |2 by means of the locally biholomorphic map (z, w) → (z2 , w) and hence is spherical. Clearly, M2 extends to ! " M2 = (z, w) ∈ C2 : Im w = |z|4 , which is a closed non-singular real-analytic hypersurface in C2 . At the same time, the Levi form of M2 at every point of the form (0, u), u ∈ R, vanishes. Theorem 4.2 thus shows that M2 cannot be mapped onto a tube hypersurface by any holomorphic automorphism of C2 .

4.5 Globalization of Spherical Tube Hypersurfaces

79

Our proof of Theorem 4.2 requires two propositions. Proposition 4.8. [56] Let F be a real-analytic function on a domain Ω ⊂Rn . As sume that F satisfies on Ω a system of the form (3.5). Then if the matrix Fαβ is degenerate at a point in Ω , it is degenerate everywhere on Ω . Proof. We start with the following lemma. Lemma 4.1. [56] Let F be a C∞ -smooth function defined on a open set Ω ⊂ Rn . Assume that F satisfies on Ω a system of the form (3.5). Let p = (p1 , . . . , pn ) be a point in Ω and s := F(p), sα := Fα (p). Define G := F −

n

∑ sα (xα − pα ) − s.

(4.63)

α =1

Then the function G satisfies on Ω a system of the same form with zero initial conditions at p, namely   ˇ γ +H ˇ αβ , ˇ αγ Gβ + D ˇ γ Gα + C Gαβ = Gγ Eˇ γ Gα Gβ + D β αβ (4.64) G(p) = 0, Gα (p) = 0, with ˇ γ := Cγ + Eγ sα sβ + Dγα sβ + ˇ α := Eα , D ˇ α := Dα + Eα sβ + 1 δ α (Eν sν ) , C E β β αβ αβ 2 β   γ γ γ γ γ Dβ sα + δβ Dνα sν + δα Dνβ sν + sα δβ + sβ δα (Eν sν ) ,   ˇ αβ := Hαβ + Eν sα sβ + Dνα sβ + Dν sα + Cν sν . H β αβ Lemma 4.1 is obtained by an elementary calculation, which we omit.  We will now prove the proposition. Let the matrix Fαβ (p) be degenerate for     some p ∈ Ω . Consider the function G defined in (4.63). Clearly, Fαβ ≡ Gαβ on   Ω , hence to prove the proposition it is sufficient to show that the matrix Gαβ is everywhere degenerate. Considering the function G(Cx) instead of G(x) for an appropriate C ∈ GL(n, R), we can assume that the symmetric matrix   matrix  ˇ is degenerˇ := H ˇ αβ = Gαβ (p) is diagonal (see Proposition 4.1). Since H H ˇ αα = 0 for ate, we can further assume that there exists 1 ≤ m < n such that H ˇ αα = 0 for α = m, . . . , n. α = 1, . . . , m − 1 and H Differentiating the (α , β )th equation of system (4.64) with respect to xν , we find ˇν H ˇ Gαβ ν (p) = C αβ νν , which by interchanging the indices implies ˇν H ˇν ˇ ˇβ ˇ ˇ C αβ νν = Cβ α Hνν = Cαν Hβ β .

(4.65)

80

4 General Methods for Solving Defining Systems

Hence, we have ˇ βαν = 0, C

α = 1, . . . , n, β = 1, . . . , m − 1, ν = m, . . . , n.

(4.66)

Similarly, for the fourth-order partial derivatives of G we obtain     μ ˇ ˇγ C ˇ νγ ˇ νH ˇ μH ˇ να H ˇ αμ H ˇ νν + D ˇ μμ. ˇ βμ +D ˇ αμ H ˇ βν + D ˇ αν H + D Gαβ ν μ (p) = C H μ μ β αβ β Setting α = ν = μ ≤ m − 1, β ≥ m, interchanging the indices, and using (4.65), (4.66), we see ˇ α = 0, α = 1, . . . , m − 1, β = m, . . . , n. D (4.67) β We will now show by induction on | j| := j1 + . . . + jn that

∂ | j| G j j ∂ x11 . . . ∂ xnn

(p) = 0 if at least one of jm , . . . , jn is non-zero.

(4.68)

Identities (4.68) clearly hold for | j| = 1, 2, thus we assume | j| ≥ 3. If at least one of jm , . . . , jn is non-zero, we have on Ω

∂ | j| G ∂ x1j1 . . . ∂ xnjn

=

∂ | j|−2 Gα0 β0

∂ x11 . . . ∂ xnn

for some 1 ≤ α0 , β0 ≤ n, with β0 ≥ m, and some non-negative integers 1 , . . . , n such that 1 + . . . + n = | j| − 2 (cf. Proof 2 of Proposition 3.1). By (4.66), (4.67) the (α0 , β0 )th equation of system (4.64) simplifies as follows: Gα0 β0 =

m−1

∑ Gγ

γ =1

 γ  ˇ αγ Gβ + Eˇ Gα0 Gβ0 + D 0 0 n

∑

γ =m

 ˇγ ˇ αγ Gβ + D ˇ γ Gα + C Gγ Eˇ γ Gα0 Gβ0 + D 0 0 0 β α 0

 0 β0

.

Since each summand in the right-hand side of the above identity contains Gμ with μ ≥ m, the induction hypothesis implies (4.68) as required. Due to identities (4.68), the Taylor series of G at p is independent of xα with α ≥ m. Since G is real-analytic on Ω , it follows that G is independent of xα for  α ≥ m on Ω . Therefore, the matrix Gαβ degenerates at every point in Ω . The proof of the proposition is complete.  The second proposition required for the proof of Theorem 4.2 is as follows. Proposition 4.9. [56] Let F be a C∞ -smooth function on an open set Ω ⊂ Rn . Assume that F satisfies on Ω a system of the form (3.5). Suppose further that at some point q = (q1 , . . . , qn ) ∈ Ω we have Fn (q) = 0 and represent the hypersurface ˜ 0 , x1 , . . ., xn−1 ), x0 = F(x1 , . . ., xn ) near the point (F(q), q) ∈ Rn+1 in the form xn = F(x ∞ ˜ ˜ where F is C -smooth on a neighborhood Ω of the point q˜ := (F(q), q1 , . . . , qn−1 ).

4.5 Globalization of Spherical Tube Hypersurfaces

81

Then on Ω˜ the function F˜ satisfies a system of the form (3.5), namely the system n−1 ∂ F˜ ∂ 2 F˜ =∑ ∂ xα ∂ xβ γ =0 ∂ xγ



˜ ˜ ∂ F˜ ∂ F˜ ˜γ ˜ γα ∂ F + D ˜ γ ∂F +C ˜ +D E˜ γ β ∂x αβ + Hαβ , ∂ xα ∂ xβ ∂ xβ α

α , β = 0, . . . , n − 1, with ˜ 00 := − 1 Cnnn , D ˜ α0 := Dαn , D ˜ 0α := −Hnα , E˜ 0 := −Hnn, E˜ α := Cαnn , D 2 ˜ 000 := −2Dnn , C ˜ α00 := Eα , C ˜ 00α = C ˜ 0α 0 := −Cnnα , ˜ α := Cα − 1 δ α Cnnn , C D β nβ 2 β ˜ α := Dα − δ α Dnn , C ˜ 0 := −Hαβ , C ˜ γ := Cγ − δαγ Cn − δ γ Cnnα , ˜α =C C nβ αβ αβ β β0 β β 0β αβ ˜ 0α = H ˜ α 0 := −Dnα , H ˜ αβ := −Cn , ˜ 00 := −En , H H αβ

α , β , γ = 1, . . . , n − 1.

Proof. The proposition is proved by direct calculation utilizing the identities 1 ∂ F˜ , =  ˜ 1 , . . . , xn−1 ) ∂ x0 Fn x1 , . . . , xn−1 , F(x   ˜ 1 , . . . , xn−1 ) Fα x1 , . . . , xn−1 , F(x ∂ F˜   , =− ˜ 1 , . . . , xn−1 ) ∂ xα Fn x1 , . . . , xn−1 , F(x

α = 1, . . . , n − 1,

which are valid on a neighborhood of q, ˜ and equations (3.5) for the function F.



We will now prove Theorem 4.2. Proof. We assume without loss of generality that M passes through the origin and near the origin is given in a standard representation with the function F satisfying a defining system on a domain Ω ⊂ Rn . Let Ω 0 := Ω and define V0 ⊂ MR to be the graph of F over Ω0 . Fix a point a ∈ MRext and a continuous path Γ : [0, 1] → MRext with Γ (0) = 0 and Γ (1) = a. Choose a partition 0 = t0 < t1 < . . . < tk−1 < tk = 1 of [0, 1] with the property that there exist domains V in MRext , with  = 1, . . . , k, satisfying for each  the following conditions: (i) V contains Γ (t ), (ii) V ∩ V−1 = 0, / and (iii) there exists 0 ≤ i ≤ n such that V is a graph xi = F  , with F  being a realanalytic function on a domain Ω in the space Rni of the variables x j , j = 0, . . . , n, j = i (here Rn0 = Rn is the space of the variables x1 , . . . , xn ). First, suppose i1 = 0. Then Ω0 ∩ Ω1 is a non-empty open set and F ≡ F 1 on Ω0 ∩ Ω1 . Thus, the function  F(x), x ∈ Ω0 , F (x) = F 1 (x), x ∈ Ω1 is real-analytic on the domain Ω0 ∪ Ω1 and satisfies on it the same defining sys-  tem as F satisfies on Ω0 . Since M is spherical, the matrix Fαβ (0) = Fαβ (0)

82

4 General Methods for Solving Defining Systems

  is non-degenerate. Then by Proposition 4.8 the matrix Fαβ is non-degenerate everywhere on Ω0 ∪ Ω1 . Arguing as at the beginning of the proof of Theorem 3.1 and using Theorem 2.1, we see that the tube hypersurface Π −1 (V0 ∪ V1 ) ⊂ M ext is spherical. Now, suppose i1 ≥ 1. Fix a point b ∈ V0 ∩ V1 , with b = (b0 , b1 , . . . , bn ), and let c := (b1 , . . . , bn ) be the projection of b to Rn . It then follows that Fi1 (c) = 0, and we represent the hypersurface V0 near b as ˜ 0 , x1 , . . . , xi −1 , xi +1 , . . . , xn ). xi1 = F(x 1 1 Let σ be the set of indices {0, 1, . . . , i1 − 1, i1 + 1, . . ., n}. By Proposition 4.9 the realanalytic function F˜ satisfies a system of the form (3.5) in the variables xα , α ∈ σ , on a neighborhood Ω˜ 1 ⊂ Rni1 of the point b˜ := (b0 , b1 , . . . , bi1 −1 , bi1 +1 , . . . , bn ), which is the projection of b to Rni1 . Since b˜ lies in Ω 1 , we can choose Ω˜ 1 to be a subset of Ω1 . Clearly, F˜ ≡ F 1 on Ω˜ 1 , and therefore F 1 satisfies on Ω1 the same system of the form (3.5) as F˜ satisfies on Ω˜ 1 . Since Π −1 (V0 ) is spherical and b lies in V0 , the matrix

2

2 1 ∂ F ∂ F˜ ˜ ˜ (b) = (b) ∂ xα ∂ xβ ∂ xα ∂ xβ α ,β ∈σ α ,β ∈σ   2 1 is non-degenerate. Hence, by Proposition 4.8 the matrix ∂ F /∂ xα ∂ xβ α ,β ∈σ is non-degenerate everywhere on Ω1 . Arguing as at the beginning of the proof of Theorem 3.1 and using Theorem 2.1, we see that the hypersurface Π −1 (V1 ) ⊂ M ext is spherical, thus the hypersurface Π −1 (V0 ∪V1 ) ⊂ M ext is spherical as well. Moving along the chain # of sets  V one can show by the above argument that the k −1 ext tube hypersurface Π =0 V ⊂ M is spherical. Since a is an arbitrary point ext ext in MR , it follows that M is spherical.  We note that alternative proofs of Theorems 4.1 and 4.2 can be found in recent paper [41] (see Theorem 9.3 in Section 9.2).

Chapter 5

Strongly Pseudoconvex Spherical Tube Hypersurfaces

Abstract In this chapter we obtain an explicit classification of closed strongly pseudoconvex spherical tube hypersurfaces in Cn+1 up to affine equivalence. This classification was originally discovered by Dadok and Yang in 1985. We proceed as outlined in Chapter 4 by considering all possible defining systems. Observe that in this case the matrix H is positive-definite (we always assume H = In ), thus defining systems of type II do not occur. Our arguments generally differ from those by Dadok and Yang.

5.1 Defining Systems of Type I As we showed in Section 4.2, if F(x) is the solution of a defining system of type I, the hypersurface x0 = F(x) near the origin can be written in the form (4.25). We start by finding the function G2 . If N = n, we have G2 = 1, thus we assume N < n. ˜ 2 -symmetric (see (4.22), (4.23)), hence The matrices A2α for α = N + 1, . . . , n are H they are in fact symmetric. Since the matrices A2α pairwise commute, there exists C ∈ Ø(n − N + 1, R) such that   C−1 A2α C = diag λα0 , λαN+1 , . . . , λαn μ

with λα ∈ R, μ ∈ Σ = {0, N + 1, . . . , n}, α = N + 1, . . . , n. Now (4.24) yields that G2 has the form   G2 =

∑ dμ exp

μ ∈Σ

n

∑

α =N+1

μ

λα xα ,

  where d μ ∈ R for μ ∈ Σ . Since the matrix G2αβ (0) and (5.1) implies G2αβ (0) =

μ

μ

∑ d μ λα λβ ,

μ ∈Σ

α ,β =N+1,...,n

(5.1) is non-degenerate

α , β = N + 1, . . . , n,

A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 5, 

83

84

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

 μ it immediately follows that rank R = n − N, where R := λα μ ∈Σ , α =N+1,...,n is an (n − N + 1) × (n − N)-matrix. Choosing n − N linearly independent rows of R, we see that after a linear change of the variables xN+1 , . . . , xn the function G2 takes the form   n

a0 exp

∑

α =N+1

+

bα xα

n

∑

α =N+1

aα exα

(5.2)

for some a μ , bα ∈ R, μ ∈ Σ , α = N + 1, . . . , n. The initial conditions G2α (0) = 0, α = N + 1, . . ., n, now imply a0 bα + aα = 0, α = N + 1, . . . , n. (5.3)   identities (5.3) yield Together with the non-degeneracy of G2αβ (0) α ,β =N+1,...,n   a μ = 0 for all μ ∈ Σ . Furthermore, the non-degeneracy of G2αβ (0) and α ,β =N+1,...,n

identities (5.3) imply that the following vectors in RN−n are linearly independent:

α = N + 1, . . ., n.

(bN+1 , . . . , bα −1 , bα − 1, bα +1, . . . , bn ) ,

(5.4)

In what follows we find the form of the function G1 . If N = 0, we have G1 = 1, ˜ 1 -symmetric (see thus we assume N ≥ 1. The matrices A1α for α = 1, . . . , N are H (4.18), (4.19)) and pairwise commute. Since χH˜ 1 = 1, by Propositions 4.2, 4.3 and Remark 4.1 we obtain that for every α = 1, . . . , N there exists Cα ∈ GL(N + 1, R) such that either Cα−1 A1α Cα is diagonal or Cα−1 A1α Cα =



Xα1

0

0 Xα2

,

(5.5)

where Xα1 is a diagonal matrix and Xα2 is either a 2 × 2-matrix of the form ηα I2 + S(δα ) having two non-real mutually conjugate eigenvalues ηα ± iδα (see (4.14)), or a matrix of size κα × κα whose Jordan normal form is one of the following:   0 0 01 ξα Iκα + (5.6) 0 T2 with T2 := 0 0 and 2 ≤ κα ≤ N + 1, 

ξα Iκα +

00 0 T3



⎞ 010 with T3 := ⎝ 0 0 1 ⎠ and 3 ≤ κα ≤ N + 1, 000 ⎛

where ξα ∈ R is distinct from every eigenvalue of Xα1 . Furthermore, we have  IN−1 0 T ˜1 Cα H Cα = 0 H , where H is the matrix defined in (4.12). We will now consider four cases.

(5.7)

5.1 Defining Systems of Type I

85

Case (i). Suppose that all eigenvalues of every matrix A1α are real and that all these matrices are diagonalizable. In this case one can find C ∈ GL(N + 1, R) such that C−1 A1α C is diagonal for every α . Therefore, using (4.20) and arguing as for the function G2 above, we see that after a linear change of the variables x1 , . . . , xN the function G1 takes the form   a0 exp

N

∑ bα xα

α =1

+

N

∑ aα exα ,

(5.8)

α =1

with aμ , bα ∈ R, aμ = 0 for μ = 0, . . . , N, α = 1, . . . , N, where analogously to (5.3) we have a0 bα + aα = 0, α = 1, . . . , N. Furthermore, the following vectors in RN are linearly independent (cf. the linear independence of vectors (5.4)):    b1 , . . . , bα −1 , bα − 1, bα +1, . . . , bN , α = 1, . . . , N. It now follows from (4.25), (5.2), (5.8) and the properties of the constants occurring in (5.2), (5.8) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface n

∑ exα = 1.

α =0

Case (ii). Suppose that for some α0 the matrix A1α0 has non-real eigenvalues. Then for α = α0 we have  1 Yα 0 −1 1 Cα0 Aα Cα0 = (5.9) 0 Yα2 , where Yα2 are 2 × 2-matrices. Since the symmetric matrices Xα10 , Yα1 , with α = α0 , pairwise commute, they can be simultaneously diagonalized by a real orthogonal transformation, thus we can assume that Cα0 is such that all these matrices are diagonal. Further, the matrices Yα2 for α = α0 are H -symmetric, hence  s −t Yα2 = α α t α uα for some sα ,tα , uα . Since we have Xα20 = ηα0 I2 + S(δα0 ), with δα0 = 0, and since each Yα2 commutes with Xα20 , it follows that uα = sα for all α = α0 . Therefore, each matrix Yα2 is either a scalar matrix (for tα = 0) or has two non-real mutually conjugate eigenvalues (for tα = 0). In particular, we have   sα + itα ηα0 + iδα0 0 0 −1 2 −1 2 C0 Xα0 C0 = , C0 Yα C0 = 0 0 sα − itα ηα0 − iδα0 for α = α0 , where

86

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

 C0 :=

11 . −i i

Setting sα0 := ηα0 , tα0 := δα0 , from formula (4.20) we therefore obtain G1 =



N−2

N

μ

∑ dμ exp ∑ λα xα

μ =0



α =1

 N  + exp ∑ sα xα × α =1

 N  N    dN−1 cos ∑ tα xα + dN sin ∑ tα xα α =1

(5.10)

α =1

μ

for some λ α ∈ R, μ= 0, . . . , N − 2, α = 1, . . . , N, and d μ ∈ R, μ = 0, . . . , N. Since is non-degenerate and (5.10) implies the matrix G1αβ (0) α ,β =1,...,N

G1αβ (0) =

N−2

μ

μ

∑ dμ λα λβ + dN−1 sα sβ + dN sα tβ + dN tα sβ − dN−1tα tβ , α , β = 1, . . . , N,

μ =0

it follows that rank R = N, where R is the (N + 1) × N-matrix ⎛ 0 ⎞ λ1 . . . λN0 ⎜ .. .. .. ⎟ ⎜ . . ⎟ ⎜ N−2 . N−2 ⎟ R := ⎜ λ ⎟. . . . λ ⎜ 1 ⎟ N ⎝ s1 . . . sN ⎠ t1 . . . tN Since the last row of R is non-zero, there is a set of N linearly independent rows of R that contains the last row. Then after a linear change of the variables x1 , . . . , xN the function G1 takes either the form  N  N−2   a0 exp ∑ bα xα + ∑ aα exα + exN−1 aN−1 cos xN + aN sin xN , α =1

(5.11)

α =1

or the form N−1



N

∑ aα exα + exp ∑ bα xα

α =1

α =1

  a0 cos xN + aN sin xN

(5.12)

for some aμ , bα ∈ R, μ = 0, . . . , N, α = 1, . . . , N. Suppose first that G1 takes the form (5.11). The initial conditions G1α (0) = 0, α = 1, . . . , N, then imply a0 bα + aα = 0, α = 1, . . . , N.   Together with the non-degeneracy of G1αβ (0) aμ

= 0 for μ = 0, . . . , N − 2 and

(aN−1 )2

α ,β =1,...,N + (aN )2 >

(5.13) identities (5.13) yield 0. Furthermore, the

5.1 Defining Systems of Type I

87

  non-degeneracy of G1αβ (0)

α ,β =1,...,N

and identities (5.13) imply that the follow-

ing vectors in RN are linearly independent (in the top vector 1 occurs at position α ): (0, . . . , 0, 1, 0, . . . , 0, −1, 0), α = 1, . . . , N − 2, and (b1 , . . . , bN−2 , bN−1 − 1, bN ), (0, . . . , 0, 0, 1).

(5.14)

It now follows from (4.25), (5.2), (5.11) and the properties of the constants occurring in (5.2), (5.11) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to any connected component of the hypersurface sin x0 =

n

∑ exα .

(5.15)

α =1

Suppose now that G1 takes the form (5.12). The initial conditions G1α (0) = 0, α  = 1, . .. , N, then imply identities (5.13). Together with the non-degeneracy of G1αβ (0) these identities yield aμ = 0 for μ = 0, . . . , N − 1. Furthermore, α ,β =1,...,N   and identities (5.13) imply that the vecthe non-degeneracy of G1αβ (0) α ,β =1,...,N

tors in RN    b1 , . . . , bα −1 , bα − 1, bα +1, . . . , bN , α = 1, . . . , N − 1, and (0, . . . , 0, 1)

(5.16)

are linearly independent. It now follows from (4.25), (5.2), (5.12) and the properties of the constants occurring in (5.2), (5.12) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to any connected component of hypersurface (5.15) as well. Case (iii). Suppose that all eigenvalues of every matrix A1α are real, that for some α0 the Jordan normal form of the matrix A1α0 contains a 2 × 2-cell, and that there are no matrices whose Jordan normal form contains a 3 × 3-cell. Then for α = α0 identities (5.9) hold, where Yα2 are matrices of size κ × κ with 2 ≤ κ ≤ N + 1 (here κ := κα0 ). We suppose, as before, that Cα0 is such that the matrices Yα1 , with α = α0 , are diagonal. Note that all matrices Xα20 , Yα2 are Hα0 -symmetric, where  Hα0 :=

Iκ −2

0

0

H

.

(5.17)

Suppose first that there exists α1 = α0 such that Yα21 has more than one eigenvalue. Then one can find D ∈ GL(κ , R) preserving Hα0 and reducing Yα21 to a blockdiagonal form in which different blocks have distinct eigenvalues and all blocks, except possibly for the lowest one, are scalar matrices. Since the matrices D−1 Xα20 D, D−1Yα2 D, with α = α0 , pairwise commute, each of D−1 Xα20 D and D−1Yα2 D, with α = α0 , α1 , splits accordingly. Thus, it is sufficient to consider the case where each Yα2 has exactly one eigenvalue. Further, there exists C ∈ GL(κ , R) such that

88

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

C−1 Xα20 C = ξα0 Iκ +



00 0 T2



 ,

C T Hα0 C =

Iκ −2

0

0

H

,

where T2 is defined in (5.6) and H has the form  0ρ ρσ with ρ , σ ∈ R, ρ = 0. Since each of the matrices C−1Yα2C commutes with C−1 Xα20 C and is CT Hα0 C-symmetric, we obtain ⎛

0 0 ρ sα

⎞

C−1Yα2C = να Iκ + ⎝ sTα 0 πα ⎠ , 00 0 where sα is a column-vector of length κ − 2 and να , πα ∈ R for α = α0 . If sα = 0 for some α , then κ > 2 and the eigenspace of Yα2 has dimension κ − 2. Since none of the matrices Yα2 has a 3 × 3-cell in its Jordan normal form, we obtain that in fact sα = 0 for all α = α0 . Setting να0 := ξα0 , from formula (4.20) we therefore derive      G1 =

N−2

N

μ =0

α =1

μ

∑ dμ exp ∑ λα xα

+ exp

N

∑ να x α

α =1

dN−1 +

N

∑ tα xα

(5.18)

α =1

μ

for some λα ∈ R with μ = 0, . . . , N − 2, α = 1, . . . , N, some dμ ∈ R with μ = 0, . .. , N − 1, and some tα ∈ R with α = 1, . . . , N. Since the matrix  G1αβ (0) is non-degenerate and (5.18) implies α ,β =1,...,N

G1αβ (0) =

N−2

μ

μ

∑ dμ λα λβ + dN−1 να νβ + να tβ + tα νβ ,

μ =0

α , β = 1, . . . , N,

it follows that rank R = N, where R is the (N + 1) × N-matrix ⎞ ⎛ 0 λ1 . . . λN0 ⎟ ⎜ .. .. .. ⎟ ⎜. . . ⎟ ⎜ R := ⎜ λ N−2 . . . λ N−2 ⎟ . ⎟ ⎜ 1 N ⎝ ν1 . . . νN ⎠ t1 . . . tN If tα = 0 for α = 1, . . . , N, then the first N rows of R are linearly independent, thus after a linear change of the variables x1 , . . . , xN the function G1 takes the form N

∑ aα exα ,

α =1

5.1 Defining Systems of Type I

89

with aα ∈ R, α = 1, . . . , N, which is impossible since grad G1 (0) = 0. Hence, the last row of R is in fact non-zero, thus there is a set of N linearly independent rows of R that contains the last row. Then after a linear change of the variables x1 , . . . , xN the function G1 takes either the form  N−2  N   a0 exp ∑ bα xα + ∑ aα exα + exN−1 aN−1 + xN α =1

or the form

N−1

(5.19)

α =1



N

∑ aα exα + exp ∑ bα xα

α =1

α =1

  a0 + xN

(5.20)

for some aμ , bα ∈ R, μ = 0, . . . , N − 1, α = 1, . . . , N. Suppose first that G1 takes the form (5.19). The initial conditions G1α (0) = 0, α = 1, . . . , N, then imply a0 bα + aα = 0, α = 1, . . . , N − 1, a0 bN + 1 = 0. 

(5.21)

 G1αβ (0)

identities (5.21) yield   aμ = 0 for μ = 0, . . . ,N −2. Furthermore, the non-degeneracy of G1αβ (0)

Together with the non-degeneracy of

α ,β =1,...,N

α ,β =1,...,N

and identities (5.21) imply that vectors (5.14) are linearly independent. It now follows from (4.25), (5.2), (5.19) and the properties of the constants occurring in (5.2), (5.19) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 =

n

∑ exα .

(5.22)

α =1

Suppose now that G1 takes the form (5.20). The initial conditions G1α (0) = 0, α  = 1, . .. , N, then imply identities (5.21). Together with the non-degeneracy of these identities yield aμ = 0 for μ = 0, . . . , N − 1. Furthermore, G1αβ (0) α ,β =1,...,N   and identities (5.21) imply that vectors the non-degeneracy of G1αβ (0) α ,β =1,...,N

(5.16) are linearly independent. It now follows from (4.25), (5.2), (5.20) and the properties of the constants occurring in (5.2), (5.20) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to hypersurface (5.22) as well. Case (iv). Suppose that all eigenvalues of every matrix A1α are real and for some α0 the Jordan normal form of the matrix A1α0 contains a 3 × 3-cell. Then for α = α0 identities (5.9) hold, where Yα2 are matrices of size κ × κ with 3 ≤ κ ≤ N + 1 (here κ := κα0 ). We suppose, as before, that Cα0 is such that the matrices Yα1 , with α = α0 , are diagonal. Furthermore, as in case (iii), it is sufficient to consider the situation where each Yα2 has exactly one eigenvalue.

90

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

There exists C ∈ GL(κ , R) such that  0 0 −1 2 C Xα0 C = ξα0 Iκ + 0 T3 ,

 C Hα0 C = T

Iκ −3

0

0

H

,

where T3 is defined in (5.7), Hα0 is defined in (5.17) and H has the form ⎛ ⎞ 0 0 ρ ⎝0 ρ σ⎠ ρσ η with ρ , σ , η ∈ R, ρ > 0. Since each of the matrices C−1Yα2C commutes with C−1 Xα20 C and is CT Hα0 C-symmetric, we obtain ⎞ ρ sα ⎜ sTα 0 ζα πα ⎟ ⎟ C−1Yα2C = να Iκ + ⎜ ⎝ 0 0 0 ζα ⎠ , 00 0 0 ⎛

000

where sα is a column-vector of length κ − 3 and να , ζα , πα ∈ R for α = α0 . Setting να0 := ξα0 , from formula (4.20) we therefore derive     G1 =

N−κ

N

μ =0

α =1

μ

∑ dμ exp ∑ λα xα

N

∑ να xα

+ exp

×

α =1

 dN−κ +1 +



N

(5.23)

∑ tα xα + Q (x1 , . . . , xN )

α =1

μ

for some λα ∈ R with μ = 0, . . . , N − κ , α = 1, . . . , N, some d μ ∈ R with μ = 0, . . . , N − κ + 1, some tα ∈ R with α = 1, . . . , N, and some quadratic form Q. Furthermore, there exist vectors w1 , . . . , wN ∈ Rκ −2 such that N

Q = ± ∑ Eαβ xα xβ ,

(5.24)

α ,β =1

where Eαβ := wα , wβ , with ·, · being the bilinear form on Rκ −2 defined for a vector w = (w1 , . . . , wκ −2 ) as w, w := ρ We write Q as follows: Q=

m

κ −3

∑ w2α + w2κ −2.

α =1



N

∑ εγ ∑

γ =1

α =1

2 γ tα x α

,

(5.25)

5.1 Defining Systems of Type I

91

  where 0 ≤ m ≤ N, εα = ±1 for α = 1, . . . , m, and the matrix tβα

α =1,...,m, β =1,...,N

has rank m. Clearly, representation (5.24) yields m ≤ κ − 2.   is non-degenerate and (5.23), (5.25) imply Since the matrix G1αβ (0) α ,β =1,...,N

G1αβ (0) =

N−κ

μ

m

μ

γ γ

∑ dμ λα λβ +dN−κ +1να νβ +να tβ +tα νβ + 2 ∑ εγ tα tβ , α , β = 1, . . . , N,

μ =0

α =1

it follows that rank R = N, where R is the (N − κ + m + 3) × N-matrix ⎛ 0 ⎞ λ1 . . . λN0 ⎜ .. ⎟ .. .. ⎜. ⎟ ⎜ N−κ . . N−κ ⎟ ⎜λ . . . λN ⎟ ⎜ 1 ⎟ ⎜ ν1 . . . νN ⎟ ⎜ ⎟. R := ⎜ ⎟ . . . tN ⎜ t1 ⎟ 1 ⎜ t1 ⎟ . . . tN ⎜ 1 ⎟ ⎜. ⎟ . . .. .. ⎝ .. ⎠ . . . tNm

t1m

Then we have m ≥ κ − 3. If m = κ − 3, then all rows of R are linearly independent, thus after a linear change of the variables x1 , . . . , xN the function G1 takes the form   N−κ +1

∑

α =1

aα exα + exN−κ +2 aN−κ +2 + xN−κ +3 +

κ −3

∑ εγ x2N−κ +γ +3

γ =1

,

with aα ∈ R, α = 1, . . . , N − κ + 2, which is impossible since grad G1 (0) = 0. Thus m = κ − 2. Hence, m ≥ 1 and there is a set of N linearly independent rows of R that contains the last m rows. Then after a linear change of the variables x1 , . . . , xN the function G1 takes either the form   N−κ +1

∑

α =1

aα exα + exN−κ +2 a0 +

N

∑

α =N−κ +3

bα xα +

κ −2

∑ εγ x2N−κ +γ +2

(5.26)

γ =1

for some aμ , bα ∈ R, μ = 0, . . . , N − κ + 1, α = N − κ + 3, . . . , N, or the form  a0 exp

N

∑

α =1

 bα xα

+

N−κ

∑ aα exα +

α =1

exN−κ +1



aN−κ +1 + xN−κ +2 +

κ −2

∑

γ =1



εγ x2N−κ +γ +2

for some aμ , bα ∈ R, μ = 0, . . . , N − κ + 1, α = 1, . . . , N, or the form

(5.27)

92

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces N−κ +1

∑

α =1

 aα exα

+ exp

N

∑

α =1

 bα xα

a0 + xN−κ +2 +

κ −2

∑

γ =1



εγ x2N−κ +γ +2

(5.28)

for some aμ , bα ∈ R, μ = 0, . . . , N − κ + 1, α = 1, . . . , N. Suppose first that G1 takes the form (5.26). Then the initial conditions G1α (0) = 0, α = 1, . . . , N, imply the identities aμ = 0 and bα = 0 for μ = 0, . . . , N − κ + 1, α =N− . . , N, and we obtain a contradiction with the non-degeneracy of the  κ + 3, .  matrix G1αβ (0) . α ,β =1,...,N

Suppose next that G1 takes the form (5.27). The initial conditions G1α (0) = 0, α = 1, . . . , N, then imply a0 bα + aα = 0,

α = 1, . . . , N − κ + 1,

a0 bN−κ +2 + 1 = 0, bα

(5.29)

α = N − κ + 3, . . . , N.   identities (5.29) yield Together with the non-degeneracy of G1αβ (0) α ,β =1,...,N   aμ = 0 for μ = 0, . . . ,N − κ . Furthermore, the non-degeneracy of G1αβ (0) = 0,

α ,β =1,...,N

and identities (5.29) imply that the following vectors in RN−κ +2 are linearly independent (in the top vector 1 occurs at position α ): (0, . . . , 0, 1, 0, . . . , 0, −1, 0), α = 1, . . . , N − κ , and (b1 , . . . , bN−κ , bN−κ +1 − 1, bN−κ +2 ), (0, . . . , 0, 0, 1). It now follows from (4.25), (5.2), (5.27) and the properties of the constants occurring in (5.2), (5.27) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 =

n−κ +2

∑

α =1

n

exα + ∑

α =n−κ +3

x2α .

(5.30)

Suppose finally that G1 takes the form (5.28). The initial conditions G1α (0) = 0, α  = 1, . .. , N, then imply identities (5.29). Together with the non-degeneracy of G1αβ (0) identities (5.29) yield aμ = 0 for μ = 0, . . . , N − κ + 1. Furα ,β =1,...,N   thermore, the non-degeneracy of G1αβ (0) and identities (5.29) imply α ,β =1,...,N

that the following vectors in RN−κ +2 are linearly independent: (b1 , . . . , bα −1 , bα − 1, bα +1, . . . , bN−κ +2 ), α = 1, . . . , N − κ + 1, and (0, . . . , 0, 1). It now follows from (4.25), (5.2), (5.28) and the properties of the constants occurring in (5.2), (5.28) that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to hypersurface (5.30) as well.

5.3 Classification

93

5.2 Defining Systems of Type III For a defining system of type III in the strongly pseudoconvex case we have D = 0. Hence, by (iv) of Lemma 3.2 the matrices Cα pairwise commute for α = 1, . . . , n. Since by (ii) of Lemma 3.2 these matrices are symmetric, they can be simultaneously diagonalized by a real orthogonal transformation, i.e. there exists C ∈ O(n, R) such that C−1 Cα C is diagonal for α = 1, . . . , n. Hence, by Proposition 4.1 the function ˆ F(x) := F(Cx) satisfies a defining system of type III, where the corresponding maˆ α are diagonal for α = 1, . . . , n. From (i) of Lemma 3.2 we now obtain trices C ˆ α are equal to zero except possibly for that for every α all entries of the matrix C α ˆ λα := Cαα . Hence, the defining system for the function Fˆ is as follows: Fˆαα = λα Fˆα + 1,

α = 1, . . . , n,

Fˆαβ = 0,

α , β = 1, . . . , n,

α = β ,

Fˆα (0) = 0, α = 1, . . . , n.

ˆ F(0) = 0,

Therefore, the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 =

m

n

α =1

α =m+1

∑ exα + ∑

x2α ,

where m is the number of indices α for which λα = 0.

5.3 Classification The results of Sections 5.1, 5.2 yield the following theorem. Theorem 5.1. [27] Let M be a closed strongly pseudoconvex spherical tube hypersurface in Cn+1 . Then M is affinely equivalent to a tube hypersurface with the base given by one of the following equations: (1) x0 =

m

n

α =1

α =m+1

∑ exα + ∑

(2) sin x0 = (3)

n

∑ exα ,

α =1

x2α , 0 ≤ m ≤ n, 0 < x0 < π ,

n

∑ exα = 1.

α =0

For each tube hypersurface in Theorem 5.1 it is not hard to explicitly find a locally diffeomorphic CR-map into either the quadric Q||·||2 (see (1.4)) or the

94

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

hypersurface S||·||2 (see (1.6)). The tube hypersurface with base (1) for a fixed m is mapped into Q||·||2 by the locally biholomorphic map1 z∗α = exp

z  α

2

α = 1, . . . , m,

,

zα z∗α = √ , 2

α = m + 1, . . ., n,



1 n w = i z0 − ∑ z2α 2 α =m+1

(5.31)

 .

Observe that this map is biholomorphic for m = 0, thus the quadric Q||·||2 is CR-equivalent to the tube hypersurface with the base x0 =

n

∑ x2α .

α =1

This last observation has an analogue for any quadric Qg . Indeed, if g(z, z) = ∑nα =1 εα |zα |2 for εα ∈ {±1, 0}, then the tube hypersurface with the base x0 =

n

∑ εα x2α

α =1

is mapped onto the quadric Qg by means of the biholomorphic map zα z∗α = √ , 2 

α = 1, . . . , n,

1 n w = i z0 − ∑ εα z2α 2 α =1

 .

Thus, every codimension one quadric can be realized as a tube hypersurface. Further, the tube hypersurface with base (2) is mapped into Q||·||2 by the locally biholomorphic map  iz0 + zα ∗ , α = 1, . . . , n, zα = exp 2 (5.32) w = eiz0 . Finally, the tube hypersurface with base (3) is mapped into S||·||2 by the locally biholomorphic map 1

Here and in similar situations below where confusion is possible, we use asterisks to indicate the coordinates of the image of a point under a map.

5.3 Classification

95

z∗α = exp

z  α

, α = 1, . . . , n, 2 (5.33) z  0 . w = exp 2 Maps (5.31)–(5.33) can be used to show that the spherical tube hypersurfaces given by Theorem 5.1 are all pairwise affinely non-equivalent (thus the number of affine equivalence classes of closed strongly pseudoconvex spherical tube hypersurface in Cn+1 is equal to n + 3). Let us show, for example, that no hypersurface with base (1) is affinely equivalent to hypersurface with base (2). Let M(1) be a hypersurface with a base of the form (1) for some 0 ≤ m ≤ n, and M(2) the hypersurface with base (2). Let further  f(1) : M(1) → Q(1) := Q||·||2 \

 {zα = 0} ,

α =1

 f (2) : M(2) → Q(2) := Q||·||2 \

m 

n 

 {zα = 0}

α =1

be the respective locally biholomorphic surjective maps given by formulas (5.31) and (5.32). Suppose that there exists an affine transformation ϕ of the form (3.1) that −1 maps M(1) onto M(2) . Then the locally defined map Φ := f(2) ◦ ϕ ◦ f(1) extends to   n+1 preserving an element of Bir Q||·||2 , i.e. to a holomorphic automorphism of CP Q||·||2 (see (1.60)). Denoting this extension by the same symbol Φ , we have

Φ ◦ f(1) = f(2) ◦ ϕ

(5.34)

on Cn+1 . It then follows that Φ maps Q(1) onto Q(2) . Let p∞ be the point of Q||·||2 lying in the infinite hyperplane in CPn+1 . If Φ preserves p∞ , then the restriction of Φ to Q||·||2 is a map of the form z∗ = λ Uz + a, w∗ = λ 2 w + 2iλ Uz, a + i||a||2 + b, where λ > 0, U ∈ Un , a ∈ Cn , b ∈ R, and ·, · denotes the Hermitian scalar prod m uct on Cn . Since Φ maps Q(1) onto Q(2) , it maps T(1) := Q||·||2 {zα = 0} α =1  n onto T(2) := Q||·||2 α =1 {zα = 0} , which implies m = n. Therefore a = 0, and comparing the w-components of the maps in both sides of (5.34) we obtain iλ 2 z0 + b ≡ exp(iϕ0 ), where ϕ0 is an affine function. This is clearly impossible. Next, suppose that Φ does not preserve p∞ . Let p(1) := Φ −1 (p∞ ) and p(2) := Φ (p∞ ). Clearly, p(1) lies in T(1) , p(2) lies in T(2) , and Φ maps T(1) \ {p(1)}

96

5 Strongly Pseudoconvex Spherical Tube Hypersurfaces

onto T(2) \ {p(2)}. It then follows, as before, that m = n. Furthermore, in the homogeneous coordinates Z = (ζ0 : ζ1 : . . . : ζn+1 ) in CPn+1 (see the end of Section 1.2) the map Φ is given by an (n + 2) × (n + 2)-matrix as follows ⎞ ⎛ a0b Z → ⎝ 0 S 0 ⎠ Z, c0d where a, b, c, d ∈ C, b = 0, and S is an n × n-matrix that has a single non-zero entry in each row and each column. Passing to the coordinates z, w in Cn+1 and comparing the w-components of the maps in both sides of (5.34), we obtain idz0 + c ≡ (ibz0 + a) exp(iϕ0 ), where ϕ0 is a non-constant affine function. This is not possible either, thus we have shown that M(1) and M(2) are not affinely equivalent. The above method for proving that M(1) and M(2) are not affinely equivalent is based on considering the locally defined map Φ , which turns out to be extendable to an automorphism of CPn+1 arising from the action of the group of PSU± || · ||2 H (see the part of Section 1.2 that follows formulas (1.48)). This method can be applied to any pair of maps in (5.31)–(5.33) to show that the corresponding hypersurfaces are not affinely equivalent. Analogous methods work for (n − 1, 1)- and (n − 2, 2)-spherical hypersurfaces considered in Chapters 6 and 7. We note that it is possible to give a simpler proof of the pairwise affine non-equivalence of the spherical tube hypersurfaces in certain families by utilizing affine geometry techniques (see Sections 8.2 and 8.3).

Chapter 6

(n − 1, 1)-Spherical Tube Hypersurfaces

Abstract In this chapter we obtain an affine classification of closed (n − 1, 1)spherical tube hypersurfaces in Cn+1 with n ≥ 2. We proceed as outlined in Chapter 4 and investigate defining systems of three types. It is convenient for us to start with systems of type II, then investigate systems of type III, and finalize our classification by considering systems of type I. We begin the chapter with useful linear-algebraic statements.

6.1 Real Canonical Forms of Pair of Matrices Q, X , where Q is Symmetric and X is Q-Symmetric Let Q be a real non-degenerate symmetric matrix and X a real Q-symmetric matrix. In this section we are dealing with the problem of simultaneously reducing Q and X (in the sense specified in the footnote on p. 56) to reasonably simple forms by real transformations. We note that for complex matrices Q and X, where Q is Hermitian (i.e. QT = Q) and X is Q-Hermitian (i.e. X T Q = QX), some canonical forms are known (see, e.g. [8], [96] and references therein). In the real case certain canonical forms for pairs of symmetric matrices were obtained in [28] (see also [46]), and one can derive some forms of Q and X by applying the results of [28] to the symmetric matrices Q and QX. In view of Proposition 4.2, it is sufficient to assume that the Jordan normal form of X consists of either a single cell having a real eigenvalue or two cells of the same size corresponding to two non-real mutually conjugate eigenvalues. In this section we concentrate on the former case and produce canonical forms of Q and X alternative to those implied by the results of [28] (for the case of matrices with non-real eigenvalues see Remark 6.1). The canonical forms described in the proposition below are particularly suited to dealing with defining systems (cf. (i) of Proposition 4.3). Proposition 6.1. Let Q be a real non-degenerate symmetric matrix of size k × k for which the number of negative eigenvalues does not exceed the number of positive A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 6, 

97

6 (n − 1, 1)-Spherical Tube Hypersurfaces

98

ones and X a real Q-symmetric matrix whose Jordan normal form consists of a single cell with eigenvalue λ ∈ R. Then there exists a linear transformation of Rk that takes Q and X into ⎛ ⎜ ⎜ ⎜ ⎜ k Q1 := ⎜ ⎜ ⎜ ⎜ ⎝

⎞

H −H

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 ..

. H −H

0

..

.

and ⎛

λ Rk, τ

τ + λ −τ ⎜ τ −τ + λ ⎜ ⎜ 0 1 ⎜ ⎜ 0 1 ⎜ ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎜ := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 1 λ 0 0 0

0 0 −1 0 0 0 λ 1 1 λ 1 0

⎞

0 0 0 −1 0 λ

0 ..

. λ 1 1 0 0

0

1 λ 0 0 0

−1 0 0 0 λ 1 1 λ 1 0

⎜ ⎜ ⎜ ⎜ ⎜ k Q2 := ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

and

0 0 −1 0 λ

..

respectively, if k is even (here τ = ±1), and into ⎛

⎞

1 H

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0

−H

..

. H

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−H

..

.

.

6.1 Real Canonical Forms of Pair of Matrices

⎛

R

k,λ

λ ⎜1 ⎜ ⎜1 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜ := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 λ 0 0 0

−1 0 0 0 λ 1 1 λ 1 0

99

⎞

0 0 −1 0 λ

..

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 . λ 1 1 0 0

1 λ 0 0 0

−1 0 0 0 λ 1 1 λ 1 0

0 0 −1 0 λ

..

.

respectively, if k is odd, where H is the matrix defined in (4.12). Proof. Without loss of generality we assume λ = 0. Since all considerations for the cases of even k and odd k are very similar, we will only give a proof for even k. Let v1 , . . . , vk be a Jordan basis for the matrix X, where we have Xv = v−1 ,

v0 := 0,

 = 1, . . . , k.

We denote by [·, ·] the bilinear form on Rk corresponding to Q. Since X is Q-symmetric, the matrix of this form in the basis v1 , . . . , vk is ⎛ ⎞ 0 0 . . . 0 a1 ⎜ 0 0 . . . a1 a 2 ⎟ ⎜ ⎟ ⎜ .. .. . .. ⎟ , ⎜ . . . . . .. . ⎟ ⎜ ⎟ ⎝ 0 a1 . . . ak−2 ak−1 ⎠ a1 a2 . . . ak−1 ak where a := [v , vk ],  = 1, . . . , k, with a1 = 0. k Further, it is straightforward to observe that Rk,0 τ is Q1 -symmetric for τ = ±1. k,0 Let u1 , . . . , uk be a Jordan basis for the matrix Rτ and let ·, · denote the bilinear form on Rk corresponding to Qk1 . The matrix of this form in the basis u1 , . . . , uk is ⎞ ⎛ 0 0 . . . 0 b1 ⎜ 0 0 . . . b1 b 2 ⎟ ⎟ ⎜ ⎜ .. .. . .. ⎟ , ⎜ . . . . . .. . ⎟ ⎟ ⎜ ⎝ 0 b1 . . . bk−2 bk−1 ⎠ b1 b2 . . . bk−1 bk where b := u , uk ,  = 1, . . . , k, b1 = 0. Hence, to prove the proposition for even k we need to choose τ and u1 , . . . , uk such that b = a for all .

6 (n − 1, 1)-Spherical Tube Hypersurfaces

100

It is straightforward to see that the general form of the vectors u = (u, 1 , . . . , u, k ) is as follows: ⎞ ⎛ k−2   u = ⎝0, . . . , 0, c1 , c1 , . . . , c , c ⎠ , ⎛

⎞

2−2

  u k + = ⎝ ∗ , . . . , ∗ , (−1)−1 τ c1 + c2 , c2 , c2+1 , c2+1 , . . . , c k + , c k + ⎠ , 2

2

(6.1)

2

 = 1, . . . , k/2, where cm , m = 1, . . . , k, with c1 = 0, are real parameters and the starred positions are occupied by linear functions in c1 , . . . , c2−1 . We will now show that u k +, 2−2l−1 − u k +, 2−2l = (−1)−l−1τ cl+1 + . . . , 2

2

 = 1, . . . , k/2,

l = 0, . . . ,  − 1,

(6.2)

where the dots indicate terms depending only on c1 , . . . , cl . We prove (6.2) by induction on . Observe that (6.2) for l = 0 becomes u k +, 2−1 − u k +, 2 = (−1)−1 τ c1 2 2 as shown in the second line of (6.1). For  = 1 we have l = 0, and therefore (6.2) holds for  = 1. Let  ≥ 2 and first assume l ≤  − 3. It then follows from the form of Rk,0 τ that u k +, 2−2l−4 + u k +, 2−2l−1 − u k +, 2−2l = u k +(−1), 2(−1)−2l , 2

2

2

2

u k +, 2−2l−4 = u k +(−1), 2(−1)−2l−1 , 2

2

and (6.2) is implied by the induction hypothesis. Next, the form of Rk,0 τ gives

τ u k +, 1 − u k +, 2 + u k +, 3 − u k +, 4 = u k +(−1), 2 , 2 2 2 2 2 (6.3)

τ u k +, 1 − u k +, 2 = u k +(−1), 1 , 2

2

2

which yields (6.2) for l =  − 2. Finally, it is not hard to observe that u k +(−1), 1 = c + . . ., where the dots indicate terms depending only on c1 , . . . , c−1 , 2 and (6.2) for l =  − 1 follows from the second identity in (6.3). This completes the proof of (6.2). Setting  = k/2 in (6.2), we obtain k

uk, k−2l−1 − uk, k−2l = (−1) 2 −l−1 τ cl+1 + . . . ,

l = 0, . . . , k/2 − 1.

Let us now compute b = u , uk ,  = 1, . . . , k. For  = 1, . . . , k/2 we get

(6.4)

6.1 Real Canonical Forms of Pair of Matrices 

k

b = (−1) 2 −

∑ (−1)α −1cα

α =1

101

  uk, k−2(−α )−1 − uk, k−2(−α ) .

Combined with (6.4), these identities yield b1 = τ c21 ,

b = 2τ c1 c + . . . ,

 = 1, . . . , k/2,

where the dots indicate terms depending only on c1 , . . . , c−1 . Hence, we can choose τ , c1 = 0, c2 , . . . , c k such that b = a for  = 1, . . . , k/2. 2

Further, write u k + for  = 1, . . . , k/2 as 2



u k + = uk + , c2+1 , c2+1 , . . . , c k + , c k + , 2

2

2

2

where uk + is the projection of u k + to the space of the first 2 coordinates, and 2

2

denote by ·, · the restriction of the bilinear form ·, · to this space. Then for  = 1, . . . , k/2 we get b k + = u k + , uk  + (−1) 2 2

k 2 −



∑ (−1)α −1c2+α

α =1

  uk, 2+2α −1 − uk, 2+2α .

(6.5)

For  = 1, . . . , k/2 − 1 identities (6.4), (6.5) imply b k + = τ c1 c k + + uk + , uk  + . . ., 2

2

(6.6)

2

where the dots indicate terms depending only on c1 , . . . , c k +−1 (this meaning is 2 reserved for dots for the rest of the proof). For  = 1, . . . , k/2 − 1 we have

uk + , uk  = (−1)−1 u k +, 2−1uk, 2−1 − u k +, 2 uk, 2 + . . . . 2

2

2

(6.7)

Further, u k +, 2−1 − u k +, 2 = (−1)−1τ c1 (see (6.1), (6.2)), and (6.4) immediately 2 2 gives uk, 2 = uk, 2−1 + . . .. Also, it is easy to see that uk, 2−1 = c k + + . . .. Hence, 2 (6.1), (6.6), (6.7) imply b k + = 2τ c1 c k + + . . . , 2

2

 = 1, . . . , k/2 − 1.

(6.8)

Finally, for  = k/2 we have k

b k + = bk = (−1) 2 −1 2

k

 2  2 + .... uk, k−1 − uk, k

(6.9)

Since uk, k−1 = (−1) 2 −1 τ c1 + ck (see (6.1)), identity (6.9) yields that (6.8) holds for  = k/2 as well. It then follows that one can find c k + such that b k + = a k + for 2 2 2  = 1, . . . , k/2. This completes the proof of the proposition.



6 (n − 1, 1)-Spherical Tube Hypersurfaces

102

Remark 6.1. By slightly modifying the method of the proof of Proposition 6.1, it is also possible to produce canonical forms – suitable for working with defining systems – for Q-symmetric matrices whose Jordan normal form consists of two cells of the same size k×k corresponding to two non-real mutually conjugate eigenvalues. These canonical forms for general k are complicated, and below we will only write them out for k = 2, 3 (for k = 1 see (ii) of Proposition 4.3). We stress that only the cases k = 1, 2, 3 are required in this chapter and in Chapter 7 for classifying closed (n − 1, 1)- and (n − 2, 2)-spherical tube hypersurfaces. Let Q be a real non-degenerate symmetric matrix of size 4 × 4 with two positive and two negative eigenvalues and X a real Q-symmetric matrix whose Jordan normal form consists of two cells of size 2 × 2 corresponding to eigenvalues η ± iδ with η , δ ∈ R, δ = 0. Then there exists a linear transformation of R4 that takes Q and X into    H 0 (6.10) 0 H and



η I4 +

aI2 + S(δ − b) −bI2 + S(−a) −bI2 + S(−a) −aI2 + S(δ + b)

 ,

respectively, for some a, b ∈ R with a2 + b2 > 0, where S( · ) is defined in (4.14). Further, let Q be a real non-degenerate symmetric matrix of size 6 × 6 with three positive and three negative eigenvalues and X a real Q-symmetric matrix whose Jordan normal form consists of two cells of size 3 × 3 corresponding to eigenvalues η ± iδ with η , δ ∈ R, δ = 0. Then there exists a linear transformation of R6 that takes Q and X into ⎛  ⎞ H 0 0 ⎝ 0 H 0 ⎠

0 0 and

⎛

H

aI2 + S(δ − b) cI2 + S(−d) −bI2 + S(−a)

⎞

⎟ ⎜ ⎟ ⎜ S(δ ) −dI2 + S(−c) ⎟ , η I6 + ⎜ cI2 + S(−d) ⎠ ⎝ −bI2 + S(−a) −dI2 + S(−c) −aI2 + S(δ + b) respectively, for some a, b, c, d ∈ R with c2 + d 2 > 0. Remark 6.2. Before proceeding with our classification results, we make a general remark that applies to the remainder of this chapter and to all of Chapter 7. Sometimes we write equations that define disconnected hypersurfaces (see, e.g. (5.15)). On such occasions it is meant that we in fact consider any connected component of the hypersurface (cf. (2) in Theorem 5.1).

6.3 Defining Systems of Type III

103

6.2 Defining Systems of Type II We proceed as in Section 4.3. Clearly, for (n − 1, 1)-spherical tube hypersurfaces defining systems of type II arise only for n = 2. To use formulas (4.29), (4.30), we need to find the function G. According to formula (4.28), to determine G(w1 ) one has to calculate exp(Aw1 ), where   0 1 A= i/2 p for some p ∈ C. Let λ1 , λ2 be the eigenvalues of A. First, suppose λ1 = λ2 . In this case A is diagonalizable, and formula (4.28) together with the initial conditions G(0) = 1, G (0) = 0 implies λ2 eλ1 w1 − λ1eλ2 w1 . G= λ2 − λ1 Now (4.29), (4.30) yield that the graph x0 = F(x) extends to a closed hypersurface in R3 which is affinely equivalent to the hypersurface sin x0 = ex1 sin x2 . Now, suppose λ1 = λ2 =: λ . Since A is not a scalar matrix, it follows that the Jordan normal form of A consists of a 2 × 2-cell. Therefore, formula (4.28) together with the initial conditions G(0) = 1, G (0) = 0 implies G = (1 − λ w1 )eλ w1 . Now (4.29), (4.30) yield that the graph x0 = F(x) extends to a closed hypersurface in R3 which is affinely equivalent to the hypersurface x0 cosx1 + x2 sin x1 = 0.

6.3 Defining Systems of Type III Our line of argument is similar (but not entirely identical) to that presented in Section 4.4. For a defining system of type III one of the following holds: (i) N + K = 1, (ii) N = K = 0. Accordingly, two cases will be considered. Case (i). Let N + K = 1. We then have L = 0 and ⎞ ⎛ 0 T (τ ) ⎟ ⎜ 0 ⎟ ⎜ D=⎜ ⎟, .. ⎝ . ⎠

0

0

6 (n − 1, 1)-Spherical Tube Hypersurfaces

104

where T (τ ) is the matrix defined in (4.13) and τ is either 1 or −1. Since the matrices Λα commute and are symmetric (see (4.31)), they can be simultaneously diagonalized by a real orthogonal transformation. Using Proposition 4.1 and the relations of Lemma 3.2, we then see that there exists C ∈ GL(n, R) such that the function ˆ = D for which the ˆ F(x) := F(Cx) satisfies a defining system of type III with D ˆ corresponding matrices Cα have the form ⎛ ⎞ a b t3 . . . tn ⎜ −b a + 2b t3 . . . tn ⎟ ⎜ ⎟ ⎜ 0⎟ ˆ C1 = ⎜ t3 −t3 s3 ⎟, ⎜ .. ⎟ .. .. ⎝ . . ⎠ . tn −tn 0 sn ⎞ ⎛ b −a − 2b −t3 . . . −tn ⎜ a + 2b −2a − 3b −t3 . . . −tn ⎟ ⎟ ⎜ ⎜ t3 −s3 0 ⎟ ˆ C2 = ⎜ −t3 ⎟, ⎟ ⎜ .. .. .. ⎠ ⎝ . . . (6.11) −tn tn 0 −sn ⎛ ⎞ tα −tα 0 . . . 0 sα 0 . . . 0 ⎜ tα −tα 0 . . . 0 sα 0 . . . 0 ⎟ ⎜ ⎟ ⎜0 0 0 ⎟ ⎜ ⎟ ⎜ . . ⎟ .. ⎜ .. .. . 0 ⎟ ⎜ ⎟ ⎟ , α = 3, . . . , n, ˆα =⎜ 0 0 C 0 ⎜ ⎟ ⎜ s −s ⎟ λα α ⎜ α ⎟ ⎜0 0 ⎟ 0 ⎜ ⎟ ⎜ . . .. ⎟ ⎝ .. .. . ⎠ 0 0 0 0 where the parameters satisfy the conditions s2α − (a + b)sα − λα tα + τ = 0,

α = 3, . . . , n

(6.12)

ˆ α the number λα occurs at position α on the diagonal. Dropping and in the matrix C hats we get the following defining system: n

F11 = 2τ F1 (F1 + F2 ) + aF1 − bF2 + ∑ tγ Fγ + 1, γ =3

n

F12 = −τ (F12 − F22 ) + bF1 + (a + 2b)F2 − ∑ tγ Fγ , γ =3

n

F22 = −2τ F2 (F1 + F2 ) − (a + 2b)F1 − (2a + 3b)F2 + ∑ tγ Fγ − 1, F1α = −F2α = τ Fα (F1 + F2) + tα (F1 + F2 ) + sα Fα , Fαα = sα (F1 + F2 ) + λα Fα + 1,

γ =3

α = 3, . . . , n, α = 3, . . . , n,

6.3 Defining Systems of Type III

105

α , β = 3, . . . , n,

Fαβ = 0,

α = 1, . . . , n.

Fα (0) = 0,

F(0) = 0,

α = β ,

Without loss of generality we assume λα = 0 for α = 3, . . . , N and λα = 0 for α = N + 1, . . . , n, where 2 ≤ N ≤ n. As in (4.33), we let u := x1 + x2, (6.13) v := x1 − x2. Expressing F in the variables u, v, x3 , . . . , xn , we obtain the function (cf. (4.34)) Fˆ := F ((u + v)/2, (u − v)/2, x3 , . . . , xn ) .

(6.14)

Further, by Proposition 4.6 the function ˆ ∂u Ψ := 2 ∂ F/

(6.15)

depends only on v, and (4.35) yields that on some interval containing the origin Ψ satisfies the equation

Ψ  = τΨ 2 + (a + b)Ψ + 1,

Ψ (0) = 0.

(6.16)

Let P(v) be the function defined by the conditions P = τΨ , P(0) = 0. According to (4.37), the function P satisfies P = (P )2 + (a + b)P + τ ,

P(0) = 0,

P (0) = 0.

Then the function Q defined in (4.39) satisfies the linear equation (cf. (4.40)) Q = (a + b)Q − τ Q, and we have (cf. (4.44))

Q(0) = 1,

Q (0) = 0,

Ψ = −τ Q /Q.

(6.17) (6.18)

The defining system yields (cf. (4.45))

∂ 2 Fˆ ∂ Fˆ = λα + sα Ψ + 1, α = 3, . . . , n, 2 ∂ xα ∂ xα ∂ 2 Fˆ = 0, ∂ xα ∂ xβ

α , β = 3, . . . , n,

(6.19)

α = β ,

and therefore we obtain (cf. (4.49))

∂ Fˆ = (sα Ψ + 1)xα + Φα (v), α = 3, . . . , N , ∂ xα sα Ψ + 1 eλα xα ∂ Fˆ =− + eλα xα Φα (v) + , α = N + 1, . . . , n ∂ xα λα λα

(6.20)

6 (n − 1, 1)-Spherical Tube Hypersurfaces

106

for some functions Φα with Φα (0) = 0, α = 3, . . . , n. Next, the defining system implies (cf. (4.50))

∂ 2 Fˆ ∂ Fˆ = (τΨ + sα ) + tα Ψ , ∂ v∂ xα ∂ xα

α = 3, . . . , n.

(6.21)

Plugging identities (6.20) into both sides of (6.21) and using (6.12), (6.16), we obtain the equations (cf. (4.53)) (Φα ) = (τΨ + sα )Φα + tαΨ , α = 3, . . . , N ,   1 , α = N + 1, . . ., n, (Φα ) = (τΨ + sα ) Φα + λα Φα (0) = 0, α = 3, . . . , n.

(6.22)

Further, (6.20) yields (cf. (4.57), (4.58)) N

N

Ψ sα Ψ + 1 2 Fˆ = u + R(v) + ∑ xα + ∑ Φα xα − 2 2 α =3 α =3

n n sα Ψ + 1 eλα xα 1 λ α xα e xα + ∑ Φα + ∑ − 1 ∑ 2 λα α =N +1 α =N +1 λα α =N +1 λα n

(6.23)

for some function R with R(0) = 0. Plugging (6.23) into the equations for the function Fˆ that arise from the defining system and using (6.12), (6.16), (6.22), we obtain (cf. (4.59)) R = (2τΨ + a + b)R +

N

∑ tα Φα −

α =3



R(0) = 0,

n tα sα a + 3b +∑ 2 α =N +1 λα



n

tα , α =N +1 λα

Ψ−∑

(6.24)

n

sα . 2 λ α =N +1 α

R (0) = − ∑

Let μ1 , μ2 be the roots of the characteristic polynomial μ 2 − (a + b)μ + τ of linear differential equation (6.17). It follows from (6.12) that for α = 3, . . . , N one of the roots coincides with sα . We will now consider three cases. Case (i.a). Suppose that μ1 , μ2 are real and μ1 = μ2 . Without loss of generality we assume sα = μ1 if α = 3, . . . , K and sα = μ2 if α = K + 1, . . . , N for some 2≤K ≤N . From (6.17), (6.18) we obtain

Ψ=

eμ1 v − e μ2 v . μ1 eμ2 v − μ2eμ1 v

Then from (6.22) using relations (6.12), we get

(6.25)

6.3 Defining Systems of Type III

Φα =

107

tα ((1 + (μ2 − μ1 )v)eμ1 v − eμ2 v ) , α = 3, . . . , K , (μ2 − μ1 ) ( μ1 eμ2 v − μ2 eμ1 v )

tα ((1 + (μ1 − μ2 )v)eμ2 v − eμ1 v ) , α = K + 1, . . . , N , (μ2 − μ1 ) ( μ1 eμ2 v − μ2 eμ1 v )   1 (μ1 − μ2 )esα v Φα = − 1 , α = N + 1, . . ., n. λα μ1 eμ2 v − μ2 eμ1 v

Φα =

(6.26)

Finally, (6.24) implies K

R=

∑ tα2

α =3

2(μ1 − μ2 )(μ1 eμ2 v − μ2 eμ1 v ) N

∑

v2 eμ1 v + (6.27)

tα2

α =K +1 v2 eμ2 v + 2(μ1 − μ2 )(μ1 eμ2 v − μ2 eμ1 v )

L1 (v)eμ1 v + L2 (v)eμ2 v , μ1 eμ2 v − μ2 eμ1 v

where L1 , L2 are affine functions. Representation (6.23) and formulas (6.25), (6.26), (6.27) now yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 = ex1 x2 +

K

N

α =3

α =K +1

∑ x2α + ex1 ∑

n

x2α + ∑

α =N +1

exα .

(6.28)

Case (i.b). Suppose μ1 = μ2 = (a + b)/2. Then τ = 1 and sα = (a + b)/2 for α = 3, . . . , N . From (6.17), (6.18) we obtain

Ψ=

2v . 2 − (a + b)v

(6.29)

Then from (6.22) we get tα v 2 , α = 3, . . . , N , 2 − (a + b)v   1 2e(sα −(a+b)/2)v − 1 , α = N + 1, . . ., n. Φα = λα 2 − (a + b)v

Φα =

(6.30)

Further, (6.24) implies N

R=

∑ tα2

α =3

12(2 − (a + b)v)

v4 +

P(v) , 2 − (a + b)v

(6.31)

6 (n − 1, 1)-Spherical Tube Hypersurfaces

108

where P is a polynomial of degree at most 3. Representation (6.23) and formulas (6.29), (6.30), (6.31) now yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 = x21 − x22 +

N

∑

α =3

n

x2α + ∑

exα

(6.32)

α =N +1

if tα = 0 for α = 3, . . . , N and degP < 3, to the hypersurface x0 = x1 x2 + x32 +

N

∑

α =3

n

x2α + ∑

exα

(6.33)

α =N +1

if tα = 0 for α = 3, . . . , N and degP = 3, and to the hypersurface x0 = x1 x2 +

if N ≥ 3 and

N n x42 + x22 x3 + x23 + ∑ x2α + ∑ exα 12 α =4 α =N +1

(6.34)

N

∑ tα2 = 0.

α =3

Case (i.c). Suppose that μ1 , μ2 are not real, hence μ1 = (a + b)/2 + iδ , μ2 = (a + b)/2 − iδ , for some δ ∈ R∗ . We then have τ = 1, δ 2 = 1 − (a + b)2 /4, and N = 2 (i.e. λα = 0 for α = 3, . . . , n). From (6.17), (6.18) we obtain

Ψ=

sin δ v . δ cos δ v − a+b 2 sin δ v

(6.35)

Then from (6.22) we get 1 Φα = λα



 δ e(sα −(a+b)/2)v − 1 , α = 3, . . . , n. δ cos δ v − a+b 2 sin δ v

(6.36)

Next, (6.24) implies R=

L1 (v) cos δ v + L2(v) sin δ v , δ cos δ v − a+b 2 sin δ v

(6.37)

where L1 , L2 are affine functions. Representation (6.23) and formulas (6.35), (6.36), (6.37) now yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 cos x1 + x2 sin x1 +

n

∑ exα = 0.

α =3

6.3 Defining Systems of Type III

109

Case (ii). Let N = K = 0. In this case L = 1, D = 0, and (iv) of Lemma 3.2 implies that the matrices Cα pairwise commute for α = 1, . . . , n. Further, by (ii) of Lemma 3.2 these matrices are H-symmetric. Since χH = 1, we obtain by Propositions 4.2, 4.3, 6.1 and Remark 4.1 that for every α = 1, . . . , n there exists Cα ∈ GL(n, R) such that either Cα−1 Cα Cα is diagonal or   1 Xα 0 −1 Cα Cα Cα = (6.38) 0 Xα2 , where Xα1 is a diagonal matrix and Xα2 is either a 2 × 2-matrix of the form ηα I2 + S(δα ) having two non-real mutually conjugate eigenvalues ηα ± iδα , or a matrix of size κα × κα of one of the following forms:     0 0 with 2 ≤ κ ≤ n, ξ I + 0 0 with 3 ≤ κ ≤ n, ξα Iκα + α α κα α 0 R3,0 0 R2,0 τ λ where τ = ±1, ξα ∈ R is distinct from every eigenvalue of Xα1 , and the matrices Rk, τ for even k and Rk,λ for odd k are defined in Proposition 6.1. Furthermore, we have   In−2 0 T Cα HCα = (6.39) 0 H .

We will now consider three cases. Case (ii.a). Suppose first that all eigenvalues of every matrix Cα are real and that all these matrices are diagonalizable. In this case one can find C ∈ GL(n, R) such that C−1 Cα C is diagonal for every α and CT HC is equal to the matrix in the right-hand side of (6.39). Using Proposition 4.1 and the relations of Lemma 3.2, we ˆ then see that the function F(x) := F(Cx) satisfies a defining system of the form (cf. Section 5.2) Fˆαα = λα Fˆα + 1,

α = 1, . . . , n − 1,

Fˆnn = λn Fˆn − 1,

α , β = 1, . . . , n,

Fˆαβ = 0, ˆ F(0) = 0,

α = β ,

Fˆα (0) = 0, α = 1, . . . , n,

α . Therefore, the graph x = F(x) extends to a closed hypersurface ˆ αα where λα := C 0 n+1 in R which is affinely equivalent to the hypersurface

x0 =

k

∑

α =1

if λn = 0, and to the hypersurface

n−1

exα + ∑ x2α − exn α =k+1

(6.40)

6 (n − 1, 1)-Spherical Tube Hypersurfaces

110

x0 =

k

n−1

α =1

α =k+1

∑ exα + ∑

x2α − x2n

(6.41)

if λn = 0, where k is the number of indices 1 ≤ α ≤ n − 1 for which λα = 0. Case (ii.b). Suppose that for some α0 the matrix Cα0 has non-real eigenvalues. Then for α = α0 we have   1 Yα 0 −1 Cα0 Cα Cα0 = (6.42) 0 Yα2 , where Yα2 are 2 × 2-matrices. Since the symmetric matrices Xα10 , Yα1 , with α = α0 , pairwise commute, they can be simultaneously diagonalized by a real orthogonal transformation, thus we can assume that Cα0 is such that all these matrices are diagonal. Further, we have Xα20 = ηα0 I2 + S(δα0 ) for some ηα0 , δα0 ∈ R, δα0 = 0, and arguing as in case (ii) in Section 5.1, we observe that Yα2 = sα I2 + S(tα ) for some sα ,tα ∈ R, α = α0 . Using Proposition 4.1 we now see that there exists C ∈ GL(n, R) such that the ˆ = 0 for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ the corresponding matrices Cα have the block-diagonal form ⎛ ⎞ να I2 + S(μα ) 0 ⎜ ⎟ λα3 ⎟ ˆα =⎜ C ⎜ ⎟ . . ⎝ ⎠ . 0 n λα β

for some λα , να , μα ∈ R, α = 1, . . . , n, β = 3, . . . , n. From (i) of Lemma 3.2 we now obtain ⎞ ⎞ ⎛ ⎛ aI2 + S(−b) 0 bI2 + S(a) 0 ⎟ ⎟ ˆ2 =⎜ ˆ1 = ⎜ C ⎠, C ⎠, ⎝ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ˆ Cα = ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0

0

⎞

0 ..

0

λα 0

0

..

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ , α = 3, . . . , n, ⎟ ⎟ ⎟ ⎟ ⎠

0

.

0

. 0

ˆ α the number λα occurs at position α on the where a2 + b2 > 0 and in the matrix C diagonal. Hence, the defining system for Fˆ is as follows: Fˆ11 = −Fˆ22 = aFˆ1 − bFˆ2 + 1, Fˆ12 = bFˆ1 + aFˆ2 ,

6.3 Defining Systems of Type III

111

Fˆαα = λα Fˆα + 1,

α = 3, . . . , n,

Fˆαβ = 0,

α = 3, . . . , n,

ˆ F(0) = 0,

β = 1, . . . , n,

α = β ,

Fˆα (0) = 0, α = 1, . . . , n.

Therefore, the hypersurface x0 = F(x) is affinely equivalent to an open subset of the hypersurface x0 = G(x1 , x2 ) +

k

∑

α =3

n

exα + ∑ x2α

(6.43)

α =k+1

for some 2 ≤ k ≤ n, where G is the solution of the system G11 = −G22 = aG1 − bG2 + 1, G12 = bG1 + aG2,

(6.44)

G(0) = 0, G1 (0) = 0, G2 (0) = 0 near the origin. We deal with system (6.44) in the way we dealt with defining systems of type II in Section 4.3. Namely, we extend G to a function G holomorphic on a neighborhood of the origin in C2 . The function G satisfies

∂ 2G ∂ 2G ∂G ∂G =− 2 =a −b + 1, 2 ∂ z1 ∂ z2 ∂ z1 ∂ z2 ∂ 2G ∂G ∂G =b +a , ∂ z1 ∂ z2 ∂ z1 ∂ z2 G (0) = 0, Next, let

∂G ∂G (0) = 0, (0) = 0. ∂ z1 ∂ z2

w1 := iz1 + z2 , w2 := −iz1 + z2 .

Expressing G in the variables w1 , w2 , we obtain the function   i i 1 1 ˆ G := G − w1 + w2 , w1 + w2 . 2 2 2 2 The function Gˆ satisfies

∂ 2 Gˆ ∂ Gˆ 1 = (b − ia) − , 2 ∂ w1 2 ∂ w1 ∂ 2 Gˆ ∂ Gˆ 1 = (b + ia) − , 2 ∂ w2 2 ∂ w2

(6.45)

6 (n − 1, 1)-Spherical Tube Hypersurfaces

112

∂ 2 Gˆ = 0, ∂ w1 ∂ w2 ∂ Gˆ ∂ Gˆ (0) = 0, (0) = 0. Gˆ(0) = 0, ∂ w1 ∂ w2 Then Gˆ = Gˆ1 + Gˆ2 , where Gˆ1 and Gˆ2 are the solutions of the first and second equations of system (6.45), respectively. Therefore, we have G = Gˆ1 (iz1 + z2 ) + Gˆ2 (−iz1 + z2 ) , hence on a neighborhood of the origin in R2 the following holds: G = Gˆ1 (ix1 + x2 ) + Gˆ2 (−ix1 + x2 ) = 2 Re Gˆ1 (ix1 + x2 ) . Solving the first equation of system (6.45) we obtain 

 1 1 . w1 + 1 − e(b−ia)w1 Gˆ1 = 2(b − ia) b − ia

(6.46)

(6.47)

Formulas (6.43), (6.46), (6.47) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the hypersurface x0 = ex1 sin x2 +

k

n

α =3

α =k+1

∑ exα + ∑

x2α

(6.48)

for some 2 ≤ k ≤ n. Case (ii.c). Suppose that all eigenvalues of every matrix Cα are real and that for some α0 the Jordan normal form of the matrix Cα0 contains either a 2 × 2cell or 3 × 3-cell. Then for α = α0 identities (6.42) hold, where Yα2 are matrices of size κ × κ with κ := κα0 . As in case (ii.b), we can suppose that Cα0 is such that the matrices Yα1 , with α = α0 , are diagonal. Note that all matrices Yα2 are Hα0 symmetric, where Hα0 is defined in (5.17). Since each Yα2 commutes with Xα20 , we obtain ⎛ ⎞ Zα σα −σα ⎜ ⎟ ζα Yα2 = ⎝ σαT πα ⎠,

σαT −ζα πα + 2ζα where Zα is a (κ − 2) × (κ − 2)-symmetric matrix, σα is a column-vector of length κ − 2, and πα , ζα ∈ R for α = α0 . Furthermore, since the matrices Yα2 pairwise commute, it follows that the matrices Zα pairwise commute as well, hence we can assume that Cα0 is such that Zα is diagonal for all α = α0 . Using Proposition 4.1 we now see that there exists C ∈ GL(n, R) such that the ˆ = 0 for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ the corresponding matrices Cα have the form

6.3 Defining Systems of Type III

113

⎛

να

μα

ρα3 . . . ραn

⎞

⎟ ⎜ ⎜ −μ ν + 2μ ρ 3 . . . ρ n ⎟ α α α α ⎜ α⎟ ⎟ ⎜ ⎟ ˆα =⎜ 3 C 3 3 ⎟ ⎜ ρα − ρ λ 0 α α ⎟ ⎜ ⎟ ⎜ . . . .. .. ⎠ ⎝ .. n n n ρα − ρα 0 λα β

β

for some λα , ρα , να , μα ∈ R, α = 1, . . . , n, β = 3, . . . , n. From Lemma 3.2 we now ˆ α have in fact the form (6.11), where the parameters satisfy obtain that the matrices C the conditions s2α − (a + b)sα − λα tα = 0, α = 3, . . . , n. (6.49) Dropping hats we get the following defining system: n

F11 = aF1 − bF2 + ∑ tγ Fγ + 1, γ =3

n

F12 = bF1 + (a + 2b)F2 − ∑ tγ Fγ , γ =3

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 + ∑ tγ Fγ − 1, γ =3

F1α = −F2α = tα (F1 + F2 ) + sα Fα ,

α = 3, . . . , n,

Fαα = sα (F1 + F2 ) + λα Fα + 1,

α = 3, . . . , n,

Fαβ = 0,

α , β = 3, . . . , n,

F(0) = 0,

Fα (0) = 0,

α = β ,

α = 1, . . . , n.

Without loss of generality we assume λα = 0 for α = 3, . . . , N and λα = 0 for α = N + 1, . . . , n, where 2 ≤ N ≤ n. We deal with this defining system in the way we dealt with the defining system arising in case (i). We introduce the variables u, v by formulas (6.13), the function Fˆ by formula (6.14), and the function Ψ by formula (6.15). The defining system yields that Ψ depends only on v and on some neighborhood of the origin satisfies the equation Ψ  = (a + b)Ψ + 1, Ψ (0) = 0. (6.50) Next, the defining system yields identities (6.19), which lead to identities (6.20) for some functions Φα with Φα (0) = 0, α = 3, . . . , n. Further, the defining system implies ∂ 2 Fˆ ∂ Fˆ = sα + tαΨ , α = 3, . . . , n. (6.51) ∂ v∂ xα ∂ xα Plugging identities (6.20) into both sides of (6.51) and using (6.49), (6.50), we obtain the equations

6 (n − 1, 1)-Spherical Tube Hypersurfaces

114

(Φα ) = sα Φα + tαΨ , α = 3, . . . , N ,   1 , α = N + 1, . . . , n, (Φα ) = sα Φα + λα

Φα (0) = 0,

(6.52)

α = 3, . . . , n.

Next, (6.20) implies representation (6.23) for some function R with R(0) = 0. Plugging (6.23) into the equations for the function Fˆ that arise from the defining system and using (6.49), (6.50), (6.52), we obtain   N n n a + 3b tα sα tα   +∑ Ψ−∑ , R = (a + b)R + ∑ tα Φα − 2 α =N +1 λα λ α =3 α =N +1 α (6.53) n s α R(0) = 0, R (0) = − ∑ . 2 α =N +1 λα We will now consider two cases. Case (ii.c.1). Assume a + b = 0. It then follows from (6.49) that sα = 0 for α = 3, . . . , N . From (6.50) we obtain Ψ = v. (6.54) Then from (6.52) we get

Φα =

tα 2 v , 2

α = 3, . . . , N ,

es α v − 1 Φα = , α = N + 1, . . . , n. λα

(6.55)

Further, (6.53) implies N

R=

∑ tα2

α =3

v4 + P(v), (6.56) 24 where P is a polynomial of degree at most 3. Representation (6.23) and formulas (6.54), (6.55), (6.56) now yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to hypersurface (6.32) if tα = 0 for α = 3, . . . , N and deg P < 3, to hypersurface (6.33) if tα = 0 for α = 3, . . . , N and deg P = 3, and to hypersurface (6.34) if N ≥ 3 and

N

∑ tα2 = 0.

α =3

Case (ii.c.2). Assume a + b = 0. From (6.49) one has sα (sα − (a + b)) = 0 for α = 3, . . . , N , thus without loss of generality we assume sα = 0 if α = 3, . . . , K and sα = a + b if α = K + 1, . . . , N for some 2 ≤ K ≤ N . Equation (6.50) yields

6.4 Defining Systems of Type I

115

Ψ=

1 (a+b)v −1 . e a+b

Then from (6.52) we get

tα (a+b)v e Φα = − (a + b)v − 1 , α = 3, . . . , K , (a + b)2



tα (a+b)v Φα = + 1 , α = K + 1 . . ., N , (a + b)v − 1 e (a + b)2

Φα =

1 sα v (e − 1), λα

(6.57)

(6.58)

α = N + 1, . . ., n.

Further, (6.53) implies N

K

R=

∑ tα2

α =3

2(a + b)2

∑

v2 +

α =K +1

tα2

2(a + b)2

v2 e(a+b)v + L1(v)e(a+b)v + L2 (v),

(6.59)

where L1 , L2 are affine functions. Representation (6.23) and formulas (6.57), (6.58), (6.59) now yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to hypersurface (6.28).

6.4 Defining Systems of Type I As we showed in Section 4.2, if F(x) is the solution of a defining system of type I, the hypersurface x0 = F(x) near the origin can be written in the form (4.25). Thus, as in Section 5.1, we need to determine the functions G1 , G2 . We will now consider two cases corresponding to the signs of the eigenvalues of the matrices H1 and H2 (see (4.19), (4.23)). Case (i). Assume that H1 is positive-definite, that is, H1 = IN . In this case we have 0 ≤ N ≤ n − 1. We will consider two situations. Case (i.a). Let N = 0. Then we have G1 = 1. Since χH˜ 2 = 1 (see (4.23)), the function G2 can be found as described in cases (i)–(iv) of Section 5.1. Formula (4.25) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to either the hypersurface n−1

∑ exα − exn = 1,

(6.60)

α =0

or the hypersurface sin x0 =

n−1

∑ exα − exn ,

α =1

or a hypersurface of one of the forms (6.40), (6.41).

(6.61)

6 (n − 1, 1)-Spherical Tube Hypersurfaces

116

Case (i.b). Let N ≥ 1. Since χH˜ 1 = 1 and χH˜ 2 = 1 (see (4.19), (4.23)), each of the functions G1 , G2 can be found as described in cases (i)–(iv) of Section 5.1. Formula (4.25) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to either the hypersurface sin x0 = ex1 sin x2 +

n

∑ exα ,

α =3

or hypersurface (6.60), or hypersurface (6.61), or a hypersurface of one of the forms (6.28), (6.40), (6.41), (6.48). Case (ii). Assume that H2 is positive-definite. In this case we have 1 ≤ N ≤ n ˜ 2 = In−N+1 . Thus, either G2 = 1 (for N = n) or G2 can be found as described and H at the beginning of Section 5.1 (for N < n). We will consider three situations. ˜ 1 = −I2 and G1 can be found as Case (ii.a). Let N = 1. In this case we have H described at the beginning of Section 5.1. Formula (4.25) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to hypersurface (6.60). Case (ii.b). Let N = 2. In this case we have χH˜ 1 = 1 and the function G1 can be found as described in cases (i)–(iv) of Section 5.1. Formula (4.25) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to either hypersurface (6.60), or hypersurface (6.61), or a hypersurface of one of the forms (6.40), (6.41). Case (ii.c). Let N ≥ 3. In this case we have χH˜ 1 = 2. By Proposition 4.2 and Remark 4.1 we obtain that for every α = 1, . . . , N there exists Cα ∈ GL(N + 1, R) such that either Cα−1 A1α Cα is diagonal or identity (5.5) holds, where Xα1 is a diagonal matrix and Xα2 is either: (1) a 2 × 2-matrix having two non-real mutually conjugate eigenvalues, or (2) a 4 × 4-matrix having only non-real eigenvalues whose Jordan normal form is diagonal, or (3) a 4 × 4-matrix having two non-real mutually conjugate eigenvalues whose Jordan normal form consists of two cells of size 2 × 2, or (4) a matrix having exactly two non-real mutually conjugate eigenvalues, each having multiplicity one, and a real eigenvalue, with the Jordan normal form corresponding to the real eigenvalue consisting of a single cell of size κ × κ for κ = 2, 3, or (5) a matrix having a real eigenvalue whose Jordan normal form consists of a single cell of size κ × κ for κ = 2, 3, 4, 5, or (6) a matrix having only real eigenvalues whose Jordan normal form consists of two cells, with the size of each cell being either 2 × 2 ˜ 1Cα is described as follows: in all the cases or 3 × 3. Furthermore, the matrix CαT H except (5) with κ = 3 we have ⎛ ˜ 1Cα = ⎝ CαT H

IN−3

0

H

0 H

⎞ ⎠,

for case (5) with κ = 3 either identity (6.62) holds or we have

(6.62)

6.5 Classification

117

⎛ ˜ 1Cα CαT H

=⎝

IN−3

0

−H

0 H

⎞ ⎠.

(6.63)

Hence, to determine the function G1 analogously to how it was done in Section 5.1, one needs to consider the following situations: either every matrix A1α is diagonalizable by means of a real transformation or for some α0 the matrix A1α0 can be reduced to the form (5.5) with Xα20 of the kind described in (1)–(6) above. The argument for finding G1 is elementary but involves much harder and longer calculations than those presented in Section 5.1. We omit these calculations but emphasize that together with formula (4.25) they yield that in case (ii.c) the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to one of the previously found hypersurfaces.

6.5 Classification The results of Sections 6.2, 6.3, 6.4 yield the following theorem. Theorem 6.1. [64] Let M be a closed (n − 1, 1)-spherical tube hypersurface in Cn+1 with n ≥ 2. Then M is affinely equivalent to a tube hypersurface with the base given by one of the following equations: (1) x0 =

(2) x0 =

(3) x0 =

m

n−1

α =1

α =m+1

m

n−1

α =1

α =m+1

m

n−2

α =1

α =m+1

∑ exα + ∑ ∑ exα + ∑ ∑ exα + ∑

(4) sin x0 =

(5) sin x0 =

(6)

(7)

x2α − x2n ,

0 ≤ m ≤ n − 1,

x2α − exn ,

0 ≤ m ≤ n − 1,

x2α + exn−1 sin xn ,

0 ≤ m ≤ n − 2,

n−1

∑ exα − exn ,

α =1 n−2

∑ exα + exn−1 sin xn ,

α =1

n−3

∑ exα + xn−2 cos xn−1 + xn sin xn−1 = 0,

α =0 n−1

∑ exα − exn = 1,

α =0

6 (n − 1, 1)-Spherical Tube Hypersurfaces

118

(8) x0 =

(9) x0 = (10) x0 =

m



α =1

α =m+1

m

n−2

α =1

α =m+1

m

n−3

α =1

α =m+1

∑ exα + ∑ ∑ exα + ∑ ∑ exα + ∑

n−2

x2α + exn−1 ∑ x2α + exn−1 xn ,

0 ≤ m ≤  ≤ n − 2, 2 − m ≤ n − 2,

x2α + xn−1xn + x3n ,

0 ≤ m ≤ n − 2,

α =+1

x2α + x2n−2 + xn−1 xn + xn−2 x2n +

x4n , 0 ≤ m ≤ n − 3. 12

For each equation in Theorem 6.1 it is not hard to explicitly find a locally diffeomorphic CR-map from the corresponding tube hypersurface into either the quadric  Qn−1,1 := (z, w) ∈ Cn+1 : Im w =

n−1

∑ |zα |2 − |zn|2

α =1

 ,

(6.64)

or the hypersurface  Sn−1,1 := (z, w) ∈ Cn+1 :

n−1

∑ |zα |2 − |zn|2 + |w|2 = 1

α =1

 ,

(recall that the closures of Sn−1,1 and Qn−1,1 in CPn+1 are equivalent by means of map (1.61)). The tube hypersurface with base (1) for a fixed m is mapped into Qn−1,1 by the locally biholomorphic map z α , z∗α = exp α = 1, . . . , m, 2 zα z∗α = √ , α = m + 1, . . ., n, (6.65) 2   1 n−1 2 z2n w = i z0 − ∑ zα + 2 . 2 α =m+1 The tube hypersurface with base (2) for a fixed m is mapped into Qn−1,1 by the locally biholomorphic map z α α = 1, . . . , m and α = n, , z∗α = exp 2 zα z∗α = √ , α = m + 1, . . ., n − 1, (6.66) 2   1 n−1 2 w = i z0 − ∑ zα . 2 α =m+1 The tube hypersurface with base (3) for a fixed m is mapped into Qn−1,1 by the locally biholomorphic map

6.5 Classification

119

z∗α = exp

z

zα z∗α = √ , 2

α

2

,

α = 1, . . . , m, α = m + 1, . . ., n − 2,

     zn−1 − izn zn−1 + izn 1 1 + exp , exp =√ 2 2 2 2i      zn−1 − izn zn−1 + izn 1 1 − exp , exp z∗n = √ 2 2 2 2i   1 n−2 2 w = i z0 − ∑ zα . 2 α =m+1 z∗n−1

(6.67)

The tube hypersurface with base (4) is mapped into Qn−1,1 by map (5.32). The tube hypersurface with base (5) is mapped into Qn−1,1 by the locally biholomorphic map   iz0 + zα , α = 1, . . . , n − 2, z∗α = exp 2      iz0 + zn−1 − izn iz0 + zn−1 + izn 1 1 ∗ zn−1 = √ + exp , exp 2 2 2 2i (6.68)      1 iz0 + zn−1 − izn iz0 + zn−1 + izn 1 z∗n = √ − exp , exp 2 2 2 2i w = eiz0 . The tube hypersurface with base (6) is mapped into Qn−1,1 by the locally biholomorphic map   zα −1 + izn−1 ∗ , zα = exp α = 1, . . . , n − 2, 2   1 zn−2 + izn z∗n−1 = √ eizn−1 + , 4 2 (6.69)   1 z + iz n n−2 , z∗n = √ eizn−1 − 4 2 w=−

izn−2 + zn izn−1 . e 2

The tube hypersurface with base (7) is mapped into Sn−1,1 by map (5.33). The tube hypersurface with base (8) for fixed m and  is mapped into Qn−1,1 by the locally biholomorphic map

6 (n − 1, 1)-Spherical Tube Hypersurfaces

120

z∗α = exp

z α

2

,

α = 1, . . . , m,

zα α = m + 1, . . ., , z∗α = √ , 2 z zα n−1 , z∗α = √ exp α =  + 1, . . ., n − 2, 2 2   z zn 1 n−2 2 1 n−1 ∗ 1+ + , zn−1 = √ zα exp ∑ 2 4 2 2 α =+1   z zn 1 n−2 2 1 n−1 ∗ , 1− − zn = √ zα exp ∑ 2 4 α =+1 2 2   1  2 w = i z0 − ∑ zα . 2 α =m+1

(6.70)

The tube hypersurface with base (9) for a fixed m is mapped into Qn−1,1 by the locally biholomorphic map z α , z∗α = exp α = 1, . . . , m, 2 zα α = m + 1, . . ., n − 2, z∗α = √ , 2   1 zn−1 3z2n ∗ √ zn−1 = , + zn + 4 8 (6.71) 2   zn−1 1 3z2 + zn − n , z∗n = √ − 4 8 2   1 n−2 2 zn−1 zn z3n w = i z0 − ∑ zα − 2 − 4 . 2 α =m+1 The tube hypersurface with base (10) for a fixed m is mapped into Qn−1,1 by the locally biholomorphic map z α , z∗α = exp α = 1, . . . , m, 2 zα α = m + 1, . . ., n − 3, z∗α = √ , 2 (6.72) 4zn−2 + z2n √ z∗n−2 = , 4 2   1 zn z3n ∗ √ , zn−1 + + zn−2 zn + zn−1 = 4 12 2

6.5 Classification

121

  1 zn z3n ∗ , zn = √ −zn−1 + − zn−2 zn − 4 12 2   1 n−2 2 zn−1 zn zn−2 z2n z4n w = i z0 − ∑ zα − 2 − 4 − 96 . 2 α =m+1 One can apply the method outlined in Section 5.3 to locally biholomorphic maps (5.32), (5.33), (6.65)–(6.72) to show that for n > 2 the spherical tube hypersurfaces given by Theorem 6.1 are all pairwise affinely non-equivalent and that for n = 2 the only affinely equivalent hypersurfaces are those with base (1) for m = 1 and base (2) with m = 0. Thus, the number of affine equivalence classes of closed (n − 1, 1)-spherical tube hypersurface in Cn+1 is equal to 5n + (n2 − 1)/4 if n is odd, to 5n + n2/4 if n is even and n ≥ 4, and to 10 if n = 2 (cf. [42]).

•

Chapter 7

(n − 2, 2)-Spherical Tube Hypersurfaces

Abstract In this chapter we obtain an affine classification of closed (n − 2, 2)spherical tube hypersurfaces in Cn+1 (here n ≥ 4). The classification is presented in the first section of the chapter, with proofs given in further sections. We proceed as outlined in Chapter 4 and consider defining systems of three types. As in Chapter 6, we start with systems of type II, then investigate systems of type III, and finish by considering systems of type I.

7.1 Classification We begin by presenting a large number of examples of (n − 2, 2)-spherical tube hypersurfaces in Cn+1 with n ≥ 4. These examples are split into four classes, which we denote by (A), (B), (C), (D). It will be shown in forthcoming sections that these examples in fact give a complete classification of closed (n − 2, 2)-spherical tube hypersurfaces. (A). Hypersurfaces in this class are derived from the (n − 1, 1)-spherical tube hypersurfaces that are globally represented as graphs in the statement of Theorem 6.1. For N ≥ 2 we denote by ΓN the collection of all functions that occur in the right-hand sides of equations numbered (1), (2), (3), (8), (9), (10) in Theorem 6.1 with n replaced by N. Let n ≥ 4. For every pair of integers N, K such that N, K ≥ 2, N + K = n and every pair of functions ϕ ∈ ΓN , ψ ∈ ΓK , let Hϕ ,ψ be the closed realanalytic tube hypersurface in Cn+1 with the base given by the equation x0 = ϕ (x1 , . . . , xN ) + ψ (xN+1 , . . . , xn ). It is straightforward to see that Hϕ ,ψ is (n − 2, 2)-spherical. Indeed, the hypersurface in CN+1 with the base x0 = ϕ (x1 , . . . , xN ) is (N − 1, 1)-spherical and can be mapped into the quadric QN−1,1 (see (6.64)) by a locally biholomorphic map of the form z∗α = fϕ ,α (z1 , . . . , zN ),

α = 1, . . . , N,

w = iz0 + gϕ (z1 , . . . , zN ), A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 7, 

123

7 (n − 2, 2)-Spherical Tube Hypersurfaces

124

where the functions fϕ ,α and gϕ are holomorphic on CN (see (6.65)–(6.67), (6.70)– (6.72)). Similarly, the hypersurface in CK+1 with the base x0 = ψ (xN+1 , . . . , xn ) is (K − 1, 1)-spherical and can be mapped into the quadric   (zN+1 , . . . , zn , w) ∈ CK+1 : Im w =

n−1

∑

β =N+1

|zβ |2 − |zn |2

by a locally biholomorphic map of the form z∗β = fψ ,β (zN+1 , . . . , zn ),

β = N + 1, . . . , n,

w = iz0 + gψ (zN+1 , . . . , zn ), where the functions fψ ,β and gψ are holomorphic on CK . It is then easy to see that the hypersurface Hϕ ,ψ is mapped into the quadric   Qn−2,2 :=

(z, w) ∈ Cn+1 : Im w =

n−2

∑ |zγ |2 − |zn−1 |2 − |zn|2

γ =1

by the locally biholomorphic map z∗α = fϕ ,α (z1 , . . . , zN ),

α = 1, . . . , N − 1,

z∗N = fψ ,n−1 (zN+1 , . . . , zn ), z∗β = fψ ,β (zN+1 , . . . , zn ),

β = N + 1, . . ., n, β = n − 1,

z∗n−1 = fϕ ,N (z1 , . . . , zN ), w = iz0 + gϕ (z1 , . . . , zN ) + gψ (zN+1 , . . . , zn ). (B). This class consists of tube hypersurfaces with the bases given by the following equations: (1) sin x0 = (2) sin x0 = (3) sin x0 = (4)

n−2

∑ exα − exn−1 − exn ,

α =1 n−3

∑ exα − exn−2 + exn−1 sin xn ,

α =1 n−4

∑ exα + exn−3 sin xn−2 + exn−1 sin xn,

α =1

n−2

∑ exα − exn−1 − exn = 1.

α =0

The tube hypersurface with base (1) is mapped into Qn−2,2 by map (5.32). The tube hypersurface with base (2) is mapped into Qn−2,2 by the locally biholomorphic map

7.1 Classification

125





iz0 + zα α = 1, . . . , n − 3, , 2      1 iz0 + zn−1 − izn iz0 + zn−1 + izn 1 z∗n−2 = √ + exp , exp 2 2 2 2i   iz0 + zn−2 z∗n−1 = exp , 2      1 iz0 + zn−1 − izn iz0 + zn−1 + izn 1 ∗ √ zn = − exp , exp 2 2 2 2i z∗α = exp

w = eiz0 . The tube hypersurface with base (3) is mapped into Qn−2,2 by the locally biholomorphic map   iz0 + zα , α = 1, . . . , n − 4, z∗α = exp 2      iz0 + zn−3 − izn−2 iz0 + zn−3 + izn−2 1 1 ∗ + exp , exp zn−3 = √ 2 2 2 2i      iz0 + zn−1 − izn iz0 + zn−1 + izn 1 1 ∗ zn−2 = √ + exp , exp 2 2 2 2i      iz0 + zn−3 − izn−2 iz0 + zn−3 + izn−2 1 1 − exp , exp z∗n−1 = √ 2 2 2 2i      iz0 + zn−1 − izn iz0 + zn−1 + izn 1 1 ∗ √ zn = − exp , exp 2 2 2 2i w = eiz0 . The tube hypersurface with base (4) is mapped into  Sn−2,2 :=

(z, w) ∈ C

n+1

n−2

:

∑ |zα |

α =1

2



− |zn−1 | − |zn | + |w| = 1 2

2

2

by map (5.33) (recall that the closures of Sn−2,2 and Qn−2,2 in CPn+1 are equivalent by means of map (1.61)). (C). This class consists of tube hypersurfaces with the bases given by the following equations: (1)

n−4

∑ exα − exn−3 + xn−2 cosxn−1 + xn sin xn−1 = 0,

α =0

7 (n − 2, 2)-Spherical Tube Hypersurfaces

126

(2) (3) (4)

n−5

∑ exα + exn−4 sin xn−3 + xn−2 cos xn−1 + xn sin xn−1 = 0,

α =0 m

n−5

α =0

α =m+1

∑ exα + exn−4 ∑

x2α + exn−4 xn−3 + xn−2 cos xn−1 + xn sin xn−1 = 0, −1 ≤ m ≤ n − 5,

n−5

∑ exα + (xn−4 + x2n−3 − x2n−2 ) cos xn−1 + (xn + 2xn−3xn−2 ) sin xn−1 = 0.

α =0

The tube hypersurface with base (1) is mapped into Qn−2,2 by the locally biholomorphic map   zα −1 + izn−1 , α = 1, . . . , n − 3, z∗α = exp 2   1 zn−2 + izn , z∗n−2 = √ eizn−1 + 4 2   zn−3 + izn−1 , z∗n−1 = exp 2   1 zn−2 + izn izn−1 ∗ zn = √ e , − 4 2 w=−

izn−2 + zn izn−1 . e 2

The tube hypersurface with base (2) is mapped into Qn−2,2 by the locally biholomorphic map   zα −1 + izn−1 ∗ , α = 1, . . . , n − 4, zα = exp 2      zn−4 − izn−3 + izn−1 zn−4 + izn−3 + izn−1 1 1 ∗ √ + exp , exp zn−3 = 2 2 2 2i   1 zn−2 + izn z∗n−2 = √ eizn−1 + , 4 2      1 zn−4 − izn−3 + izn−1 zn−4 + izn−3 + izn−1 1 ∗ − exp , exp zn−1 = √ 2 2 2 2i   1 zn−2 + izn z∗n = √ eizn−1 − , 4 2 w=−

izn−2 + zn izn−1 . e 2

7.1 Classification

127

The tube hypersurface with base (3) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map   zα −1 + izn−1 ∗ , zα = exp α = 1, . . . , m + 1, 2   zn−4 + izn−1 zα −1 ∗ , zα = √ exp α = m + 2, . . ., n − 4, 2 2     zn−4 + izn−1 zn−3 1 n−5 2 1 ∗ , + 1+ zn−3 = √ ∑ zα exp 2 4 α =m+1 2 2   1 zn−2 + izn izn−1 , =√ e + 4 2     zn−4 + izn−1 zn−3 1 n−5 2 1 ∗ , − 1− zn−1 = √ ∑ zα exp 2 4 α =m+1 2 2 z∗n−2

z∗n

  1 zn−2 + izn izn−1 , =√ e − 4 2

w=−

izn−2 + zn izn−1 e . 2

The tube hypersurface with base (4) is mapped into Qn−2,2 by the locally biholomorphic map   zα −1 + izn−1 , α = 1, . . . , n − 4, z∗α = exp 2   zn−3 − izn−2 izn−1 1 e z∗n−3 = √ + zn−3 + izn−2 , 4 2   1 zn−4 + izn z2n−3 + 2izn−3zn−2 − z2n−2 izn−1 ∗ , + e + zn−2 = √ 4 8 2   zn−3 − izn−2 izn−1 1 e z∗n−1 = √ − zn−3 − izn−2 , 4 2   1 zn−4 + izn z2n−3 + 2izn−3 zn−2 − z2n−2 izn−1 ∗ , − − e zn = √ 4 8 2   izn−4 + zn iz2n−3 + 2zn−3 zn−2 − iz2n−2 izn−1 w= − e − . 2 4

7 (n − 2, 2)-Spherical Tube Hypersurfaces

128

(D). This class consists of tube hypersurfaces with the bases given by the following equations: (1) x0 =

(2) x0 =

(3) x0 =

(4) x0 =

m

∑

α =1

exα +

n−4

∑

α =m+1

m

n−4

α =1

α =m+1

m

n−3

α =1

α =m+1

m

n−5

α =1

α =m+1

∑ exα + ∑ ∑ exα + ∑ ∑ exα + ∑

x2α + xn−3 xn−2 + xn−1xn + x2n−3xn ,

0 ≤ m ≤ n − 4,

x2α + xn−3 xn−2 + xn−1xn + x2n−3xn + xn−3x2n , 0 ≤ m ≤ n − 4, x2α − x2n−2 − xn−1xn − xn−2x2n −

x4n , 12

0 ≤ m ≤ n − 3,

x2α + x2n−4 + xn−3xn−2 + xn−1 xn + xn−4 x2n + x2n−3 xn +

x4n , 12

0 ≤ m ≤ n − 5, (5) x0 =

m

n−7

α =1

α =m+1

∑ exα + ∑

x2α + x2n−6 + x2n−5 + x2n−4 + xn−3xn−2 + xn−1xn +

 √ 1+t 2 2(1 + t)xn−6 xn−3 xn−1 + √ xn−5 x2n−3 + 2 3t xn−5 x2n−1 + 3t



 −t 2 + 34t − 1 xn−4 x2n−3 + x2n−3 + x2n−1 x2n−3 + tx2n−1 , 3t

√ 0 ≤ m ≤ n − 7, 1 ≤ t ≤ 17 + 12 2,

(6) x0 =

m

∑

α =1

exα +

n−4

∑

α =m+1

x2α +xn−3 xn−2 +xn−1 xn +

3 2 x xn +10xn−2 x2n +xn−3 x3n + x5n , 20 n−3 0 ≤ m ≤ n − 4,

(7) x0 =

m

n−5

α =1

α =m+1

∑ exα + ∑ xn−2 x2n 2

+

x2α + x2n−4 + xn−3xn−2 + xn−1 xn + 2xn−4 xn−3 xn +

xn−4 x3n x2n−3 x2n xn−3 x4n x6 + + + n , 3 2 12 360

x3n−3 + 3

0 ≤ m ≤ n − 5.

The tube hypersurface with base (1) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α , z∗α = exp α = 1, . . . , m, 2 zα z∗α = √ , 2

α = m + 1, . . ., n − 4,

7.1 Classification

129

1 z∗n−3 = √ (zn−3 + zn−2 + zn−3 zn ) , 2 2   2 z 1 n−3 , z∗n−2 = √ zn−1 + zn + 2 2 2 1 z∗n−1 = √ (zn−3 − zn−2 − zn−3 zn ) , 2 2   z2n−3 1 ∗ , zn = √ zn−1 − zn + 2 2 2 

1 n−4 2 zn−3 zn−2 zn−1 zn z2n−3 zn w = i z0 − ∑ zα − 2 − 2 − 4 2 α =m+1

 .

The tube hypersurface with base (2) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α z∗α = exp α = 1, . . . , m, , 2 zα z∗α = √ , α = m + 1, . . ., n − 4, 2   1 z2 z∗n−3 = √ zn−3 + zn−2 + zn−3 zn + n , 2 2 2   z2n−3 1 ∗ + zn−3 zn , zn−1 + zn + zn−2 = √ 2 2 2   1 z2 z∗n−1 = √ zn−3 − zn−2 − zn−3 zn − n , 2 2 2   z2 1 zn−1 − zn + n−3 + zn−3zn , z∗n = √ 2 2 2   1 n−4 2 zn−3 zn−2 zn−1 zn z2n−3 zn zn−3 z2n w = i z0 − ∑ zα − 2 − 2 − 4 − 4 . 2 α =m+1 The tube hypersurface with base (3) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α , z∗α = exp α = 1, . . . , m, 2 zα α = m + 1, . . ., n − 3, z∗α = √ , 2

7 (n − 2, 2)-Spherical Tube Hypersurfaces

130

  1 zn z3n ∗ , zn−2 = √ −zn−1 + − zn−2 zn − 4 12 2   1 zn z3 z∗n−1 = √ zn−1 + + zn−2zn + n , 4 12 2 4zn−2 + z2n √ , 4 2   1 n−3 2 z2n−2 zn−1 zn zn−2 z2n z4n w = i z0 − ∑ zα + 2 + 2 + 4 + 96 . 2 α =m+1 z∗n =

The tube hypersurface with base (4) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α , z∗α = exp α = 1, . . . , m, 2 zα α = m + 1, . . ., n − 5, z∗α = √ , 2   1 z2n ∗ , zn−4 = √ zn−4 + 4 2 1 z∗n−3 = √ (zn−3 + zn−2 + zn−3 zn ) , 2 2   2 3 z z 1 z∗n−2 = √ zn−1 + zn + zn−4 zn + n−3 + n , 2 12 2 2 1 z∗n−1 = √ (zn−3 − zn−2 − zn−3 zn ) , 2 2   z2n−3 z3n 1 ∗ , + zn−1 − zn + zn−4 zn + zn = √ 2 12 2 2 

1 n−4 2 zn−3 zn−2 zn−1 zn zn−4 z2n z2n−3 zn z4n w = i z0 − ∑ zα − 2 − 2 − 4 − 4 − 96 2 α =m+1

 .

The tube hypersurface with base (5) for fixed m and t is mapped into Qn−2,2 by the locally biholomorphic map

z  α , z∗α = exp α = 1, . . . , m, 2 1 z∗α = √ zα , 2

α = m + 1, . . ., n − 7,

7.1 Classification

131

  1 1 ∗ zn−6 = √ zn−6 + 2(1 + t)zn−3 zn−1 , 2 2   √ 1 1+t 2 3t 2 ∗ , zn−5 + √ zn−3 + z zn−5 = √ 2 n−1 4 3t 2 z∗n−4 z∗n−3

1 =√ 2



1 zn−4 + 4



 −t 2 + 34t − 1 2 zn−3 , 3t

  i 1+t =− 2zn−3 + zn−2 + 2(1 + t)zn−6 zn−1 + √ zn−5 zn−3 + 4 3t  2 1 + t −t + 34t − 1 3 2 zn−4 zn−3 + zn−3 + zn−3 zn−1 , 3t 2

 √ i

2zn−1 + zn + 2(1 + t)zn−6 zn−3 + 2 3t zn−5 zn−1 + 4  1+t 2 3 z zn−1 + tzn−1 , 2 n−3   i 1+t z∗n−1 = − −2zn−3 + zn−2 + 2(1 + t)zn−6 zn−1 + √ zn−5 zn−3 + 4 3t  1+t −t 2 + 34t − 1 3 2 zn−4 zn−3 + zn−3 + zn−3 zn−1 , 3t 2

z∗n−2 = −

(7.1)

 √ i

−2zn−1 + zn + 2(1 + t)zn−6 zn−3 + 2 3t zn−5 zn−1 + 4  1+t 2 3 z zn−1 + tzn−1 , 2 n−3   2(1 + t) 1 n−4 2 zn−3 zn−2 zn−1 zn w = i z0 − zα − − − zn−6 zn−3 zn−1 − ∑ 2 α =m+1 2 2 2 √ 1+t 1 −t 2 + 34t − 1 3t 2 2 √ zn−5 zn−3 − zn−5 zn−1 − zn−4 z2n−3 − 2 4 3t 4 3t  

 1 2 2 2 2 +z zn−3 + tzn−1 . z 8 n−3 n−1

z∗n = −

The tube hypersurface with base (6) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α z∗α = exp , α = 1, . . . , m, 2

7 (n − 2, 2)-Spherical Tube Hypersurfaces

132

1 z∗α = √ zα , 2

α = m + 1, . . ., n − 4,

  1 3 z3n 2 = √ zn−3 + zn−2 + zn−3 zn + 5zn + , 20 4 2 2   1 3 2 3 5 4 ∗ 2 zn−2 = √ zn−1 + zn + zn−3 + 10zn−2zn + zn−3 zn + zn , 40 4 8 2 2   1 3 z3n ∗ 2 , zn−1 = √ zn−3 − zn−2 − zn−3 zn + 5zn − 20 4 2 2   1 3 2 3 5 4 ∗ 2 zn = √ zn−1 − zn + zn−3 + 10zn−2zn + zn−3 zn + zn , 40 4 8 2 2  3 1 n−4 2 zn−3 zn−2 zn−1 zn 5 w = i z0 − ∑ zα − 2 − 2 − 80 z2n−3 zn − 2 zn−2 z2n − 2 α =m+1 z∗n−3

 zn−3 z3n z5n . − 8 16

The tube hypersurface with base (7) for a fixed m is mapped into Qn−2,2 by the locally biholomorphic map

z  α z∗α = exp α = 1, . . . , m, , 2 1 z∗α = √ zα , α = m + 1, . . ., n − 5, 2   1 zn−3 zn z3n , + z∗n−4 = √ zn−4 + 4 24 2   z2n−3 z2n zn−3 z2n z4n 1 ∗ zn−3 = √ , + + + zn−3 + zn−2 + zn−4 zn + 2 4 4 96 2 2 z∗n−2

1 = √ 2 2

1 z∗n−1 = √ 2 2 z∗n

1 = √ 2 2







zn−2 zn zn−4 z2n z2n−3 zn zn−3 z3n z5 + + + + n zn−1 + zn + zn−4 zn−3 + 3 4 4 24 960 z2 z2 zn−3 z2n z4n − zn−3 − zn−2 − zn−4 zn − n−3 + n − 2 4 4 96

 ,

 ,

zn−2 zn zn−4 z2n z2n−3 zn zn−3 z3n z5 + + + + n zn−1 − zn + zn−4 zn−3 + 3 4 4 24 960

 ,

7.2 Defining Systems of Type II

 w = i z0 −

133

1 n−4 2 zn−3 zn−2 zn−1 zn zn−4 zn−3 zn z3n−3 zn−2 z2n − − − ∑ zα − 2 − 2 − 2 α =m+1 2 12 8

 zn−4 z3n z2n−3 z2n zn−3 z4n z6n . − − − 24 16 192 11520

In this chapter we obtain the following result. Theorem 7.1. Let M be a closed (n − 2, 2)-spherical tube hypersurface in Cn+1 with n ≥ 4. Then M is affinely equivalent to a tube hypersurface that belongs to one of the classes (A), (B), (C), (D). Theorem 7.1 was announced in paper [53] with a brief sketch of a proof. Full details were given in preprint [54]. Because of the prohibitive length of the preprint the complete proof has never been published. In this chapter it appears in print for the first time. Our proof of Theorem 7.1 is spread over the next three sections. We proceed as outlined in Chapter 4 and consider defining systems of three types. We start with systems of type II, then investigate systems of type III, and finish by considering systems of type I. Before proceeding, we note an interesting feature that makes the case k = n − 2 significantly different from the cases k = n and k = n − 1 considered earlier. Namely, class (D) for n ≥ 7 contains a family of (n − 2, 2)-spherical tube hypersurfaces that depends on a continuous parameter (see equations (5)). It will be shown in Chapter 8 that (at least for m = 0) all hypersurfaces in this family are pairwise affinely non-equivalent. Thus, the number of affine equivalence classes of closed (n − 2, 2)-spherical tube hypersurfaces is infinite for every n ≥ 7 (see Theorem 8.1 for more detail). On the other hand, this number is finite for n = 4, 5, 6 and can be determined explicitly for each of these three values using the method outlined in Section 5.3. We leave details to the reader.

7.2 Defining Systems of Type II We proceed as in Section 4.3. Clearly, for (n − 2, 2)-spherical tube hypersurfaces defining systems of type II arise only for n = 4. To use formulas (4.29), (4.30), we need to find the function G. According to formula (4.28), to determine G(w1 , w3 ) one has to calculate exp(Aw1 + Bw3 ), where ⎞ ⎞ ⎛ ⎛ 0 10 0 01 A = ⎝ i/2 p q ⎠ , B = ⎝ 0 q r ⎠ 0 qr i/2 r s are commuting matrices for some p, q, r, s ∈ C. Clearly, A, B are H -symmetric, with

7 (n − 2, 2)-Spherical Tube Hypersurfaces

134

 H :=

i/2

0

0

I2

 .

We will now consider three cases. Case (i). Assume first that each of A, B is diagonalizable. Then since A and B commute, there exists C ∈ GL(3, C) such that C −1 AC = diag(λ1 , λ2 , λ3 ) and C−1 BC = diag(ν1 , ν2 , ν3 ). Now formula (4.28) together with the initial conditions G(0) = 1, ∂ G/∂ w1 (0) = 0, ∂ G/∂ w3 (0) = 0 implies G=

1 × (λ1 − λ2 )(ν1 − ν3 ) − (λ1 − λ3)(ν1 − ν2 ) [(λ2 ν3 − λ3ν2 ) exp(λ1 w1 + ν1 w3 ) + (λ3 ν1 − λ1 ν3 ) exp(λ2 w1 + ν2 w3 )+ (λ1 ν2 − λ2 ν1 ) exp(λ3 w1 + ν3 w3 )] .

Note that in the above formula the number (λ1 − λ2)(ν1 − ν3 ) − (λ1 − λ3 )(ν1 − ν2 ) is non-zero since the matrices A, B, I3 are clearly linearly independent. Now (4.29), (4.30) yield that the graph x0 = F(x) extends to a closed hypersurface in R5 which is affinely equivalent to the hypersurface sin x0 = ex1 sin x2 + ex3 sin x4 . The tube hypersurface with the above base belongs to class (B) for n = 4. Case (ii). Assume now that one of the matrices A, B (say the matrix A) has a 2 × 2-cell in its Jordan normal form and neither of the matrices has a 3 × 3-cell. Let C ∈ GL(3, C) be a matrix such that ⎛ ⎞ λ1 1 0 C−1 AC = ⎝ 0 λ1 0 ⎠ 0 0 λ2 with λ1 , λ2 ∈ C. We will now consider two cases. Case (ii.a). First, suppose λ1 = λ2 . Since C−1 BC commutes with C−1 AC, we obtain ⎛ ⎞ ν1 η 0 C−1 BC = ⎝ 0 ν1 0 ⎠ 0 0 ν2 for some ν1 , ν2 , η ∈ C. Now formula (4.28) together with the initial conditions G(0) = 1, ∂ G/∂ w1 (0) = 0, ∂ G/∂ w3 (0) = 0 implies G=

 1 ((λ1 ν2 − λ2 ν1 )(w1 + η w3 ) + ηλ2 − ν2 ) × (ν1 − ν2 ) + η (λ2 − λ1)  exp (λ1 w1 + ν1 w3 ) + (ν1 − ηλ1 ) exp (λ2 w1 + ν2 w3 ) .

7.2 Defining Systems of Type II

135

Note that in the above formula the number (ν1 − ν2 ) + η (λ2 − λ1 ) is non-zero since the matrices A, B, I3 are linearly independent. Identities (4.29), (4.30) now yield that the graph x0 = F(x) extends to a closed hypersurface in R5 which is affinely equivalent to the hypersurface ex0 sin x1 + x2 cosx3 + x4 sin x3 = 0.

(7.2)

The tube hypersurface with the above base belongs to class (C) for n = 4. Case (ii.b). Now, suppose λ1 = λ2 =: λ . Since A is H -symmetric and H is non-degenerate, we have ⎛ ⎞ 0 ρ 0 CT H C = ⎝ ρ σ ε ⎠ , 0 ε ω where ρ , σ , ε , ω ∈ C, ρ , ω = 0. Further, C−1 BC commutes with C−1 AC and is CT H C-symmetric, hence ⎞ ⎛ abc C−1 BC = ⎝ 0 a 0 ⎠ 0ed for some a, b, c, d, e ∈ C satisfying ρ c − ω e = ε (a − d). Clearly, a and d are the only eigenvalues of B. If d = a, then ρ c = ω e, thus c = 0 if and only if e = 0. However, if c = 0, e = 0, then B has a 3 × 3-cell in its Jordan normal form, which is impossible by our assumption. It then follows that c = e = 0, which yields that the matrices A, B, I3 are linearly dependent. This contradiction shows that in fact we have d = a. If B has a 2 × 2-cell in its Jordan normal form, then arguing as in case (ii.a) we obtain that the graph x0 = F(x) extends to a closed hypersurface in R5 which is affinely equivalent to hypersurface (7.2). If B is diagonalizable, then the matrix C can be chosen to guarantee that C−1 BC is diagonal, and arguing as in case (ii.a) for η = 0 we are again led to hypersurface (7.2). Case (iii). Assume now that one of the matrices A, B (say the matrix A) has a 3 × 3-cell in its Jordan normal form. Let C ∈ GL(3, C) be a matrix such that ⎛ ⎞ λ 1 0 C−1 AC = ⎝ 0 λ 1 ⎠ 0 0λ with λ ∈ C. Since C−1 BC commutes with C−1 AC, we obtain ⎛ ⎞ νη μ C−1 BC = ⎝ 0 ν η ⎠ 0 0 ν for some ν , η , μ ∈ C (since A, B, I3 are linearly independent, we have μ = 0). Then formula (4.28) together with the initial conditions G(0) = 1, ∂ G/∂ w1 (0) = 0, ∂ G/∂ w3 (0) = 0 implies

7 (n − 2, 2)-Spherical Tube Hypersurfaces

136

 G=

 λη −ν 2 (w1 + η w3 ) − λ w1 − ν w3 + 1 exp(λ w1 + ν w3 ) . 2μ

Identities (4.29), (4.30) now yield that the graph x0 = F(x) extends to a closed hypersurface in R5 which is affinely equivalent to the hypersurface (x0 + x21 − x22 ) cos x3 + (x4 + 2x1 x2 ) sin x3 = 0.

The tube hypersurface with the above base belongs to class (C) for n = 4.

7.3 Defining Systems of Type III Our line of argument is similar (but not entirely identical) to that presented in Section 4.4. For a defining system of type III one of the following holds: (i) N = 2, K = 0 or N = 0, K = 2, (ii) N = K = 1, (iii) N + K = 1, (iv) K = N = 0. Accordingly, four cases will be considered. Case (i). Let either N = 2, K = 0 or N = 0, K = 2. We then have L = 0 and ⎛ ⎞ 0 T (τ ) ⎜ ⎟ T (τ ) ⎜ ⎟ ⎜ ⎟ 0 D=⎜ ⎟, ⎜ ⎟ . . ⎝ . ⎠

0

0

where T (τ ) is the matrix defined in (4.13) and τ is either 1 or −1. Since the matrices Λα commute and are symmetric (see (4.31)), they can be simultaneously diagonalized by a real orthogonal transformation. Using Proposition 4.1 and the relations of Lemma 3.2, we then see that there exists C ∈ GL(n, R) such that the function ˆ = D for which the ˆ F(x) := F(Cx) satisfies a defining system of type III with D ˆ α have the form corresponding matrices C ⎞ ⎛ a b p q t5 . . . tn ⎜ −b a + 2b −q p + 2q t5 . . . tn ⎟ ⎟ ⎜ ⎜ p q c d r5 . . . rn ⎟ ⎟ ⎜ ⎟ ˆ1 =⎜ C ⎜ −q p + 2q −d c + 2d r5 . . . rn ⎟ , ⎜ t5 −t5 r5 −r5 s5 0⎟ ⎟ ⎜ ⎟ ⎜ .. .. .. .. .. ⎝ . . ⎠ . . . tn

−tn

rn

−rn

0

sn

7.3 Defining Systems of Type III

⎛

b ⎜ a + 2b ⎜ ⎜ q ⎜ ˆ2 =⎜ C ⎜ p + 2q ⎜ −t5 ⎜ ⎜ .. ⎝ . ⎛

−tn

137

−a − 2b q −2a − 3b p + 2q −p − 2q d −2p − 3q c + 2d t5 −r5 .. .. . . tn −rn

−p − 2q −2p − 3q −c − 2d −2c − 3d r5 .. .

−t5 −t5 −r5 −r5 −s5

rn

0

⎞

⎞ −tn −tn ⎟ ⎟ −rn ⎟ ⎟ −rn ⎟ ⎟, 0 ⎟ ⎟ ⎟ .. ⎠ . −sn

... ... ... ...

p q c d r5 . . . rn ⎜ −p p + 2q −d c + 2d r5 . . . rn ⎟ ⎜ ⎟ ⎜ c d g h ρ5 . . . ρn ⎟ ⎜ ⎟ ⎟ ˆ3 =⎜ C ⎜ −c c + 2d −h g + 2h ρ5 . . . ρn ⎟ , ⎜ r5 −r5 ρ5 −ρ5 σ5 ⎟ 0 ⎜ ⎟ ⎜ .. ⎟ .. .. .. . . ⎝ . ⎠ . . . . rn −rn ρn −ρn 0 σn ⎛ ⎞ q −p − 2q d −c − 2d −r5 . . . −rn ⎜ p + 2q −2p − 3q c + 2d −2c − 3d −r5 . . . −rn ⎟ ⎜ ⎟ ⎜ d −c − 2d h −g − 2h −ρ5 . . . −ρn ⎟ ⎜ ⎟ ⎟ ˆ4 =⎜ C ⎜ c + 2d −2c − 3d g + 2h −2g − 3h −ρ5 . . . −ρn ⎟ , ⎜ −r5 ⎟ r − ρ ρ − σ 0 5 5 5 5 ⎜ ⎟ ⎜ .. ⎟ .. .. .. . . ⎝ . ⎠ . . . . −rn rn − ρn ρn 0 −σn ⎞ ⎛ tα −tα rα −rα 0 . . . 0 sα 0 . . . 0 ⎜ tα −tα rα −rα 0 . . . 0 sα 0 . . . 0 ⎟ ⎟ ⎜ ⎜ rα −rα ρα −ρα 0 . . . 0 σα 0 . . . 0 ⎟ ⎟ ⎜ ⎜ rα −rα ρα −ρα 0 . . . 0 σα 0 . . . 0 ⎟ ⎟ ⎜ ⎟ ⎜0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ . . . . .. ⎜ . . . . ˆ Cα = ⎜ . . . . . 0 ⎟ ⎟ , α = 5, . . . , n, ⎟ ⎜0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ s −s σ −σ λα α α α ⎟ ⎜ α ⎟ ⎜0 0 0 0 0 ⎟ ⎜ ⎜ . . .. .. .. ⎟ ⎝ .. .. . ⎠ . . 0 0

0

0

0

(7.3)

0

where the parameters satisfy the conditions s2α − (a + b)sα − (p + q)σα − λα tα + τ = 0,

α = 5, . . . , n,

sα σα − (p + q)sα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2 − (c + d)sα − (g + h)σα − λα ρα + τ = 0,

α = 5, . . . , n,

rα sα − tα σα + (p + q)(tα − ρα ) + (c + d − a − b)rα = 0,

α = 5, . . . , n,

(7.4)

7 (n − 2, 2)-Spherical Tube Hypersurfaces

138

ρα sα − rα σα + (c + d)(tα − ρα ) + (g + h − p − q)rα = 0,

α = 5, . . . , n,

n

∑ (tα ρα − rα2 ) + p2 + 4pq + 3q2 + c2 + 4cd + 3d 2−

α =5

ac − 2ad − 2bc − 3bd − gp − 2ph − 2gq − 3hq = 0, (c + d)(a + b − c − d) + (p + q)(g + h − p − q) − τ = 0

ˆ α the number λα occurs at position α on the diagonal. Dropping and in the matrix C hats we get the following defining system: n

F11 = 2τ F1 (F1 + F2 ) + aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = −τ (F12 − F22 ) + bF1 + (a + 2b)F2 + qF3 + (p + 2q)F4 − ∑ tγ Fγ , γ =5

F22 = −2τ F2 (F1 + F2) − (a + 2b)F1 − (2a + 3b)F2 − (p + 2q)F3 − (2p + 3q)F4+ n

n

∑ tγ Fγ − 1,

F33 = 2τ F3 (F3 + F4 ) + cF1 − dF2 + gF3 − hF4 + ∑ ργ Fγ + 1, γ =5

n

γ =5

F34 = −τ (F32 − F42 ) + dF1 + (c + 2d)F2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

F44 = −2τ F4 (F3 + F4) − (c + 2d)F1 − (2c + 3d)F2 − (g + 2h)F3 − (2g + 3h)F4+ n

F13 = τ (2F1 F3 + F1 F4 + F2 F3 ) + pF1 − qF2 + cF3 − dF4 + ∑ rγ Fγ , γ =5

n

∑ ργ Fγ − 1,

γ =5 n

F14 = F23 = τ (F2 F4 − F1 F3 ) + qF1 + (p + 2q)F2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

F24 = −τ (F1 F4 + F2 F3 + 2F2 F4 ) − (p + 2q)F1 − (2p + 3q)F2 − (c + 2d)F3−

n

(2c + 3d)F4 + ∑ rγ Fγ , γ =5

F1α = −F2α = τ Fα (F1 + F2) + tα (F1 + F2 ) + rα (F3 + F4) + sα Fα ,

α = 5, . . . , n,

F3α = −F4α = τ Fα (F3 + F4) + rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα ,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + σα (F3 + F4) + λα Fα + 1,

α = 5, . . . , n, α , β = 5, . . . , n,

Fαβ = 0, F(0) = 0,

α = β ,

α = 1, . . . , n.

Fα (0) = 0,

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. As in (4.33), we let u1 := x1 + x2 , v1 := x1 − x2 ,

u2 := x3 + x4 , v2 := x3 − x4 .

(7.5)

Expressing F in the variables u1 , u2 , v1 , v2 , x5 , . . . , xn , we obtain the function (cf. (4.34)) Fˆ := F ((u1 + v1 ) /2, (u1 − v1 ) /2, (u2 + v2 ) /2, (u2 − v2 ) /2, x5, . . . , xn ) .

(7.6)

7.3 Defining Systems of Type III

139

Further, by Proposition 4.6 the functions ˆ ∂ u1 , Ψ1 := 2 ∂ F/

ˆ ∂ u2 Ψ2 := 2 ∂ F/

(7.7)

depend only on v1 , v2 , and (4.35) yields that on some neighborhood of the origin Ψ1 , Ψ2 satisfy the system of equations

∂Ψ1 = τΨ12 + (a + b)Ψ1 + (p + q)Ψ2 + 1, ∂ v1 ∂Ψ1 ∂Ψ2 = = τΨ1Ψ2 + (p + q)Ψ1 + (c + d)Ψ2, ∂ v2 ∂ v1

(7.8)

∂Ψ2 = τΨ22 + (c + d)Ψ1 + (g + h)Ψ2 + 1, ∂ v2 Ψ1 (0) = 0,

Ψ2 (0) = 0.

Let P(v1 , v2 ) be the function defined by the conditions

∂P = τΨ1 , ∂ v1

∂P = τΨ2 , ∂ v2

P(0) = 0.

According to (4.37), the function P satisfies the system

∂ 2P = ∂ v21



∂P ∂ v1

2 ∂P ∂P + (a + b) + (p + q) + τ, ∂ v1 ∂ v2

∂ 2P ∂P ∂P ∂P ∂P = + (p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ 2P = ∂ v22



P(0) = 0,

∂P ∂ v2

2 ∂P ∂P + (c + d) + (g + h) + τ, ∂ v1 ∂ v2

∂P (0) = 0, ∂ v1

∂P (0) = 0. ∂ v2

The form of the above system is similar to that of defining systems of type I corresponding to strongly pseudoconvex spherical tube hypersurfaces in C3 . In order to solve it, we consider the function Q defined by formula (4.39). According to (4.40), this function satisfies the linear system

∂ 2Q ∂Q ∂Q = (a + b) + (p + q) − τ Q, ∂ v1 ∂ v2 ∂ v21 ∂ 2Q ∂Q ∂Q = (p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2

(7.9)

7 (n − 2, 2)-Spherical Tube Hypersurfaces

140

∂ 2Q ∂Q ∂Q = (c + d) + (g + h) − τ Q, 2 ∂ v1 ∂ v2 ∂ v2 Q(0) = 1,

∂Q (0) = 0, ∂ v1

∂Q (0) = 0, ∂ v2

and we have (cf. (4.44))

Ψ1 = −τ

1 ∂Q , Q ∂ v1

Ψ2 = −τ

1 ∂Q . Q ∂ v2

(7.10)

Further, the defining system yields (cf. (4.45))

∂ 2 Fˆ ∂ Fˆ = λα + sα Ψ1 + σαΨ2 + 1, α = 5, . . . , n, 2 ∂ xα ∂ xα ∂ 2 Fˆ = 0, ∂ xα ∂ xβ

(7.11)

α , β = 5, . . . , n,

α = β ,

and therefore we obtain (cf. (4.49))

∂ Fˆ = (sα Ψ1 + σαΨ2 + 1)xα + Φα (v1 , v2 ), ∂ xα

α = 5, . . . , N ,

sαΨ1 + σα Ψ2 + 1 eλα xα ∂ Fˆ =− + eλα xα Φα (v1 , v2 ) + , α = N + 1, . . . , n ∂ xα λα λα

(7.12)

for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies (cf. (4.50))

∂ 2 Fˆ ∂ Fˆ = (τΨ1 + sα ) + tαΨ1 + rα Ψ2 , α = 5, . . . , n, ∂ v1 ∂ xα ∂ xα ∂ 2 Fˆ ∂ Fˆ = (τΨ2 + σα ) + rαΨ1 + ρα Ψ2 , α = 5, . . . , n. ∂ v2 ∂ xα ∂ xα

(7.13)

Plugging identities (7.12) into both sides of (7.13) and using (7.4), (7.8), we obtain the equations (cf. (4.53))

∂ Φα = (τΨ1 + sα )Φα + tαΨ1 + rα Ψ2 , α = 5, . . . , N , ∂ v1 ∂ Φα = (τΨ2 + σα )Φα + rα Ψ1 + ραΨ2 , α = 5, . . . , N , ∂ v2   ∂ Φα 1 , = (τΨ1 + sα ) Φα + α = N + 1, . . ., n, ∂ v1 λα

(7.14)

7.3 Defining Systems of Type III

141



1 ∂ Φα = (τΨ2 + σα ) Φα + ∂ v2 λα

Φα (0) = 0,

 , α = N + 1, . . ., n,

α = 5, . . . , n.

Further, (7.12) yields (cf. (4.57), (4.58)) N

N

Ψ1 Ψ2 sα Ψ1 + σα Ψ2 + 1 2 Fˆ = u1 + u2 + R(v1 , v2 ) + ∑ xα + ∑ Φα xα − 2 2 2 α =5 α =5  n n n sα Ψ1 + σαΨ2 + 1 eλα xα 1 λα xα e xα + ∑ Φα + ∑ − 1 ∑ 2 λα α =N +1 α =N +1 λα α =N +1 λα

(7.15)

for some function R with R(0) = 0. Plugging (7.15) into the equations for the function Fˆ that arise from the defining system and using (7.4), (7.8), (7.14), we obtain (cf. (4.59)) N ∂ 2R ∂R ∂R = (2 τΨ + a + b) + (p + q) + tα Φα − 1 ∑ ∂ v1 ∂ v2 α =5 ∂ v21     n n n a + 3b p + 3q tα sα tα σα tα Ψ1 − Ψ2 − ∑ , +∑ +∑ 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα N ∂ 2R ∂R ∂R = (τΨ2 + p + q) + (τΨ1 + c + d) + ∑ rα Φα − ∂ v1 ∂ v2 ∂ v1 ∂ v2 α =5     n n n p + 3q c + 3d rα sα rα σα rα +∑ +∑ Ψ1 − Ψ2 − ∑ , (7.16) 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = (c + d) + (2τΨ2 + g + h) + ∑ ρα Φα − 2 ∂ v1 ∂ v2 α =5 ∂ v2     n n n c + 3d g + 3h ρ α sα ρα σα ρα +∑ +∑ Ψ1 − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα

R(0) = 0,

n ∂R sα (0) = − ∑ , 2 ∂ v1 λ α =N +1 α

n ∂R σα (0) = − ∑ . 2 ∂ v2 α =N +1 λα

In order to determine Fˆ from representation (7.15), we first need to find the functions Ψ1 , Ψ2 from (7.10), then the functions Φα from system (7.14), and finally the function R from system (7.16). In order to find Ψ1 , Ψ2 , we have to determine the function Q from system (7.9). We deal with system (7.9) as explained in Section 4.4 (cf. Section 5.1). Namely, we set   ∂Q ∂Q V := Q, , (7.17) ∂ v1 ∂ v2

7 (n − 2, 2)-Spherical Tube Hypersurfaces

142

and obtain the system (cf. (4.41))

∂V = B1 V , ∂ v1

∂V = B2 V , ∂ v2

V (0) = (1, 0, 0),

(7.18)

where (cf. (4.42)) ⎛

⎞ 0 1 0 B1 := ⎝ −τ a + b p + q ⎠ , 0 p+q c+d

⎛

⎞ 0 0 1 B2 := ⎝ 0 p + q c + d ⎠ . −τ c + d g + h

Relations (7.4) imply that the matrices B1 and B2 commute, and therefore the solution of (7.18) is given by (cf. (4.43)) ⎛ ⎞  1

(7.19) V = exp B1 v1 + B2 v2 ⎝ 0 ⎠ . 0 Next, we observe that the matrices B1 and B2 are H  -symmetric, where   −τ 0  H := 0 I2 . If τ = −1, then B1 , B2 are symmetric and hence can be simultaneously diagonalized by a real orthogonal transformation. It then follows from (7.19) that the function Q has the form Q=

3

∑ A j exp(L j (v1, v2 ))

(7.20)

j=1

for some real numbers A j and linear functions L j . Then straightforward (but lengthy) calculations utilizing (7.10), (7.14), (7.15), (7.16) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Now, suppose τ = 1. In this case χH  = 1, thus we will consider four cases analogous to cases (i)–(iv) in Section 5.1. Case (i.a). Suppose that all eigenvalues of B1 , B2 are real and that both these matrices are diagonalizable. Arguing as in case (i) in Section 5.1, we obtain that the function Q has the form (7.20), and analogously to the case τ = −1 above we obtain that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of class (A). Case (i.b). Suppose that one of B1 , B2 has non-real eigenvalues. Arguing as in case (ii) in Section 5.1, we obtain that the function Q has the form Q = A1 exp(L1 (v1 , v2 ))+ (7.21) (A2 sin(L2 (v1 , v2 )) + A3 cos(L2 (v1 , v2 ))) exp(L3 (v1 , v2 ))

7.3 Defining Systems of Type III

143

for some real numbers A j and linear functions L j . In this case (7.10), (7.14), (7.15), (7.16) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (i.c). Suppose that all eigenvalues of B1 , B2 are real, that for one of these matrices the Jordan normal form contains a 2 × 2-cell, and that the Jordan normal form of the other matrix does not contain a 3 × 3-cell. Arguing as in case (iii) in Section 5.1, we obtain that the function Q has the form Q = A exp(L1 (v1 , v2 )) + L (v1 , v2 ) exp(L2 (v1 , v2 ))

(7.22)

for some real number A , linear functions L j , and an affine function L . In this case (7.10), (7.14), (7.15), (7.16) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (i.d). Suppose that all eigenvalues of B1 , B2 are real and that the Jordan normal form of one of these matrices contains a 3 × 3-cell. Arguing as in case (iv) in Section 5.1, we obtain that the function Q has the form Q = Q(v1 , v2 ) exp(L (v1 , v2 ))

(7.23)

for some linear function L and a polynomial Q of degree at most 2. In this case (7.10), (7.14), (7.15), (7.16) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D). Case (ii). Let N = K = 1. We then have L = 0 and ⎞ ⎛ 0 T (1) ⎟ ⎜ T (−1) ⎟ ⎜ ⎟ ⎜ 0 D=⎜ ⎟. ⎟ ⎜ . . ⎝ . ⎠

0

0

As before, since the matrices Λα commute and are symmetric (see (4.31)), they can be simultaneously diagonalized by a real orthogonal transformation. Using Proposition 4.1 and the relations of Lemma 3.2, we then see that there exists C ∈ GL(n, R) ˆ such that the function F(x) := F(Cx) satisfies a defining system of type III with ˆ α have the form ˆ D = D for which the corresponding matrices C ⎛ ⎞ a b p q t5 . . . tn ⎜ −b a + 2b 2p + q −p t5 . . . tn ⎟ ⎜ ⎟ ⎜ p −2p − q c d r5 . . . rn ⎟ ⎜ ⎟ −d c + 2d r5 . . . rn ⎟ ˆ1 =⎜ C ⎜ −q −p ⎟, ⎜ t5 −t5 r5 −r5 s5 0⎟ ⎜ ⎟ ⎜ .. ⎟ .. .. .. .. ⎝ . . ⎠ . . . tn −tn rn −rn 0 sn

7 (n − 2, 2)-Spherical Tube Hypersurfaces

144

⎛

⎞ b −a − 2b −2p − q p −t5 . . . −tn ⎜ a + 2b −2a − 3b −3p − 2q 2p + q −t5 . . . −tn ⎟ ⎜ ⎟ ⎜ −2p − q 3p + 2q −2c − d c −r5 . . . −rn ⎟ ⎜ ⎟ 2p + q −c −d −r5 . . . −rn ⎟ ˆ2 =⎜ C ⎜ −p ⎟, ⎜ −t5 t5 −r5 r5 −s5 0 ⎟ ⎜ ⎟ ⎜ ⎟ .. .. .. .. . . ⎝ ⎠ . . . . . −tn tn −rn rn 0 −sn ⎛ ⎞ p −2p − q c d r5 . . . rn ⎜ 2p + q −3p − 2q 2c + d −c r5 . . . rn ⎟ ⎜ ⎟ ⎜ c −2c − d g h ρ5 . . . ρn ⎟ ⎜ ⎟ −c −h g + 2h ρ5 . . . ρn ⎟ ˆ3 =⎜ C ⎜ −d ⎟, ⎜ r5 −r5 ρ5 −ρ5 σ5 0⎟ ⎜ ⎟ ⎜ .. ⎟ .. .. .. .. ⎝ . ⎠ . . . .

ρn

−rn

rn

− ρn

q p d −c − 2d −r5 ⎜ −p 2p + q −c −d −r5 ⎜ ⎜ d c h −g − 2h − ρ5 ⎜ ⎜ c + 2d −d g + 2h −2g − 3h −ρ5 ˆ C4 = ⎜ ⎜ −r5 r5 − ρ5 ρ5 −σ5 ⎜ ⎜ .. .. .. .. ⎝ . . . . rn − ρn ρn 0 −rn ⎛

tα ⎜ tα ⎜ ⎜ rα ⎜ ⎜ rα ⎜ ⎜0 ⎜ ⎜ ˆ α = ⎜ ... C ⎜ ⎜0 ⎜ ⎜s ⎜ α ⎜0 ⎜ ⎜ . ⎝ .. 0

−tα −tα −rα −rα

rα rα ρα ρα

0 .. .

0 .. .

0 −sα 0 .. .

0 σα 0 .. .

0

0

σn

0

⎛

⎞ −rn −rn ⎟ ⎟ −ρ n ⎟ ⎟ −ρ n ⎟ , ⎟ 0 ⎟ ⎟ ⎟ .. ⎠ . −σn

... ... ... ...

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0 0 ⎟ ⎟ .. .. . . 0 ⎟ ⎟, ⎟ 0 0 ⎟ ⎟ −σα λα ⎟ ⎟ 0 0 ⎟ .. .. ⎟ . ⎠ . 0 0 0 −rα −rα − ρα − ρα

0 0 0 0

... ... ... ...

0 sα 0 sα 0 σα 0 σα

0 0 0 0

(7.24)

... ... ... ...

α = 5, . . . , n,

where the parameters satisfy the conditions s2α − (a + b)sα − (p + q)σα − λα tα + 1 = 0,

α = 5, . . . , n,

sα σα + (p + q)sα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2 + (c + d)sα − (g + h)σα − λα ρα − 1 = 0,

α = 5, . . . , n,

7.3 Defining Systems of Type III

145

rα sα − tα σα − (p + q)(tα + ρα ) + (c + d − a − b)rα = 0,

α = 5, . . . , n,

ρα sα − rα σα − (c + d)(tα + ρα ) + (g + h + p + q)rα = 0, α = 5, . . . , n, n

∑ (tα ρα − rα2 ) + q2 + 4pq + 3p2 − c2 + d 2+

(7.25)

ac + 2bc + bd + gp − hq = 0,

α =5

(c + d)(a + b − c − d) + (p + q)(g + h + p + q) − 1 = 0 ˆ α the number λα occurs at position α on the diagonal. Dropping and in the matrix C hats we get the following defining system: n

F11 = 2F1 (F1 + F2 ) + aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = −(F12 − F22 ) + bF1 + (a + 2b)F2 − (2p + q)F3 − pF4 − ∑ tγ Fγ , γ =5

F22 = −2F2 (F1 + F2 ) − (a + 2b)F1 − (2a + 3b)F2 + (3p + 2q)F3 + (2p + q)F4+ n

∑ tγ Fγ − 1,

γ =5

n

F33 = −2F3 (F3 + F4 ) + cF1 + (2c + d)F2 + gF3 − hF4 + ∑ ργ Fγ + 1, n

γ =5

F34 = (F32 − F42 ) + dF1 − cF2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

n

F44 = 2F4 (F3 + F4 ) − (c + 2d)F1 − dF2 − (g + 2h)F3 − (2g + 3h)F4 + ∑ ργ Fγ − 1, γ =5

n

F13 = F24 = (F2 F3 − F1 F4 ) + pF1 + (2p + q)F2 + cF3 − dF4 + ∑ rγ Fγ , γ =5n

F14 = (F1 F3 + 2F1 F4 + F2 F4 ) + qF1 − pF2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

F23 = −(F1 F3 + 2F2F3 + F2 F4 ) − (2p + q)F1 − (3p + 2q)F2 − (2c + d)F3 − cF4 − n

∑ rγ Fγ ,

γ =5

F1α = −F2α = Fα (F1 + F2 ) + tα (F1 + F2) + rα (F3 + F4 ) + sα Fα ,

α = 5, . . . , n,

F3α = −F4α = −Fα (F3 + F4 ) + rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα , α = 5, . . . , n, Fαα = sα (F1 + F2 ) + σα (F3 + F4) + λα Fα + 1,

α , β = 5, . . . , n,

Fαβ = 0, F(0) = 0,

α = 5, . . . , n,

Fα (0) = 0,

α = β ,

α = 1, . . . , n.

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. We argue analogously to case (i). First, we introduce the variables u1 , u2 , v1 , v2 by formulas (7.5), the function Fˆ by formula (7.6), and the functions Ψ1 , Ψ2 by formulas (7.7). By Proposition 4.6 the functions Ψ1 , Ψ2 depend only on v1 , v2 , and (4.35) yields that on some neighborhood of the origin Ψ1 , Ψ2 satisfy

7 (n − 2, 2)-Spherical Tube Hypersurfaces

146

∂Ψ1 = Ψ12 + (a + b)Ψ1 − (p + q)Ψ2 + 1, ∂ v1 ∂Ψ1 ∂Ψ2 =− = −Ψ1Ψ2 − (p + q)Ψ1 − (c + d)Ψ2, ∂ v2 ∂ v1

(7.26)

∂Ψ2 = −Ψ22 + (c + d)Ψ1 + (g + h)Ψ2 + 1, ∂ v2 Ψ1 (0) = 0,

Ψ2 (0) = 0.

Let P(v1 , v2 ) be the function defined by the conditions

∂P = Ψ1 , ∂ v1

∂P = −Ψ2 , ∂ v2

P(0) = 0.

(7.27)

According to (4.37), the function P satisfies the system

∂ 2P = ∂ v21



∂P ∂ v1

2 ∂P ∂P + (a + b) + (p + q) + 1, ∂ v1 ∂ v2

∂ 2P ∂P ∂P ∂P ∂P = − (p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ v1 ∂ v2   ∂ 2P ∂P 2 ∂P ∂P = − (c + d) + (g + h) − 1, 2 ∂ v2 ∂ v1 ∂ v2 ∂ v2 P(0) = 0,

∂P (0) = 0, ∂ v1

∂P (0) = 0. ∂ v2

Observe that the above system is a defining system of type I corresponding to (1, 1)-spherical tube hypersurfaces in C3 . In order to solve it, we consider the function Q defined by formula (4.39). According to (4.40), this function satisfies the linear system ∂ 2Q ∂Q ∂Q = (a + b) + (p + q) − Q, ∂ v1 ∂ v2 ∂ v21

∂ 2Q ∂Q ∂Q = −(p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ 2Q ∂Q ∂Q = −(c + d) + (g + h) + Q, ∂ v1 ∂ v2 ∂ v22 ∂Q ∂Q (0) = 0, (0) = 0, Q(0) = 1, ∂ v1 ∂ v2

(7.28)

and we have (cf. (4.44))

Ψ1 = −

1 ∂Q , Q ∂ v1

Ψ2 =

1 ∂Q . Q ∂ v2

(7.29)

As in case (i), identities (7.11) hold and lead to identities (7.12) for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies (cf. (4.50))

7.3 Defining Systems of Type III

147

∂ 2 Fˆ ∂ Fˆ = (Ψ1 + sα ) + tαΨ1 + rα Ψ2 , ∂ v1 ∂ xα ∂ xα

α = 5, . . . , n,

∂ 2 Fˆ ∂ Fˆ = (−Ψ2 + σα ) + rαΨ1 + ραΨ2 , α = 5, . . . , n. ∂ v2 ∂ xα ∂ xα

(7.30)

Plugging identities (7.12) into both sides of (7.30) and using (7.25), (7.26), we obtain the equations (cf. (4.53))

∂ Φα = (Ψ1 + sα )Φα + tαΨ1 + rαΨ2 , ∂ v1

α = 5, . . . , N ,

∂ Φα = (−Ψ2 + σα )Φα + rαΨ1 + ρα Ψ2 , α = 5, . . . , N , ∂ v2   ∂ Φα 1 , = (Ψ1 + sα ) Φα + α = N + 1, . . . , n, ∂ v1 λα   ∂ Φα 1 , = (−Ψ2 + σα ) Φα + α = N + 1, . . . , n, ∂ v2 λα Φα (0) = 0,

(7.31)

α = 5, . . . , n.

Further, (7.12) yields representation (7.15) for some function R with R(0) = 0. Plugging (7.15) into the equations for the function Fˆ that arise from the defining system and using (7.25), (7.26), (7.31), we obtain (cf. (4.59)) N ∂ 2R ∂R ∂R = (2Ψ1 + a + b) + (p + q) + ∑ tα Φα − 2 ∂ v1 ∂ v2 α =5 ∂ v1     n n n tα sα tα σα tα a + 3b 3p + q +∑ +∑ Ψ1 − − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = (−Ψ2 − p − q) + (Ψ1 + c + d) + ∑ rα Φα − ∂ v1 ∂ v2 ∂ v1 ∂ v2 α =5     n n n 3p + q c−d rα sα rα σα rα − Ψ1 − − Ψ2 − ∑ , (7.32) +∑ +∑ 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = −(c + d) + (−2Ψ2 + g + h) + ∑ ρα Φα − 2 ∂ v1 ∂ v2 α =5 ∂ v2     n n n c−d g + 3h ρα sα ρα σα ρα +∑ +∑ − Ψ1 − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα

R(0) = 0,

n ∂R sα (0) = − ∑ , 2 ∂ v1 α =N +1 λα

n ∂R σα (0) = − ∑ . 2 ∂ v2 α =N +1 λα

148

7 (n − 2, 2)-Spherical Tube Hypersurfaces

In order to determine Fˆ from representation (7.15), we first need to find the functions Ψ1 , Ψ2 from (7.29), then the functions Φα from system (7.31), and finally the function R from system (7.32). In order to find Ψ1 , Ψ2 , we have to determine the function Q from system (7.28). We deal with system (7.28) as explained in Section 4.4 (cf. Section 5.1). As in case (i), we introduce V by formula (7.17). Clearly V satisfies system (7.18), where (cf. (4.42)) ⎞ ⎞ ⎛ ⎛ 0 1 0 0 0 1 B1 := ⎝ −1 a + b p + q ⎠ , B2 := ⎝ 0 −p − q c + d ⎠ . 0 −p − q c + d 1 −c − d g + h Relations (7.25) imply that the matrices B1 and B2 commute, and therefore the solution of (7.18) is given by formula (7.19). Next, we observe that the matrices B1 and B2 are H  -symmetric, where ⎞ ⎛ 1 0 0 H  := ⎝ 0 −1 0 ⎠ . 0 0 1 Since χH  = 1, we will consider four cases analogous to cases (i.a)–(i.d) above and argue as in cases (i)–(iv) of Section 5.1, respectively. Case (ii.a). Suppose that all eigenvalues of B1 , B2 are real and that both these matrices are diagonalizable. Arguing as in case (i) in Section 5.1, we obtain that the function Q has the form (7.20), and (7.15), (7.29), (7.31), (7.32) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (ii.b). Suppose that one of B1 , B2 has non-real eigenvalues. Arguing as in case (ii) in Section 5.1, we obtain that the function Q has the form (7.21). In this case (7.15), (7.29), (7.31), (7.32) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (ii.c). Suppose that all eigenvalues of B1 , B2 are real, that for one of these matrices the Jordan normal form contains a 2 × 2-cell, and that the Jordan normal form of the other matrix does not contain a 3 × 3-cell. Arguing as in case (iii) in Section 5.1, we obtain that the function Q has the form (7.22). In this case (7.15), (7.29), (7.31), (7.32) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of class (A). Case (ii.d). Suppose that all eigenvalues of B1 , B2 are real and that the Jordan normal form of one of these matrices contains a 3 × 3-cell. Arguing as in case (iv) in Section 5.1, we obtain that the function Q has the form (7.23). In this case (7.15), (7.29), (7.31), (7.32) imply that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D).

7.3 Defining Systems of Type III

149

Case (iii). Now, suppose N + K = 1. We then have L = 1 and ⎛ ⎜ ⎜ D=⎜ ⎝

T (τ )

0 0 ..

0

.

⎞ ⎟ ⎟ ⎟, ⎠

0

where τ is either 1 or −1. The matrices Λα commute and are H  -symmetric, where    0 H  (7.33) H := 0 In−4 (see (4.31)). Since χH  = 1, we will consider three cases analogous to cases (ii.a)– (ii.c) in Section 6.3. Case (iii.a). Suppose that all eigenvalues of every matrix Λα are real and that all these matrices are diagonalizable. Simultaneously diagonalizing all matrices Λα as was done in case (ii.a) in Section 6.3 and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the function ˆ = D for which the ˆ F(x) := F(Cx) satisfies a defining system of type III with D ˆ corresponding matrices Cα have the form ⎛ ⎞ a b p q t5 . . . tn ⎜ −b a + 2b p q t5 . . . tn ⎟ ⎜ ⎟ ⎜ p −p c 0 0 . . . 0 ⎟ ⎜ ⎟ ⎟ ˆ1 =⎜ C ⎜ −q q 0 d 0 . . . 0 ⎟ , ⎜ t5 −t5 0 0 s5 ⎟ 0 ⎜ ⎟ ⎜ .. ⎟ .. .. .. . . ⎝ . . ⎠ . . . tn −tn 0 0 0 sn ⎛ ⎞ b −a − 2b −p −q −t5 . . . −tn ⎜ a + 2b −2a − 3b −p −q −t5 . . . −tn ⎟ ⎜ ⎟ ⎜ −p p −c 0 0 ... 0 ⎟ ⎜ ⎟ −q 0 −d 0 . . . 0 ⎟ ˆ2 =⎜ (7.34) C ⎜ q ⎟, ⎜ −t5 t5 0 0 −s5 0 ⎟ ⎜ ⎟ ⎜ .. ⎟ .. .. .. .. ⎝ . ⎠ . . . . −tn tn 0 0 0 −sn ⎛ ⎛ ⎞ ⎞ p −p c 0 0 . . . 0 q −q 0 −d 0 . . . 0 ⎜ p −p c 0 0 . . . 0 ⎟ ⎜ q −q 0 −d 0 . . . 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ c −c g 0 0 . . . 0 ⎟ ⎜ 0 0 0 0 0 ... 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ d −d 0 h 0 . . . 0 ⎟ ⎟ ˆ ˆ3 =⎜ C ⎜ 0 0 0 0 0 ... 0 ⎟, C ⎟, 4=⎜ ⎜0 0 0 0 0 ⎜0 0 0 0 0 0⎟ 0⎟ ⎜ ⎜ ⎟ ⎟ ⎜ .. .. .. .. ⎜ .. .. .. .. ⎟ ⎟ .. .. ⎝. . . . ⎝. . . . . ⎠ . ⎠ 0 0 0 0

0

0

0 0 0 0

0

0

7 (n − 2, 2)-Spherical Tube Hypersurfaces

150

⎛

tα −tα 0 0 0 . . . 0 sα 0 . . . ⎜ tα −tα 0 0 0 . . . 0 sα 0 . . . ⎜ ⎜ 0 0 0 0 0 ... 0 0 0 ... ⎜ ⎜ 0 0 0 0 0 ... 0 0 0 ... ⎜ ⎜ 0 0 000 ⎜ ⎜ ˆ α = ⎜ ... ... ... ... . . . C 0 ⎜ ⎜0 0 00 0 ⎜ ⎜ s −s 0 0 λα α ⎜ α ⎜0 0 00 0 ⎜ ⎜ . . . . .. ⎝ .. .. .. .. 0 . 0 0 00

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

α = 5, . . . , n,

0

where the parameters satisfy the conditions s2α − (a + b)sα − λα tα + τ = 0, c2 − (a + b)c − pg + τ

= 0,

d 2 − (a + b)d + qh + τ

=0

α = 5, . . . , n, (7.35)

ˆ α the number λα occurs at position α on the diagonal. Dropping and in the matrix C hats we get the following defining system: n

F11 = 2τ F1 (F1 + F2 ) + aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = −τ (F12 − F22 ) + bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −2τ F2 (F1 + F2 ) − (a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, F33 = cF1 + cF2 + gF3 + 1,

γ =5

F34 = 0, F44 = −dF1 − dF2 + hF4 − 1, F13 = −F23 = τ F3 (F1 + F2 ) + pF1 + pF2 + cF3, F14 = −F24 = τ F4 (F1 + F2 ) + qF1 + qF2 + dF4, F1α = −F2α = τ Fα (F1 + F2) + tα (F1 + F2 ) + sα Fα ,

α = 5, . . . , n,

F3α = 0,

α = 5, . . . , n,

F4α = 0,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + λα Fα + 1,

α = 5, . . . , n,

Fαβ = 0,

α , β = 5, . . . , n, α = β ,

F(0) = 0,

Fα (0) = 0,

α = 1, . . . , n.

7.3 Defining Systems of Type III

151

Our analysis of the above system is similar to that of the system that appears in case (i) in Section 6.3. Define λ3 := g, s3 := c, t3 := p, λ4 := h, s4 := d, t4 := q and let I and J be the subsets of the index set {3, . . . , n} such that λα = 0 for α ∈ I and λα = 0 for α ∈ J . As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13), the function Fˆ by formula (6.14), and the function Ψ by formula (6.15). The function Ψ depends only on v and on some neighborhood of the origin satisfies equation (6.16), which can be solved by means of (6.17), (6.18). The defining system yields (cf. (4.45))

∂ 2 Fˆ ∂ Fˆ = λα + (sα Ψ + 1)Hαα , α = 3, . . . , n, ∂ x2α ∂ xα ∂ 2 Fˆ = 0, α , β = 3, . . . , n, ∂ xα ∂ xβ

(7.36)

α = β ,

and therefore we obtain (cf. (4.49))

∂ Fˆ = (sα Ψ + 1)Hαα xα + Φα (v), α ∈I, ∂ xα (sα Ψ + 1)Hαα eλα xα Hαα ∂ Fˆ =− + eλα xα Φα (v) + , α ∈J ∂ xα λα λα

(7.37)

for some functions Φα with Φα (0) = 0, α = 3, . . . , n. Next, the defining system implies identities (6.21). Plugging identities (7.37) into both sides of (6.21) and using (6.16), (7.35), we obtain the equations (cf. (4.53))

α ∈I, (Φα ) = (τΨ + sα )Φα + tαΨ ,   Hαα (Φα ) = (τΨ + sα ) Φα + , α ∈J, λα Φα (0) = 0, α = 3, . . . , n.

(7.38)

Further, (7.37) yields (cf. (4.57), (4.58))

Ψ (sα Ψ + 1)Hαα 2 xα + ∑ Φα xα − Fˆ = u + R(v) + ∑ 2 2 α ∈I α ∈I  (sα Ψ + 1)Hαα eλα xα Hαα λα xα e x + Φ + − 1 α α ∑ ∑ ∑ 2 λα α ∈J α ∈J λα α ∈J λα

(7.39)

for some function R with R(0) = 0. Plugging (7.39) into the equations for the function Fˆ that arise from the defining system and using (6.16), (7.35), (7.38), we obtain (cf. (4.59)) R = (2τΨ + a + b)R + ∑ Hαα tα Φα −  α ∈I tα sα tα a + 3b +∑ Ψ−∑ , 2 λ λ α α ∈J α ∈J α

7 (n − 2, 2)-Spherical Tube Hypersurfaces

152

R (0) = − ∑

R(0) = 0,

α ∈J

sα Hαα . λα2

Let μ1 , μ2 be the roots of the characteristic polynomial μ 2 −(a+b)μ + τ of linear differential equation (6.17). As in case (i) in Section 6.3, we need to consider three situations. In each of these situations the functions Ψ , Φα , R are found as in Section 6.3. Therefore, we omit details and only state the final results. If μ1 , μ2 are real and μ1 = μ2 , then the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A), if μ1 = μ2 = (a + b)/2, then the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D), if μ1 , μ2 are not real, then the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (iii.b). Suppose that for some α0 the matrix Λα0 has non-real eigenvalues. Arguing as in case (ii.b) in Section 6.3 and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the function ˆ = D for which the ˆ F(x) := F(Cx) satisfies a defining system of type III with D ˆ corresponding matrices Cα have the form ⎛

a b p ⎜ −b a + 2b p ⎜ ⎜ p −p c ⎜ ⎜ ˆ C1 = ⎜ −q q −d ⎜ t5 −t5 0 ⎜ ⎜ .. .. .. ⎝ . . . −tn

tn

q q d c 0 .. .

t5 t5 0 0 s5

0 0

0

⎛

b −a − 2b ⎜ a + 2b −2a − 3b ⎜ ⎜ −p p ⎜ ⎜ q −q ˆ C2 = ⎜ ⎜ −t5 t5 ⎜ ⎜ .. .. ⎝ . . −tn ⎛

p ⎜ p ⎜ ⎜ c ⎜ ˆ3 =⎜ C ⎜ −d ⎜ 0 ⎜ ⎜ .. ⎝ . 0

tn

−p −p −c d 0 .. . 0

c c g −h 0 .. .

d d h g 0 .. .

0 0 0 0 0

0 0

0

⎞ tn tn ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ ⎟ .. . ⎠ sn

... ... ... ...

−q −t5 . . . −q −t5 . . . −d 0 . . . −c 0 . . . 0 −s5 .. .. . . 0 0 0

−p −p −c d 0 .. .

⎞ −tn −tn ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ ⎟ ⎠ −sn

⎛ ⎞ 0 q ⎜q 0⎟ ⎜ ⎟ ⎜d 0⎟ ⎜ ⎟ ⎜ 0⎟ ˆ = , C ⎟ 4 ⎜c ⎜0 ⎟ 0⎟ ⎜ ⎜ .. ⎟ .. ⎝. . ⎠ 0 0

... ... ... ...

(7.40)

−q −q −d −c 0 .. .

d d h g 0 .. .

−c −c −g h 0 .. .

0 0 0

0 0 0 0 0

0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ ⎟ .. . ⎠ 0

... ... ... ...

7.3 Defining Systems of Type III

⎛

153

tα −tα 0 0 0 . . . 0 sα 0 . . . ⎜ tα −tα 0 0 0 . . . 0 sα 0 . . . ⎜ ⎜ 0 0 0 0 0 ... 0 0 0 ... ⎜ ⎜ 0 0 0 0 0 ... 0 0 0 ... ⎜ ⎜ 0 0 000 ⎜ ⎜ ˆ α = ⎜ ... ... ... ... . . . C 0 ⎜ ⎜0 0 00 0 ⎜ ⎜ s −s 0 0 λα α ⎜ α ⎜0 0 00 0 ⎜ ⎜ . . . . .. ⎝ .. .. .. .. 0 . 0 0 00

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

α = 5, . . . , n,

0

where the parameters satisfy the conditions s2α − (a + b)sα − λα tα + τ = 0,

α = 5, . . . , n,

c2 − d 2 − (a + b)c − pg + qh + τ = 0,

(7.41)

(a + b)d − 2cd + qg + ph = 0 ˆ α the number λα occurs at position and d 2 + g2 + h2 > 0. As before, in the matrix C α on the diagonal. Dropping hats we get the following defining system: n

F11 = 2τ F1 (F1 + F2 ) + aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = −τ (F12 − F22 ) + bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −2τ F2 (F1 + F2 ) − (a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, F33 = −F44 = cF1 + cF2 + gF3 − hF4 + 1,

γ =5

F34 = dF1 + dF2 + hF3 + gF4 , F13 = −F23 = τ F3 (F1 + F2 ) + pF1 + pF2 + cF3 − dF4, F14 = −F24 = τ F4 (F1 + F2 ) + qF1 + qF2 + dF3 + cF4 , F1α = −F2α = τ Fα (F1 + F2) + tα (F1 + F2 ) + sα Fα ,

α = 5, . . . , n,

F3α = 0,

α = 5, . . . , n,

F4α = 0,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + λα Fα + 1,

α = 5, . . . , n,

Fαβ = 0,

α , β = 5, . . . , n, α = β ,

F(0) = 0,

Fα (0) = 0,

α = 1, . . . , n.

7 (n − 2, 2)-Spherical Tube Hypersurfaces

154

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13), the function Fˆ by formula (6.14), and the function Ψ by formula (6.15). The function Ψ depends only on v and on some neighborhood of the origin satisfies equation (6.16), which can be solved by means of (6.17), (6.18). The defining system yields

∂ 2 Fˆ ∂ Fˆ = λα + sα Ψ + 1, α = 5, . . . , n, ∂ x2α ∂ xα ∂ 2 Fˆ = 0, ∂ xα ∂ xβ

α , β = 5, . . . , n,

(7.42)

α = β ,

and therefore we obtain

∂ Fˆ = (sα Ψ + 1)xα + Φα (v), ∂ xα

α = 5, . . . , N ,

sα Ψ + 1 eλα xα ∂ Fˆ =− + eλα xα Φα (v) + , α = N + 1, . . . , n ∂ xα λα λα

(7.43)

for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies

∂ 2 Fˆ ∂ Fˆ = (τΨ + sα ) + tα Ψ , ∂ v∂ xα ∂ xα

α = 5, . . . , n.

(7.44)

Plugging identities (7.43) into both sides of (7.44) and using (6.16), (7.41), we obtain the equations (Φα ) = (τΨ + sα )Φα + tαΨ , α = 5, . . . , N ,   1 (Φα ) = (τΨ + sα ) Φα + , α = N + 1, . . ., n, λα

Φα (0) = 0,

(7.45)

α = 5, . . . , n.

Further, (7.43) yields N

N

Ψ ˆ x3 , x4 ) + ∑ sα Ψ + 1 x2α + ∑ Φα xα − Fˆ = u + R(v, 2 2 α =5 α =5  n n sα Ψ + 1 eλα xα 1 λ α xα xα + ∑ Φα + ∑ − 1 e ∑ 2 λα α =N +1 α =N +1 λα α =N +1 λα n

ˆ with R(0) ˆ for some function R, = 0. Set

(7.46)

7.3 Defining Systems of Type III

155

u˜ := x3 + x4, v˜ := x3 − x4.

(7.47)

ˆ (u˜ + v)/2, Let R˜ := R(v, ˜ (u˜ − v)/2) ˜ be the function obtained by expressing Rˆ in the variables v, u, ˜ v. ˜ Plugging (7.46) into the equations for the function Fˆ that arise from the defining system and using (6.16), (7.41), (7.45), we get ∂ 2 R˜ ∂ R˜ ∂ R˜ ∂ R˜ = (2τΨ + a + b) + (p − q) + (p + q) + 2 ∂v ∂ u˜  ∂ v˜  ∂v N n n a + 3b tα sα tα ∑ tα Φα − 2 + ∑ λα Ψ − ∑ λα , α =5 α =N +1 α =N +1

∂ 2 R˜ g + h ∂ R˜ g − h ∂ R˜ d − + Ψ, = ∂ u˜2 2 ∂ u˜ 2 ∂ v˜ 2 g + h ∂ R˜ g − h ∂ R˜ d ∂ 2 R˜ + − Ψ, =− 2 ∂ v˜ 2 ∂ u˜ 2 ∂ v˜ 2 ∂ 2 R˜ ∂ R˜ ∂ R˜ p + q Ψ, = (τΨ + c) +d + ∂ v∂ u˜ ∂ u˜ ∂ v˜ 2 ∂ 2 R˜ ∂ R˜ ∂ R˜ p − q = −d + (τΨ + c) + Ψ, ∂ v∂ v˜ ∂ u˜ ∂ v˜ 2 g − h ∂ R˜ g + h ∂ R˜ c 1 ∂ 2 R˜ = + + Ψ+ , ∂ u˜∂ v˜ 2 ∂ u˜ 2 ∂ v˜ 2 2 n ∂ R˜ sα ∂ R˜ ˜ R(0) = 0, (0) = − ∑ (0) = 0, , 2 ∂v ∂ u˜ α =N +1 λα

(7.48)

∂ R˜ (0) = 0. ∂ v˜

Observe that the functions Ψ , Φα are found as in case (i) of Section 6.3, so we will ˜ We will consider two situations. only describe the form of R. Case (iii.b.1). First, suppose g = h = 0. In this case (7.41) implies τ = 1, c = (a + b)/2, (a + b)2 /4 + d 2 = 1. Therefore, the roots of the characteristic polynomial μ 2 − (a + b)μ + τ of linear differential equation (6.17) are not real, and the functions Ψ , Φα are found as in case (i.c) of Section 6.3. The solution of system (7.48) can be written as follows: dΨ 2 (a + b)Ψ + 2 (u˜ − v˜2 ) + u˜v˜ + B(v)u˜ + D(v)v˜ + R(v), R˜ = 4 4 where B, D, R are found from the system 2Ψ + a + b p+q Ψ, B + dD + 2 2 p−q 2Ψ + a + b D = −dB + D+ Ψ, 2 2   R = (2Ψ + a + b)R + (p − q)B + (p + q)D+   N n n tα sα tα a + 3b ∑ tα Φα − 2 + ∑ λα Ψ − ∑ λα , α =5 α =N +1 α =N +1

B =

7 (n − 2, 2)-Spherical Tube Hypersurfaces

156

B(0) = 0,

D(0) = 0,

n

sα . 2 α =N +1 λα

R (0) = − ∑

R(0) = 0,

The above system yields ˜ v) ˜ cos dv + Q2(v, u, ˜ v) ˜ sin dv Q1 (v, u, R˜ = , a+b d cosdv − 2 sin dv where Q j are polynomials of degree at most 2. Now representation (7.46) implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (iii.b.2). Now, suppose g2 + h2 > 0. In this case the solution of system (7.48) can be written as follows:  (cη + d ρ )Ψ + η

˜ v) ˜ − u− ˜ R˜ = 2 Re B(v)e(ρ −iη )(u+i 2(ρ 2 + η 2 ) (cρ − d η )Ψ + ρ v˜ + R(v), 2(ρ 2 + η 2 )

(7.49)

where ρ := (g + h)/2, η := (g − h)/2, the function B is found from the equation B = (τΨ + c + id)B,

B(0) =

i(η − iρ )2 , 4(ρ 2 + η 2 )2

and the function R is found from the equation R = (2τΨ + a + b)R + 

N

∑ tα Φα −

α =5

n tα sα a + 3b (p − q)(cη + d ρ ) + (p + q)(cρ − d η ) + + ∑ 2 2 2 2(ρ + η ) α =N +1 λα



Ψ−

(p − q)η + (p + q)ρ tα − , 2(ρ 2 + η 2 ) α =N +1 λα n

∑

R(0) = −

ρη (ρ 2 + η 2 )2

,

sα ρ (cη + d ρ ) + η (cρ − d η ) − . 2 λ 2(ρ 2 + η 2 )2 α =N +1 α n

R (0) = − ∑

Let μ1 , μ2 be the roots of the characteristic polynomial μ 2 − (a + b)μ + τ of linear differential equation (6.17). As in case (i) in Section 6.3, we need to consider three situations. Case (iii.b.2.1). Suppose that μ1 , μ2 are real and μ1 = μ2 . Then we have R˜ =

 1 ˜ v)e ˜ μ1 v + Q2 (v, u, ˜ v)e ˜ μ2 v + Q1 (v, u, μ1 eμ2 v − μ2 eμ1 v

  ˜ η v˜ A1 sin(dv − η u˜ + ρ v) ˜ + A2 cos(dv − η u˜ + ρ v) ˜ ecv+ρ u+

7.3 Defining Systems of Type III

157

for some real numbers A j and polynomials Q j of degree at most 2. Representation (7.46) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iii.b.2.2). Suppose μ1 = μ2 = (a + b)/2. Then τ = 1, and we have R˜ =

 1 P(v, u, ˜ v)+ ˜ 2 − (a + b)v 

 ˜ η v˜ A1 sin(dv − η u˜ + ρ v) ˜ + A2 cos(dv − η u˜ + ρ v) ˜ e(c−(a+b)/2)v+ρ u+

for some real numbers A j and a polynomial P of degree at most 4. Representation (7.46) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iii.b.2.3). Suppose that μ1 , μ2 are not real. Then τ = 1, μ1 = (a + b)/2 + iδ , μ2 = (a + b)/2 − iδ , where δ 2 = 1 − (a + b)2/4. In this case we obtain R˜ =

 1 L1 (v, u, ˜ v) ˜ cos δ v + L2(v, u, ˜ v) ˜ sin δ v+ δ cos δ v − a+b 2 sin δ v  

˜ η v˜ ˜ + A2 cos(dv − η u˜ + ρ v) ˜ e(c−(a+b)/2)v+ρ u+ A1 sin(dv − η u˜ + ρ v)

for some real numbers A j and affine functions L j . Representation (7.46) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (iii.c). Suppose that all eigenvalues of every matrix Λα are real and that for some α0 the Jordan normal form of the matrix Λα0 contains either a 2 × 2- or a 3 × 3-cell. Arguing as in case (ii.c) in Section 6.3 and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the ˆ = D for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ the corresponding matrices Cα have the form ⎛ ⎞ a b p q t5 . . . tn ⎜ −b a + 2b p q t5 . . . tn ⎟ ⎜ ⎟ ⎜ p −p c d r 5 . . . rn ⎟ ⎜ ⎟ ⎟ ˆ1 =⎜ C ⎜ −q q −d c + 2d r5 . . . rn ⎟ , ⎜ t5 −t5 r5 −r5 s5 ⎟ 0 ⎜ ⎟ ⎜ .. ⎟ .. .. .. . . ⎝ . . ⎠ . . . tn −tn rn −rn 0 sn ⎛ ⎞ b −a − 2b −p −q −t5 . . . −tn ⎜ a + 2b −2a − 3b −p −q −t5 . . . −tn ⎟ ⎜ ⎟ ⎜ −p p −c −d −r5 . . . −rn ⎟ ⎜ ⎟ −q d −c − 2d −r5 . . . −rn ⎟ ˆ2 =⎜ C ⎜ q ⎟, ⎜ −t5 t5 −r5 r5 −s5 0 ⎟ ⎜ ⎟ ⎜ .. ⎟ .. .. .. .. ⎝ . ⎠ . . . . −tn

tn

−rn

rn

0

−sn

7 (n − 2, 2)-Spherical Tube Hypersurfaces

158

⎛

p ⎜ p ⎜ ⎜ c ⎜ ˆ3 =⎜ C ⎜ −d ⎜ r5 ⎜ ⎜ .. ⎝ . rn

−p −p −c d −r5 .. .

c d c d g h −h g + 2h ρ5 − ρ5 .. .. . . −rn ρn −ρn

r5 r5 ρ5 ρ5 σ5

0

⎛

q −q d ⎜ q −q d ⎜ ⎜ d −d h ⎜ ⎜ c + 2d −c − 2d g + 2h ˆ C4 = ⎜ ⎜ −r5 r5 − ρ5 ⎜ ⎜ .. .. .. ⎝ . . . −rn rn − ρn ⎛

tα ⎜ tα ⎜ ⎜ rα ⎜ ⎜ rα ⎜ ⎜0 ⎜ ⎜ . . ˆ Cα = ⎜ ⎜ . ⎜0 ⎜ ⎜s ⎜ α ⎜0 ⎜ ⎜ . ⎝ .. 0

−tα −tα −rα −rα

rα rα ρα ρα

0 .. .

0 .. .

0 −sα 0 .. .

0 σα 0 .. .

0

0

⎞ rn rn ⎟ ⎟ ρn ⎟ ⎟ ρn ⎟ ⎟, 0⎟ ⎟ ⎟ .. ⎠ . σn

... ... ... ...

−c − 2d −c − 2d −g − 2h −2g − 3h ρ5 .. .

−r5 −r5 − ρ5 − ρ5 −σ5

ρn

0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0 0 ⎟ ⎟ .. .. . . 0 ⎟ ⎟, ⎟ 0 0 ⎟ ⎟ −σα λα ⎟ ⎟ 0 0 ⎟ .. .. ⎟ . ⎠ 0 . 0 0 −rα −rα − ρα − ρα

0 0 0 0

... ... ... ...

0 sα 0 sα 0 σα 0 σα

0 0 0 0

⎞ −rn −rn ⎟ ⎟ −ρn ⎟ ⎟ −ρn ⎟ ⎟, 0 ⎟ ⎟ ⎟ .. ⎠ . −σn

... ... ... ...

(7.50)

... ... ... ...

α = 5, . . . , n,

where the parameters satisfy the conditions s2α − (a + b)sα − (p + q)σα − λα tα + τ = 0,

α = 5, . . . , n,

sα σα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2 − (g + h)σα − λα ρα = 0,

α = 5, . . . , n,

rα sα − tα σα − (p + q)ρα + (c + d − a − b)rα = 0, α = 5, . . . , n,

ρα sα − rα σα − (c + d)ρα + (g + h)rα = 0,

(7.51)

α = 5, . . . , n,

n

∑ (tα ρα − rα2 ) + 2cd + 2d 2 − ad − bd − ph − gq − 2hq = 0,

α =5

(c + d)(a + b − c − d) + (p + q)(g + h) − τ = 0 ˆ α the number λα occurs at position α on the diagonal. Dropping and in the matrix C hats we get the following defining system:

7.3 Defining Systems of Type III

159 n

F11 = 2τ F1 (F1 + F2 ) + aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = −τ (F12 − F22 ) + bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −2τ F2 (F1 + F2) − (a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, γ =5

n

F33 = cF1 + cF2 + gF3 − hF4 + ∑ ργ Fγ + 1, γ =5

n

F34 = dF1 + dF2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

n

F44 = −(c + 2d)F1 − (c + 2d)F2 − (g + 2h)F3 − (2g + 3h)F4 + ∑ ργ Fγ − 1, n

γ =5

F13 = −F23 = τ F3 (F1 + F2 ) + pF1 + pF2 + cF3 − dF4 + ∑ rγ Fγ , γ =5

n

F14 = −F24 = τ F4 (F1 + F2 ) + qF1 + qF2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

F1α = −F2α = τ Fα (F1 + F2) + tα (F1 + F2 ) + rα (F3 + F4) + sα Fα ,

α = 5, . . . , n,

F3α = −F4α = rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα ,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + σα (F3 + F4) + λα Fα + 1,

α = 5, . . . , n, α , β = 5, . . . , n,

Fαβ = 0,

α = β ,

α = 1, . . . , n. Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13) and the function Ψ by formula (6.15). The function Ψ depends only on v and satisfies equation (6.16), which can be solved by means of (6.17), (6.18). Further, we introduce the variables u˜ and v˜ by formulas (7.47). Expressing F in the variables u, u, ˜ v, v, ˜ x5 , . . . , xn , we obtain the function F(0) = 0,

Fα (0) = 0,

F˜ := F ((u + v)/2, (u − v)/2, (u˜ + v) ˜ /2, (u˜ − v) ˜ /2, x5 , . . . , xn ) . Next, let

˜ ∂ u. Ψ˜ := 2∂ F/ ˜

(7.52) (7.53)

The defining system implies that Ψ˜ depends only on v, v˜ and near the origin satisfies the system of equations

∂ Ψ˜ = τΨ Ψ˜ + (p + q)Ψ + (c + d)Ψ˜ , ∂v ∂ Ψ˜ = (c + d)Ψ + (g + h)Ψ˜ + 1, ∂ v˜ Ψ˜ (0) = 0. Further, the defining system yields

(7.54)

7 (n − 2, 2)-Spherical Tube Hypersurfaces

160

∂ 2 F˜ ∂ F˜ = λα + sα Ψ + σαΨ˜ + 1, α = 5, . . . , n, 2 ∂ xα ∂ xα ∂ 2 F˜ = 0, ∂ xα ∂ xβ

α , β = 5, . . . , n,

(7.55)

α = β ,

and therefore we obtain

∂ F˜ = (sα Ψ + σαΨ˜ + 1)xα + Φα (v, v), ˜ ∂ xα

α = 5, . . . , N ,

sα Ψ + σα Ψ˜ + 1 eλα xα ∂ F˜ =− + eλα xα Φα (v, v) ˜ + , α = N + 1, . . . , n ∂ xα λα λα

(7.56)

for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies

∂ 2 F˜ ∂ F˜ = (τΨ + sα ) + tαΨ + rα Ψ˜ , α = 5, . . . , n, ∂ v∂ xα ∂ xα ∂ 2 F˜ ∂ F˜ = σα + rαΨ + ραΨ˜ , ∂ v˜∂ xα ∂ xα

(7.57)

α = 5, . . . , n.

Plugging identities (7.56) into both sides of (7.57) and using (6.16), (7.51), (7.54), we obtain the equations

∂ Φα = (τΨ + sα )Φα + tαΨ + rα Ψ˜ , α ∂v ∂ Φα = σα Φα + rαΨ + ραΨ˜ , α ∂ v˜   1 ∂ Φα α , = (τΨ + sα ) Φα + ∂v λα   ∂ Φα 1 , = σα Φα + α ∂ v˜ λα Φα (0) = 0,

= 5, . . . , N , = 5, . . . , N , = N + 1, . . . , n,

(7.58)

= N + 1, . . . , n,

α = 5, . . . , n.

Further, (7.56) yields N N Ψ Ψ˜ sα Ψ + σαΨ˜ + 1 2 F˜ = u + u˜ + R(v, v) ˜ +∑ xα + ∑ Φα xα − 2 2 2 α =5 α =5

 n n sα Ψ + σαΨ˜ + 1 eλα xα 1 λα xα xα + ∑ Φα + ∑ −1 e ∑ 2 λα α =N +1 α =N +1 λα α =N +1 λα n

(7.59)

for some function R with R(0) = 0. Plugging (7.59) into the equations for the function F˜ that arise from the defining system and using (6.16), (7.51), (7.54), (7.58),

7.3 Defining Systems of Type III

161

we obtain

∂ 2R ∂R ∂R N = (2 τΨ + a + b) tα Φα − + (p + q) + ∂ v2 ∂v ∂ v˜ α∑ =5     n n n a + 3b p−q tα sα tα σα ˜ tα +∑ +∑ Ψ− − Ψ−∑ , 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α ∂ 2R ∂R N = (τΨ + c + d) + rα Φα − ∂ v∂ v˜ ∂ v˜ α∑ =5     n n n rα sα rα σα ˜ rα p−q +∑ − Ψ − d+ ∑ Ψ−∑ , 2 α =N +1 λα α =N +1 λα α =N +1 λα

(7.60)

∂ 2R ∂R N = (g + h) ρα Φα − + 2 ∂ v˜ ∂ v˜ α∑ =5     n n n g + 3h ρα sα ρα σα ˜ ρα +∑ d+ ∑ Ψ− Ψ−∑ , λ 2 λ λ α α α =N +1 α =N +1 α =N +1 α R(0) = 0,

n ∂R sα , (0) = − ∑ 2 ∂v α =N +1 λα

n ∂R σα . (0) = − ∑ 2 ∂ v˜ α =N +1 λα

In order to determine F˜ from representation (7.59), we first need to find the functions Ψ , Ψ˜ from (6.16), (7.54), then the functions Φα from system (7.58), and finally the function R from system (7.60). Let μ1 , μ2 be the roots of the characteristic polynomial μ 2 − (a + b)μ + τ of linear differential equation (6.17). As in case (i) in Section 6.3, we need to consider three situations. In each of these situations the function Ψ is found as in Section 6.3. We will now describe the form of Ψ˜ . In the first two situations we will have to distinguish the following two cases: g + h = 0 and g + h = 0. Case (iii.c.1). Suppose that μ1 , μ2 are real and μ1 = μ2 . Case (iii.c.1.1). First, assume g + h = 0. It follows from (7.51) that in this case c + d is a root of μ 2 − (a + b)μ + τ , so we set μ1 := c + d. Then we have 

 1 p + q  μ1 v p + q μ2 v ˜ Ψ= e . (p + q)v + (μ1 − μ2 )v˜ − e + μ1 eμ2 v − μ2 eμ1 v μ1 − μ2 μ1 − μ2 Now a straightforward (but lengthy) calculation utilizing (7.58), (7.59), (7.60) yields that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iii.c.1.2). Now, assume g + h = 0. Then we have

Ψ˜ =

1 (g + h)(μ1 eμ2 v − μ2 eμ1 v )

 (μ1 − μ2 )e(c+d)v+(g+h)v˜+  (c + d − μ1 )eμ2 v − (c + d − μ2 )eμ1 v .

162

7 (n − 2, 2)-Spherical Tube Hypersurfaces

Now formulas (7.58), (7.59), (7.60) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iii.c.2). Suppose μ1 = μ2 = (a + b)/2. In this case τ = 1. Case (iii.c.2.1). First, assume g + h = 0. Then we have (p + q)v2 + 2v˜ Ψ˜ = . 2 − (a + b)v Now formulas (7.58), (7.59), (7.60) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D). Case (iii.c.2.2). Now, assume g + h = 0. Then we have

Ψ˜ =

 

1 (a + b − 2(c + d))v + 2 e(c+d−(a+b)/2)v+(g+h)v˜ − 1 . (g + h)(2 − (a + b)v)

Now formulas (7.58), (7.59), (7.60) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iii.c.3). Suppose that μ1 , μ2 are not real. Then τ = 1, μ1 = (a + b)/2 + iδ , μ2 = (a + b)/2 − iδ , where δ 2 = 1 − (a + b)2 /4. In this situation conditions (7.51) imply g + h = 0, and we obtain

Ψ˜ =

 1 δ e(c+d−(a+b)/2)v+(g+h)v˜−  (g + h)(δ cos δ v − a+b sin δ v) 2 (c + d − (a + b)/2) sin δ v − δ cos δ v .

Now formulas (7.58), (7.59), (7.60) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Case (iv). Let N = K = 0. In this case L = 2, D = 0, and (iv) of Lemma 3.2 implies that the matrices Cα pairwise commute for α = 1, . . . , n. Further, by (ii) of Lemma 3.2 these matrices are H-symmetric. Since χH = 2, we obtain by Propositions 4.2, 4.3, 6.1 and Remarks 4.1, 6.1 that for every α = 1, . . . , n there exists Cα ∈ GL(n, R) such that either Cα−1 Cα Cα is diagonal and we have ⎛ ⎞ In−4 0 ⎠, H (7.61) CαT HCα = ⎝ 0 H or identity (6.38) holds, where Xα1 is a diagonal matrix and Xα2 is a matrix of a special form of size κα × κα with 2 ≤ κα ≤ n. All possibilities for Xα2 are listed below along with the corresponding matrices CαT HCα . We split these possibilities into six types (cf. (1)–(6) in case (ii.c) in Section 6.4). Type 1. Any 2 × 2 matrix of the form ηα I2 + S(δα ) having two non-real mutually conjugate eigenvalues ηα ± iδα . Here (7.61) holds.

7.3 Defining Systems of Type III

163

Type 2. Any 4 × 4 matrix of the form  ηα I2 + S(δα )

0



ηα I2 + S(δα )

0

having non-real eigenvalues ηα ± iδα and ηα ± iδα . Here (7.61) holds. Type 3. Any 4 × 4 matrix of the form 

ηα I4 +

aα I2 + S(δα − bα ) −bα I2 + S(−aα )



−aα I2 + S(δα + bα )

−bα I2 + S(−aα )

having two non-real mutually conjugate eigenvalues ηα ± iδα , where a2α + b2α > 0. Here (7.61) holds. Type 4. Any matrix of one of the following forms: ⎛ ⎞ ξα Iκα −4 0 ⎜ ⎟ 2,ξ ⎝ ⎠ with κα ≥ 4, Rτ α 0 ηα I2 + S(δα ) ⎛ ⎝

ξα Iκα −5

R3,ξα

⎞

0 ηα I2 + S(δα )

0

⎠ with κα ≥ 5

having two non-real mutually conjugate eigenvalues ηα ± iδα and a real eigenvalue ξα distinct from every eigenvalue of Xα1 , where τ = ±1 and the matrices Rτk,λ for even k and Rk,λ for odd k are defined in Proposition 6.1. Here (7.61) holds. Type 5. Any matrix of one of the following forms:   0 0 with k = 2, 4, ξα Iκα + 0 Rk,0 τ 

ξα Iκα +

0 0 0 Rk,0

 with k = 3, 5,

having a real eigenvalue ξα distinct from every eigenvalue of Xα1 , where κα ≥ k, τ = ±1. Here for k = κα = 2 identity (7.61) holds, for k = 2, κα ≥ 3 either (7.61) holds or we have ⎞ ⎛ 0 In−κα −2 ⎟ ⎜ H ⎟, CαT HCα = ⎜ (7.62) ⎠ ⎝ Iκα −2 0 H for k = κα = 3 either identity (7.62) holds or we have

7 (n − 2, 2)-Spherical Tube Hypersurfaces

164

⎛ CαT HCα

=⎝

In−4

0

H

−H

0

⎞ ⎠,

(7.63)

for k = 3, κα ≥ 4 either one of (7.62), (7.63) holds or we have ⎞ ⎛ 0 In−4 ⎠, −H CαT HCα = ⎝  0 H and for k = 4, 5 identity (7.63) holds. Type 6. Any matrix of one of the following forms: ⎛ ⎞ ξα Iα −2 0 ⎜ ⎟ 2,ξ Rτ α ⎜ ⎟ (a) ⎜ ⎟ with α , α ≥ 2,  ξα Iα −2 ⎝ ⎠ 2,ξ  0 Rτ  α ⎞ ⎛ ξα Iα −2 0 ⎟ ⎜ 2,ξ ⎟ ⎜ Rτ α (b) ⎜ ⎟ with α ≥ 2, α ≥ 3,  ξα Iα −3 ⎠ ⎝  3, ξ α 0 R ⎛ ⎜ (c) ⎜ ⎝

ξα Iα −3

0

R3,ξα

⎞

0 ξα Iα −3

⎟ ⎟ with α ,  ≥ 3 α ⎠ 

R3,ξα

having real eigenvalues ξα , ξα distinct from every eigenvalue of Xα1 , where τ , τ  = ±1. In this case we have ⎛ ⎞ In−α −2 0 ⎜ ⎟ H ⎟. CαT HCα = ⎜ ⎝ ⎠ I −2

0

α

H

Case (iv.a). Suppose that all eigenvalues of every matrix Cα are real and that all these matrices are diagonalizable. Arguing as in case (ii.a) in Section 6.3 (cf. Section 5.2), we obtain that one can find C ∈ GL(n, R) such that the function ˆ F(x) := F(Cx) satisfies a defining system of the form Fˆαα = λα Fˆα + 1, Fˆn−1 n−1 = λn−1 Fˆn−1 − 1, Fˆnn = λn Fˆn − 1,

α = 1, . . . , n − 2,

7.3 Defining Systems of Type III

165

α , β = 1, . . . , n,

Fˆαβ = 0, ˆ F(0) = 0,

α = β ,

Fˆα (0) = 0, α = 1, . . . , n,

α . Therefore, the graph x = F(x) extends to a closed hypersurface ˆ αα where λα := C 0 n+1 in R which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.b). Suppose that for some α0 the matrix Xα20 is of either type 2 or type 3. It follows that for α = α0 identities (6.42) hold, where Yα2 are 4 × 4-matrices. Since the symmetric matrices Xα10 , Yα1 , with α = α0 , pairwise commute, they can be simultaneously diagonalized by a real orthogonal transformation, thus we can assume that Cα0 is such that all these matrices are diagonal. Further, since every matrix Yα2 , with α = α0 , is symmetric with respect to matrix (6.10) and commutes with Xα20 , using Proposition 4.1 we see that there exists C ∈ GL(n, R) such that the ˆ = 0 for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ α have the block-diagonal form the corresponding matrices C ⎞ ⎛ να I2 + S(μα ) ρα I2 + S(σα ) ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎜ ρα I2 + S(σα ) ζα I2 + S(πα ) ⎟ ⎜ ˆα =⎜ 5 C ⎟ λα ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ . 0

λαn

β

for some λα , να , μα , ρα , σα , ζα , πα ∈ R, α = 1, . . . , n, β = 5, . . . , n. From (i) of Lemma 3.2 we now obtain ⎛ ⎛ ⎞ ⎞ aI2 + S(−b) cI2 + S(−d) bI2 + S(a) dI2 + S(c) ⎜ ⎜ 0⎟ 0⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ˆ 1 = ⎜cI2 + S(−d) pI2 + S(−q) ˆ 2 = ⎜dI2 + S(c) qI2 + S(p) C ⎟, C ⎟, ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ ⎛

0 cI2 + S(−d)

pI2 + S(−q)

⎜ ⎜ ⎜ ˆ C3 = ⎜ pI2 + S(−q) ⎜ ⎝ ⎛

rI2 + S(−s)

0

0 ⎜ .. ⎜ . 0 ⎜ ⎜ 0 ⎜ ˆα =⎜ C λα ⎜ ⎜ 0 ⎜ ⎜ .. ⎝ 0 .

⎞

0 dI2 + S(c)

qI2 + S(p)

⎜ ⎜ ⎜ ⎟ ˆ ⎟ , C4 = ⎜qI2 + S(p) ⎜ ⎟ ⎝ ⎠

0⎟ ⎟

sI2 + S(r)

0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠ 0

⎛

α = 5, . . . , n,

⎞

0⎟ ⎟

⎟ ⎟, ⎟ ⎠

7 (n − 2, 2)-Spherical Tube Hypersurfaces

166

where the parameters satisfy the condition (d − ic)2 + (q − ip)2 − (d − ic)(s − ir) − (b − ia)(q − ip) = 0

(7.64)

ˆ α the number λα occurs at position α on the diagonal. Hence, and in the matrix C the defining system for Fˆ is as follows: Fˆ11 = −Fˆ22 = aFˆ1 − bFˆ2 + cFˆ3 − d Fˆ4 + 1, Fˆ33 = −Fˆ44 = pFˆ1 − qFˆ2 + rFˆ3 − sFˆ4 + 1, Fˆ12 = bFˆ1 + aFˆ2 + d Fˆ3 + cFˆ4 , Fˆ34 = qFˆ1 + pFˆ2 + sFˆ3 + rFˆ4 , Fˆ13 = −Fˆ24 = cFˆ1 − d Fˆ2 + pFˆ3 − qFˆ4 , Fˆ14 = Fˆ23 = d Fˆ1 + cFˆ2 + qFˆ3 + pFˆ4 , Fˆαα = λα Fˆα + 1, Fˆαβ = 0, ˆ F(0) = 0,

(7.65)

α = 5, . . . , n, α = 5, . . . , n,

Fˆα (0) = 0,

β = 1, . . . , n,

α = β ,

α = 1, . . . , n.

Therefore, the hypersurface x0 = F(x) is affinely equivalent to an open subset of the hypersurface x0 = G(x1 , x2 , x3 , x4 ) +

k

n

α =5

α =k+1

∑ exα + ∑

x2α

(7.66)

for some 4 ≤ k ≤ n, where G satisfies the equations in the first six lines of system (7.65) near the origin. We deal with the system for G in the way we dealt with defining systems of type II in Section 4.3. Namely, we extend G to a function G holomorphic on a neighborhood of the origin in C4 . The function G satisfies a system identical to that for G with the derivatives with respect to xα replaced by the corresponding derivatives with respect to zα . Next, let w1 := iz1 + z2 , w3 := iz3 + z4 , w2 := −iz1 + z2 , w4 := −iz3 + z4 . Expressing G in the variables w1 , w2 , w3 , w4 , we obtain the function   i i 1 1 i i 1 1 Gˆ := G − w1 + w2 , w1 + w2 , − w3 + w4 , w3 + w4 . 2 2 2 2 2 2 2 2 The function Gˆ satisfies

∂ 2 Gˆ ∂ Gˆ ∂ Gˆ 1 = (b − ia) + (d − ic) − , 2 ∂ w1 ∂ w3 2 ∂ w1 ∂ 2 Gˆ ∂ Gˆ ∂ Gˆ = (d − ic) + (q − ip) , ∂ w1 ∂ w3 ∂ w1 ∂ w3

7.3 Defining Systems of Type III

167

∂ 2 Gˆ ∂ Gˆ ∂ Gˆ 1 = (q − ip) + (s − ir) − , 2 ∂ w1 ∂ w3 2 ∂ w3 ∂ 2 Gˆ ∂ Gˆ ∂ Gˆ 1 = (b + ia) + (d + ic) − , 2 ∂ w2 ∂ w4 2 ∂ w2 ∂ 2 Gˆ ∂ Gˆ ∂ Gˆ = (d + ic) + (q + ip) , ∂ w2 ∂ w4 ∂ w2 ∂ w4 ∂ 2 Gˆ ∂ Gˆ ∂ Gˆ 1 = (q + ip) + (s + ir) − , 2 ∂ w2 ∂ w4 2 ∂ w4

(7.67)

∂ 2 Gˆ = 0, ∂ wα ∂ wβ

α = 1, 3, β = 2, 4,

∂ Gˆ Gˆ(0) = 0, (0) = 0, ∂ wα

α = 1, 2, 3, 4.

Then Gˆ = Gˆ1 + Gˆ2 , where Gˆ1 and Gˆ2 are the solutions of the first and second parts of system (7.67), respectively. Therefore, we have G = Gˆ1 (iz1 + z2 , iz3 + z4 ) + Gˆ2 (−iz1 + z2 , −iz3 + z4 ) , hence on a neighborhood of the origin in R4 the following holds: G = Gˆ1 (ix1 + x2 , ix3 + x4 ) + Gˆ2 (−ix1 + x2 , −ix3 + x4 ) = 2 Re Gˆ1 (ix1 + x2 , ix3 + x4 ) .

(7.68)

To solve the first part of system (7.67), we set  A :=

b − ia d − ic d − ic q − ip



 ,

B :=

d − ic q − ip q − ip s − ir

 .

(7.69)

Note that condition (7.64) exactly means that the matrices A and B commute. It is now not hard to see that if each of A and B is diagonalizable, then the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). On the other hand, if the Jordan form of at least one of A, B consists of a 2 × 2-cell, then the solution of the first part of system (7.67) has the form Gˆ1 = L1 (w1 , w3 ) exp(L (w1 , w3 )) + L2 (w1 , w3 ), where L j are affine functions and L is a linear function. Now formulas (7.66), (7.68) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (C). Remark 7.1. Before considering further cases, we will make a useful observation. Suppose that for some α0 the matrix Xα0 is of one of the following types: type

7 (n − 2, 2)-Spherical Tube Hypersurfaces

168

1, type 4, type 5 with k = κα0 = 2, type 5 with k = 2, 3, κα0 ≥ 3 where (7.62) holds, type 6 with ξα0 = ξα 0 . Since the matrices Cα pairwise commute, it follows that for α = α0 identities (6.42) are valid, where Yα2 have size κ × κ with κ := κα0 . Proposition 4.1 and the relations of Lemma 3.2 then imply that the funcˆ = 0 which is split into two ˆ tion F(x) := F(Cα0 x) satisfies a defining system with D subsystems corresponding to the following two groups of variables: {x1 , . . . , xn−κ } and {xn−κ +1, . . . , xn }. In addition, we have  ˆ = H

ˆ1 H

0

0

ˆ2 H

 ,

(7.70)

ˆ 2 has size κ × κ , and ˆ 1 has size (n − κ ) × (n − κ ), the matrix H where the matrix H χHˆ 1 = χHˆ 2 = 1. It then immediately follows from the results obtained in case (ii) in Section 6.3 that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Note that the above observation also applies in the situation where Xα0 is of type 2 with four pairwise distinct eigenvalues, which is included in case (iv.b) considered above. In this situation the matrices A, B defined in (7.69) are diagonalizable. Further, suppose that for some α0 the matrix Xα0 is of type 5 with κα0 = 3 and either k = 2 where (7.61) holds or k = 3 where (7.63) holds. As before, it ˆ is easy to see that the function F(x) := F(Cα0 x) satisfies a defining system with ˆ = 0 which is split into two subsystems corresponding to the groups of variables D ˆ splits into two blocks as {x1 , . . . , xn−3 } and {xn−2 , xn−1 , xn }. In this case the matrix H ˆ 1 is of size (n − 3) × (n − 3) and is positive-definite, H ˆ 2 is of size in (7.70), where H 3 × 3 and has one positive and two negative eigenvalues. In this situation the hypersurface x0 = F(x) is affinely equivalent to an open subset of a hypersurface of the form x0 = −G(x1 , x2 , x3 ) +

m

n

α =4

α =m+1

∑ exα + ∑

x2α ,

3 ≤ m ≤ n,

where G is the solution of a defining system of type III without quadratic terms, corresponding to (2, 1)-spherical tube hypersurfaces in C4 . Systems of this kind were considered in case (ii) of Section 6.3, and we obtain that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D). We will now consider the remaining cases. In particular, if for some α0 the matrix Xα0 is of type 5, it will be assumed that κα0 ≥ 4 and (7.62) cannot not hold. Case (iv.c). Suppose that for some α0 the matrix Xα20 is either of type 6 (a) with τ = τ  , ξα0 = ξα 0 or of type 6 (c) with ξα0 = ξα 0 . It follows that for α = α0 identities (6.42) hold, where Yα2 are κ × κ -matrices with κ := κα0 . As before, since the symmetric matrices Xα10 , Yα1 , with α = α0 , pairwise commute, they can be simultaneously diagonalized by a real orthogonal transformation, thus we can assume that Cα0 is such that all these matrices are diagonal. Further, every matrix Yα2 commutes with Xα20 and is symmetric with respect to the matrix

7.3 Defining Systems of Type III

⎛ ⎜ ⎜ ⎝

169

Iκ −α −2

0

H

0

Iα −2

⎞ ⎟ ⎟. ⎠

(7.71)

H

Using Proposition 4.1 together with the relations of Lemma 3.2, from these two facts ˆ we see that there exists C ∈ GL(n, R) such that the function F(x) := F(Cx) satisfies ˆα ˆ = 0 for which the corresponding matrices C a defining system of type III with D have the form (7.3), where the parameters satisfy the conditions s2α − (a + b)sα − (p + q)σα − λα tα = 0,

α = 5, . . . , n,

sα σα − (p + q)sα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2 − (c + d)sα − (g + h)σα − λα ρα = 0,

α = 5, . . . , n,

rα sα − tα σα + (p + q)(tα − ρα ) + (c + d − a − b)rα = 0, α = 5, . . . , n,

ρα sα − rα σα + (c + d)(tα − ρα ) + (g + h − p − q)rα = 0, α = 5, . . . , n,

(7.72)

n

∑ (tα ρα − rα2 ) + p2 + 4pq + 3q2 + c2 + 4cd + 3d 2−

α =5

ac − 2ad − 2bc − 3bd − gp − 2ph − 2gq − 3hq = 0,

(c + d)(a + b − c − d) + (p + q)(g + h − p − q) = 0. Dropping hats we get the following defining system: n

F11 = aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = bF1 + (a + 2b)F2 + qF3 + (p + 2q)F4 − ∑ tγ Fγ , γ =5

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 − (p + 2q)F3 − (2p + 3q)F4 + ∑ tγ Fγ − 1, γ =5

n

F33 = cF1 − dF2 + gF3 − hF4 + ∑ ργ Fγ + 1, γ =5

n

F34 = dF1 + (c + 2d)F2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

n

F44 = −(c + 2d)F1 − (2c + 3d)F2 − (g + 2h)F3 − (2g + 3h)F4 + ∑ ργ Fγ − 1, γ =5

n

F13 = pF1 − qF2 + cF3 − dF4 + ∑ rγ Fγ , γ =5

n

F14 = F23 = qF1 + (p + 2q)F2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

n

F24 = −(p + 2q)F1 − (2p + 3q)F2 − (c + 2d)F3 − (2c + 3d)F4 + ∑ rγ Fγ , γ =5

F1α = −F2α = tα (F1 + F2 ) + rα (F3 + F4 ) + sα Fα ,

α = 5, . . . , n,

F3α = −F4α = rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα ,

α = 5, . . . , n,

7 (n − 2, 2)-Spherical Tube Hypersurfaces

170

Fαα = sα (F1 + F2 ) + σα (F3 + F4 ) + λα Fα + 1,

α = 5, . . . , n,

Fαβ = 0,

α , β = 5, . . . , n, α = β ,

F(0) = 0,

α = 1, . . . , n.

Fα (0) = 0,

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. We argue analogously to case (i). First, we introduce the variables u1 , u2 , v1 , v2 by formulas (7.5), the function Fˆ by formula (7.6), and the functions Ψ1 , Ψ2 by formulas (7.7). It follows from the defining system that Ψ1 , Ψ2 depend only on v1 , v2 and on some neighborhood of the origin satisfy the equations

∂Ψ1 = (a + b)Ψ1 + (p + q)Ψ2 + 1, ∂ v1 ∂Ψ1 ∂Ψ2 = = (p + q)Ψ1 + (c + d)Ψ2, ∂ v2 ∂ v1

(7.73)

∂Ψ2 = (c + d)Ψ1 + (g + h)Ψ2 + 1, ∂ v2 Ψ1 (0) = 0,

Ψ2 (0) = 0.

Let P(v1 , v2 ) be the function defined by the conditions

∂P = Ψ1 , ∂ v1

∂P = Ψ2 , ∂ v2

P(0) = 0.

(7.74)

The function P satisfies

∂ 2P ∂P ∂P = (a + b) + (p + q) + 1, 2 ∂ v1 ∂ v2 ∂ v1 ∂ 2P ∂P ∂P = (p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ 2P ∂P ∂P = (c + d) + (g + h) + 1, ∂ v1 ∂ v2 ∂ v22 P(0) = 0,

∂P (0) = 0, ∂ v1

(7.75)

∂P (0) = 0. ∂ v2

As in case (i), identities (7.11) hold and lead to identities (7.12) for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies

∂ 2 Fˆ ∂ Fˆ = sα + tαΨ1 + rα Ψ2 , α = 5, . . . , n, ∂ v1 ∂ xα ∂ xα ∂ 2 Fˆ ∂ Fˆ = σα + rα Ψ1 + ραΨ2 , α = 5, . . . , n. ∂ v2 ∂ xα ∂ xα

(7.76)

7.3 Defining Systems of Type III

171

Plugging identities (7.12) into both sides of (7.76) and using (7.72), (7.73), we obtain the equations

∂ Φα = sα Φα + tαΨ1 + rα Ψ2 , α ∂ v1 ∂ Φα = σα Φα + rα Ψ1 + ραΨ2 , α ∂ v2   ∂ Φα 1 , = sα Φα + α ∂ v1 λα   1 ∂ Φα = σα Φα + α , ∂ v2 λα Φα (0) = 0,

= 5, . . . , N , = 5, . . . , N , = N + 1, . . . , n,

(7.77)

= N + 1, . . . , n,

α = 5, . . . , n.

Further, (7.12) yields representation (7.15) for some function R with R(0) = 0. Plugging (7.15) into the equations for the function Fˆ that arise from the defining system and using (7.72), (7.73), (7.77), we obtain N ∂ 2R ∂R ∂R = (a + b) + (p + q) + ∑ tα Φα − 2 ∂ v1 ∂ v2 α =5 ∂ v1     n n n a + 3b p + 3q tα sα tα σα tα Ψ1 − Ψ2 − ∑ , +∑ +∑ 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = (p + q) + (c + d) + ∑ rα Φα − ∂ v1 ∂ v2 ∂ v1 ∂ v2 α =5     n n n rα sα rα σα rα p + 3q c + 3d +∑ +∑ Ψ1 − Ψ2 − ∑ , (7.78) 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = (c + d) + (g + h) + ρα Φα − ∂ v1 ∂ v2 α∑ ∂ v22 =5     n n n ρ α sα ρα σα ρα c + 3d g + 3h +∑ +∑ Ψ1 − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα

R(0) = 0,

n ∂R sα (0) = − ∑ , 2 ∂ v1 λ α =N +1 α

n ∂R σα (0) = − ∑ . 2 ∂ v2 α =N +1 λα

In order to determine Fˆ from representation (7.15), we first need to find the functions Ψ1 , Ψ2 from (7.74), then the functions Φα from system (7.77), and finally the function R from system (7.78). In order to find the functions Ψ1 and Ψ2 , we need to determine the function P from system (7.75). Observe that system (7.75) is a defining system of type III corresponding to strongly pseudoconvex spherical tube hypersurfaces in C3 . Systems of this form were considered in Section 5.2. There are solutions of three kinds, thus we will consider three cases accordingly.

7 (n − 2, 2)-Spherical Tube Hypersurfaces

172

Case (iv.c.1). Suppose first that the function P has the form P = exp(L1 (v1 , v2 )) + exp(L2 (v1 , v2 )) + L3 (v1 , v2 ), where L j are affine functions. Now formulas (7.15), (7.77), (7.78) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of class (A). Case (iv.c.2). Suppose next that the function P has the form P = exp(L1 (v1 , v2 )) + (L (v1 , v2 ))2 + L2 (v1 , v2 ), where L j are affine functions and L is a linear function. Again, formulas (7.15), (7.77), (7.78) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of class (A). Case (iv.c.3). Suppose finally that the function P has the form P = (L1 (v1 , v2 ))2 + (L2 (v1 , v2 ))2 , where L j are linear functions. Now formulas (7.15), (7.77), (7.78) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of either class (A) or class (D). Case (iv.d). Suppose that for some α0 the matrix Xα20 is of type 6 (a) with τ = −τ  , ξα0 = ξα 0 . It follows that for α = α0 identities (6.42) hold, where Yα2 are κ × κ -matrices with κ := κα0 . As before, since the symmetric matrices Xα10 , Yα1 , with α = α0 , pairwise commute, they can be simultaneously diagonalized by a real orthogonal transformation, thus we can assume that Cα0 is such that all these matrices are diagonal. Further, every matrix Yα2 commutes with Xα20 and is symmetric with respect to matrix (7.71). Using Proposition 4.1 together with the relations of Lemma 3.2, from these two facts we see that there exists C ∈ GL(n, R) such that the ˆ = 0 for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ α have the form (7.24), where the parameters satisfy the corresponding matrices C the conditions s2α − (a + b)sα − (p + q)σα − λα tα = 0,

α = 5, . . . , n,

sα σα + (p + q)sα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2

α = 5, . . . , n,

+ (c + d)sα − (g + h)σα − λα ρα = 0,

rα sα − tα σα − (p + q)(tα + ρα ) + (c + d − a − b)rα = 0, α = 5, . . . , n,

ρα sα − rα σα − (c + d)(tα + ρα ) + (g + h + p + q)rα = 0, α = 5, . . . , n, n

∑ (tα ρα − rα2 ) + q2 + 4pq + 3p2 − c2 + d 2+

α =5

ac + 2bc + bd + gp − hq = 0,

(c + d)(a + b − c − d) + (p + q)(g + h + p + q) = 0. Dropping hats we get the following defining system:

(7.79)

7.3 Defining Systems of Type III

173 n

F11 = aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = bF1 + (a + 2b)F2 − (2p + q)F3 − pF4 − ∑ tγ Fγ , γ =5

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 + (3p + 2q)F3 + (2p + q)F4 + ∑ tγ Fγ − 1, γ =5

n

F33 = cF1 + (2c + d)F2 + gF3 − hF4 + ∑ ργ Fγ + 1, γ =5 n

F34 = dF1 − cF2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

n

F44 = −(c + 2d)F1 − dF2 − (g + 2h)F3 − (2g + 3h)F4 + ∑ ργ Fγ − 1, n

γ =5

F13 = F24 = pF1 + (2p + q)F2 + cF3 − dF4 + ∑ rγ Fγ , n

γ =5

F14 = qF1 − pF2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

n

F23 = −(2p + q)F1 − (3p + 2q)F2 − (2c + d)F3 − cF4 − ∑ rγ Fγ , F1α = −F2α = tα (F1 + F2 ) + rα (F3 + F4 ) + sα Fα ,

γ =5

α = 5, . . . , n,

F3α = −F4α = rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα ,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + σα (F3 + F4) + λα Fα + 1,

α = 5, . . . , n, α , β = 5, . . . , n,

Fαβ = 0, F(0) = 0,

α = β ,

α = 1, . . . , n.

Fα (0) = 0,

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. We argue analogously to case (ii). First, we introduce the variables u1 , u2 , v1 , v2 by formulas (7.5), the function Fˆ by formula (7.6), and the functions Ψ1 , Ψ2 by formulas (7.7). It follows from the defining system that Ψ1 , Ψ2 depend only on v1 , v2 and on some neighborhood of the origin satisfy the equations

∂Ψ1 = (a + b)Ψ1 − (p + q)Ψ2 + 1, ∂ v1 ∂Ψ1 ∂Ψ2 =− = −(p + q)Ψ1 − (c + d)Ψ2, ∂ v2 ∂ v1 ∂Ψ2 = (c + d)Ψ1 + (g + h)Ψ2 + 1, ∂ v2 Ψ1 (0) = 0,

Ψ2 (0) = 0.

Let P(v1 , v2 ) be the function defined by conditions (7.27). It satisfies

(7.80)

7 (n − 2, 2)-Spherical Tube Hypersurfaces

174

∂ 2P ∂P ∂P = (a + b) + (p + q) + 1, 2 ∂ v1 ∂ v2 ∂ v1 ∂ 2P ∂P ∂P = −(p + q) + (c + d) , ∂ v1 ∂ v2 ∂ v1 ∂ v2 ∂ 2P ∂P ∂P = −(c + d) + (g + h) − 1, ∂ v1 ∂ v2 ∂ v22 P(0) = 0,

∂P (0) = 0, ∂ v1

(7.81)

∂P (0) = 0. ∂ v2

As in case (i), we obtain that identities (7.11) hold and lead to identities (7.12) for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, as in case (iv.c), the defining system implies identities (7.76). Plugging (7.12) into both sides of (7.76) and using (7.79), (7.80), we obtain equations (7.77). Further, (7.12) yields representation (7.15) for some function R with R(0) = 0. Plugging (7.15) into the equations for the function Fˆ that arise from the defining system and using (7.77), (7.79), (7.80), we obtain N ∂ 2R ∂R ∂R = (a + b) + (p + q) + tα Φα − ∑ ∂ v1 ∂ v2 α =5 ∂ v21     n n n 3p + q a + 3b tα sα tα σα tα +∑ +∑ Ψ1 − − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα N ∂ 2R ∂R ∂R = −(p + q) + (c + d) + ∑ rα Φα − ∂ v1 ∂ v2 ∂ v1 ∂ v2 α =5     n n n 3p + q c−d rα sα rα σα rα +∑ +∑ − Ψ1 − − Ψ2 − ∑ , (7.82) 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α N ∂ 2R ∂R ∂R = −(c + d) + (g + h) + ∑ ρα Φα − 2 ∂ v1 ∂ v2 α =5 ∂v    2  n n n ρα sα ρα σα ρα c−d g + 3h +∑ +∑ − Ψ1 − Ψ2 − ∑ , 2 α =N +1 λα 2 α =N +1 λα α =N +1 λα

R(0) = 0,

n ∂R sα (0) = − ∑ , 2 ∂ v1 λ α =N +1 α

n ∂R σα (0) = − ∑ . 2 ∂ v2 α =N +1 λα

In order to determine Fˆ from representation (7.15), we first need to find the functions Ψ1 , Ψ2 from (7.27), then the functions Φα from system (7.77), and finally the function R from system (7.82). In order to find the functions Ψ1 and Ψ2 , we need to determine the function P from system (7.81). Observe that system (7.81) is a defining system of type III without quadratic terms corresponding to (1, 1)-spherical tube hypersurfaces in C3 . Systems of this form were considered in case (ii) of Section

7.3 Defining Systems of Type III

175

6.3. Going over all types of solutions of system (7.81), after substantial computations utilizing formulas (7.15), (7.77), (7.82) we obtain that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base a hypersurface of either class (A) or class (C) or class (D). Case (iv.e). Suppose that for some α0 the matrix Xα20 is either of type 5 where κα0 ≥ 4 and (7.62) cannot not hold, or of type 6 (b) with ξα0 = ξα 0 . It follows that for α = α0 identities (6.42) are valid, where Yα2 are κα0 × κα0 -matrices. As before, we can assume that Cα0 is such that all Yα1 , with α = α0 , are diagonal. Further, every matrix Yα2 commutes with Xα20 and is symmetric with respect to CαT0 HCα0 . Using Proposition 4.1, from these two facts we see that there exists D ∈ GL(n, R) ˇ such that the function F(x) := F(Dx) satisfies a defining system of type III without ˇ α have the form quadratic terms for which the corresponding matrices C ⎞ να μα ηα πα ζα5 . . . ζαn ⎜ − μα να + 2μα ηα πα ζα5 . . . ζαn ⎟ ⎟ ⎜ ⎟ ⎜ ηα −ηα ⎟ ⎜ ⎟ ⎜ πα ˇ α = ⎜ − πα C ⎟. ⎟ ⎜ ζ5 5 −ζα Λα ⎟ ⎜ α ⎟ ⎜ . .. ⎠ ⎝ .. . n n ζα −ζα ⎛

β

Here ζα , να , μα , ηα , πα ∈ R and Λα are pairwise commuting H  -symmetric matrices, where H  is defined in (7.33), α = 1, . . . , n, β = 5, . . . , n. Since χH  = 1, we will consider three cases analogous to cases (ii.a)–(ii.c) in Section 6.3. Case (iv.e.1). Suppose that all eigenvalues of every matrix Λα are real and that all these matrices are diagonalizable. Simultaneously diagonalizing all matrices Λα , as was done in case (ii.a) in Section 6.3, and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the ˆ = 0 for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ the corresponding matrices Cα have the form (7.34), where the parameters satisfy the conditions s2α − (a + b)sα − λα tα = 0,

α = 5, . . . , n,

c2 − (a + b)c − pg = 0,

(7.83)

d 2 − (a + b)d + qh = 0. Dropping hats we get the following defining system: n

F11 = aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, F33 = cF1 + cF2 + gF3 + 1,

γ =5

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176

F34 = 0, F44 = −dF1 − dF2 + hF4 − 1, F13 = −F23 = pF1 + pF2 + cF3, F14 = −F24 = qF1 + qF2 + dF4 , F1α = −F2α = tα (F1 + F2) + sα Fα ,

α = 5, . . . , n,

F3α = 0,

α = 5, . . . , n,

F4α = 0,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + λα Fα + 1,

α = 5, . . . , n,

Fαβ = 0,

α , β = 5, . . . , n, α = β ,

F(0) = 0,

α = 1, . . . , n.

Fα (0) = 0,

We argue analogously to case (iii.a). Define λ3 := g, s3 := c, t3 := p, λ4 := h, s4 := d, t4 := q and let I and J be the subsets of the index set {3, . . . , n} such that λα = 0 for α ∈ I and λα = 0 for α ∈ J . As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13), the function Fˆ by formula (6.14), and the function Ψ by formula (6.15). The function Ψ depends only on v and on some neighborhood of the origin satisfies equation (6.50). The defining system yields identities (7.36), which lead to identities (7.37) for some functions Φα with Φα (0) = 0, α = 3, . . . , n. Next, the defining system implies identities (6.51). Plugging identities (7.37) into both sides of (6.51) and using (6.50), (7.83), we obtain the equations

α ∈I, (Φα ) = sα Φα + tαΨ ,   Hαα (Φα ) = sα Φα + , α ∈J, λα Φα (0) = 0,

(7.84)

α = 3, . . . , n.

Further, (7.37) implies representation (7.39) for some function R with R(0) = 0. Plugging (7.39) into the equations for the function Fˆ that arise from the defining system and using (6.50), (7.83), (7.84), we obtain R = (a + b)R + ∑ Hαα tα Φα − α ∈I   a + 3b tα sα tα Ψ−∑ , +∑ 2 λ λ α α ∈J α ∈J α R(0) = 0,

R (0) = − ∑

α ∈J

sα Hαα . λα2

As in case (ii.c) in Section 6.3, we need to consider the following two situations: a + b = 0 and a + b = 0. In each of these situations the functions Ψ , Φα , R are found

7.3 Defining Systems of Type III

177

as in Section 6.3, and therefore we omit details and only state the final results. If a + b = 0, the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D), if a + b = 0, the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.2). Suppose that for some α0 the matrix Λα0 has non-real eigenvalues. Arguing as in case (ii.b) in Section 6.3 and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the function ˆ = 0 for which the ˆ F(x) := F(Cx) satisfies a defining system of type III with D ˆ corresponding matrices Cα have the form (7.40), where the parameters satisfy the conditions s2α − (a + b)sα − λα tα = 0, α = 5, . . . , n, c2 − d 2 − (a + b)c − pg + qh = 0,

(7.85)

(a + b)d − 2cd + qg + ph = 0 and d 2 + g2 + h2 > 0. Dropping hats we get the following defining system: n

F11 = aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, F33 = −F44 = cF1 + cF2 + gF3 − hF4 + 1,

γ =5

F34 = dF1 + dF2 + hF3 + gF4 , F13 = −F23 = pF1 + pF2 + cF3 − dF4 , F14 = −F24 = qF1 + qF2 + dF3 + cF4 , F1α = −F2α = tα (F1 + F2) + sα Fα ,

α = 5, . . . , n,

F3α = 0,

α = 5, . . . , n,

F4α = 0,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + λα Fα + 1,

α = 5, . . . , n,

Fαβ = 0,

α , β = 5, . . . , n, α = β ,

F(0) = 0,

Fα (0) = 0,

α = 1, . . . , n.

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. We argue analogously to case (iii.b). As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13), the function Fˆ by formula (6.14), and the function Ψ by formula (6.15). The function Ψ depends only on v and on some neighborhood of the origin satisfies equation (6.50).

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178

The defining system yields identities (7.42), which lead to identities (7.43) for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies

∂ 2 Fˆ ∂ Fˆ = sα + tα Ψ , ∂ v∂ xα ∂ xα

α = 5, . . . , n.

(7.86)

Plugging identities (7.43) into both sides of (7.86) and using (6.50), (7.85), we obtain the equations (Φα ) = sα Φα + tαΨ , α = 5, . . . , N ,   1 , α = N + 1, . . . , n, (Φα ) = sα Φα + λα

Φα (0) = 0,

(7.87)

α = 5, . . . , n.

ˆ Further, (7.43) yields representation (7.46) for some function Rˆ with R(0) = 0. ˆ We now introduce the variables u, ˜ v˜ by formulas (7.47) and express R in v, u, ˜ v˜ as ˆ (u˜ + v)/2, R˜ := R(v, ˜ (u˜ − v)/2). ˜ Plugging (7.46) into the equations for the function ˆ F that arise from the defining system and using (6.50), (7.85), (7.87), we obtain

∂ 2 R˜ ∂ R˜ ∂ R˜ ∂ R˜ = (a + b) + (p − q) + (p + q) + 2 ∂v ∂v  ∂ u˜ ∂ v˜ N n n a + 3b tα sα tα + t Φ − Ψ − , α α ∑ ∑ ∑ 2 α =N +1 λα α =5 α =N +1 λα ∂ 2 R˜ g + h ∂ R˜ g − h ∂ R˜ d = − + Ψ, ∂ u˜2 2 ∂ u˜ 2 ∂ v˜ 2 ∂ 2 R˜ g + h ∂ R˜ g − h ∂ R˜ d + − Ψ, =− 2 ∂ v˜ 2 ∂ u˜ 2 ∂ v˜ 2

(7.88)

∂ 2 R˜ ∂ R˜ ∂ R˜ p + q Ψ, =c +d + ∂ v∂ u˜ ∂ u˜ ∂ v˜ 2 ∂ 2 R˜ ∂ R˜ ∂ R˜ p − q = −d +c + Ψ, ∂ v∂ v˜ ∂ u˜ ∂ v˜ 2 g − h ∂ R˜ g + h ∂ R˜ c 1 ∂ 2 R˜ = + + Ψ+ , ∂ u˜∂ v˜ 2 ∂ u˜ 2 ∂ v˜ 2 2 ˜ R(0) = 0,

n ∂ R˜ sα (0) = − ∑ , 2 ∂v λ α =N +1 α

∂ R˜ (0) = 0, ∂ u˜

∂ R˜ (0) = 0. ∂ v˜

Observe that the functions Ψ and Φα are found as in case (ii.c) of Section 6.3, and ˜ we will only describe the form of R. Relations (7.85) imply g2 + h2 > 0. It then follows that the solution of system (7.88) can be written in the form (7.49), where ρ := (g + h)/2, η := (g − h)/2, the function B is found from the equation

7.3 Defining Systems of Type III

179

B = (c + id)B,

B(0) =

i(η − iρ )2 , 4(ρ 2 + η 2 )2

and the function R is found from the equation R = (a + b)R + 

N

∑ tα Φα −

α =5

n a + 3b (p − q)(cη + d ρ ) + (p + q)(cρ − d η ) tα sα + + ∑ 2 2(ρ 2 + η 2 ) α =N +1 λα



Ψ−

tα (p − q)η + (p + q)ρ , − 2(ρ 2 + η 2 ) α =N +1 λα n

∑

R(0) = −

ρη , 2 (ρ + η 2 )2

sα ρ (cη + d ρ ) + η (cρ − d η ) − . 2 2(ρ 2 + η 2 )2 α =N +1 λα n

R (0) = − ∑

We will now consider two cases. Case (iv.e.2.1). First, suppose a + b = 0. Then we have 

˜ η v˜ ˜ + A2 cos(dv − η u˜ + ρ v) ˜ ecv+ρ u+ + P(v, u, ˜ v) ˜ R˜ = A1 sin(dv − η u˜ + ρ v) for some real numbers A j and a polynomial P of degree at most 4. Representation (7.46) then implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.2.2). Now, suppose a + b = 0. Then we have

 ˜ η v˜ R˜ = A1 sin(dv − η u˜ + ρ v) ˜ + A2 cos(dv − η u˜ + ρ v) ˜ ecv+ρ u+ + ˜ v)e ˜ (a+b)v + Q2 (v, u, ˜ v) ˜ Q1 (v, u, for some real numbers A j and polynomials Q j of degree at most 2. Representation (7.46) then again implies that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.3). Suppose that all eigenvalues of every matrix Λα are real and that for some α0 the Jordan normal form of the matrix Λα0 contains either a 2 × 2- or a 3 × 3-cell. Arguing as in case (ii.c) in Section 6.3 and using Proposition 4.1 together with the relations of Lemma 3.2, we see that there exists C ∈ GL(n, R) such that the ˆ = D for which ˆ function F(x) := F(Cx) satisfies a defining system of type III with D ˆ the corresponding matrices Cα have the form (7.50), where the parameters satisfy the conditions s2α − (a + b)sα − (p + q)σα − λα tα = 0,

α = 5, . . . , n,

sα σα − (c + d)σα − λα rα = 0,

α = 5, . . . , n,

σα2 − (g + h)σα − λα ρα = 0,

α = 5, . . . , n,

7 (n − 2, 2)-Spherical Tube Hypersurfaces

180

rα sα − tα σα − (p + q)ρα + (c + d − a − b)rα = 0, α = 5, . . . , n,

ρα sα − rα σα − (c + d)ρα + (g + h)rα = 0,

α = 5, . . . , n, (7.89)

n

∑ (tα ρα − rα2 ) + 2cd + 2d 2 − ad − bd − ph − gq − 2hq = 0,

α =5

(c + d)(a + b − c − d) + (p + q)(g + h) = 0. Dropping hats we get the following defining system: n

F11 = aF1 − bF2 + pF3 − qF4 + ∑ tγ Fγ + 1, γ =5

n

F12 = bF1 + (a + 2b)F2 − pF3 + qF4 − ∑ tγ Fγ , γ =5

n

F22 = −(a + 2b)F1 − (2a + 3b)F2 + pF3 − qF4 + ∑ tγ Fγ − 1, γ =5

n

F33 = cF1 + cF2 + gF3 − hF4 + ∑ ργ Fγ + 1, γ =5

n

F34 = dF1 + dF2 + hF3 + (g + 2h)F4 − ∑ ργ Fγ , γ =5

n

F44 = −(c + 2d)F1 − (c + 2d)F2 − (g + 2h)F3 − (2g + 3h)F4 + ∑ ργ Fγ − 1, n

F13 = −F23 = pF1 + pF2 + cF3 − dF4 + ∑ rγ Fγ , γ =5

γ =5

n

F14 = −F24 = qF1 + qF2 + dF3 + (c + 2d)F4 − ∑ rγ Fγ , γ =5

F1α = −F2α = tα (F1 + F2 ) + rα (F3 + F4 ) + sα Fα ,

α = 5, . . . , n,

F3α = −F4α = rα (F1 + F2 ) + ρα (F3 + F4 ) + σα Fα ,

α = 5, . . . , n,

Fαα = sα (F1 + F2 ) + σα (F3 + F4) + λα Fα + 1,

α = 5, . . . , n, α , β = 5, . . . , n,

Fαβ = 0, F(0) = 0,

Fα (0) = 0,

α = β ,

α = 1, . . . , n.

Without loss of generality we assume λα = 0 for α = 5, . . . , N and λα = 0 for α = N + 1, . . . , n, where 4 ≤ N ≤ n. We argue analogously to case (iii.c). As in case (i) in Section 6.3, we introduce the variables u and v by formulas (6.13) and the function Ψ by formula (6.15). The function Ψ depends only on v and satisfies equation (6.50). Further, we introduce the variables u˜ and v˜ by formulas (7.47) and the function F˜ by formula (7.52). Next, we introduce the function Ψ˜ by formula (7.53). The defining system implies that Ψ˜ depends only on v, v˜ and near the origin satisfies the system of equations

7.3 Defining Systems of Type III

181

∂ Ψ˜ = (p + q)Ψ + (c + d)Ψ˜ , ∂v ∂ Ψ˜ = (c + d)Ψ + (g + h)Ψ˜ + 1, ∂ v˜

(7.90)

Ψ˜ (0) = 0. Further, the defining system yields identities (7.55), which lead to identities (7.56) for some functions Φα with Φα (0) = 0, α = 5, . . . , n. Next, the defining system implies ∂ 2 F˜ ∂ F˜ = sα + tαΨ + rαΨ˜ , α = 5, . . . , n, ∂ v∂ xα ∂ xα (7.91) ∂ 2 F˜ ∂ F˜ = σα + rαΨ + ραΨ˜ , α = 5, . . . , n. ∂ v˜∂ xα ∂ xα Plugging identities (7.56) into both sides of (7.91) and using (6.50), (7.89), (7.90), we obtain the equations

∂ Φα = sα Φα + tαΨ + rα Ψ˜ , α = 5, . . . , N , ∂v ∂ Φα = σα Φα + rαΨ + ραΨ˜ , α = 5, . . . , N , ∂ v˜   1 ∂ Φα α = N + 1, . . ., n, , = sα Φα + ∂v λα   1 ∂ Φα α = N + 1, . . ., n, , = σα Φα + ∂ v˜ λα Φα (0) = 0,

(7.92)

α = 5, . . . , n.

Further, (7.56) yields representation (7.59) for some function R with R(0) = 0. Plugging (7.59) into the equations for the function F˜ that arise from the defining system and using (6.50), (7.89), (7.90), (7.92), we obtain

∂ 2R ∂R ∂R N + (p + q) + = (a + b) tα Φα − 2 ∂v ∂v ∂ v˜ α∑ =5     n n n p−q tα sα tα σα ˜ tα a + 3b Ψ− − Ψ−∑ , +∑ +∑ 2 α =N +1 λα 2 α =N +1 λα λ α =N +1 α ∂ 2R ∂R N rα Φα − = (c + d) + ∂ v∂ v˜ ∂ v˜ α∑ =5     n n n p−q rα sα rα σα ˜ rα − Ψ − d+ ∑ Ψ−∑ , +∑ 2 α =N +1 λα λ λ α α =N +1 α =N +1 α

(7.93)

7 (n − 2, 2)-Spherical Tube Hypersurfaces

182

∂ 2R ∂R N + = (g + h) ρα Φα − 2 ∂ v˜ ∂ v˜ α∑ =5     n n n g + 3h ρ α sα ρα σα ˜ ρα +∑ d+∑ Ψ− Ψ−∑ , λ 2 λ α α α =N +1 α =N +1 α =N +1 λα R(0) = 0,

n ∂R sα , (0) = − ∑ 2 ∂v λ α =N +1 α

n ∂R σα . (0) = − ∑ 2 ∂ v˜ α =N +1 λα

In order to determine F˜ from representation (7.59), we first need to find the functions Ψ , Ψ˜ from (6.50), (7.90), then the functions Φα from system (7.92), and finally the function R from system (7.93). The function Ψ is found as in case (ii.c) in Section 6.3, so we will now describe the form of Ψ˜ . We will consider the cases a + b = 0 and a + b = 0. In each of these cases we have to distinguish the following two situations: g + h = 0 and g + h = 0. Case (iv.e.3.1). First, suppose a + b = 0. Case (iv.e.3.1.1). Assume g + h = 0. Then identities (7.89) imply c + d = 0, and we have p+q 2 v + v. Ψ˜ = ˜ 2 Now formulas (7.59), (7.92), (7.93) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of either class (A) or class (D). Case (iv.e.3.1.2). Assume g + h = 0. Then we have

Ψ˜ =

 1 (c+d)v+(g+h)v˜ e − (c + d)v − 1 . g+h

Now formulas (7.59), (7.92), (7.93) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.3.2). Now, suppose a + b = 0. Case (iv.e.3.2.1). Assume g + h = 0. We will consider two cases. Case (iv.e.3.2.1.a). Let c + d = 0. Then we have  p + q (a+b)v Ψ˜ = − (a + b)v − 1 + v. ˜ e (a + b)2 Now formulas (7.59), (7.92), (7.93) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.3.2.1.b). Let c + d = 0. Then identities (7.89) imply c + d = a + b, and we have    1 Ψ˜ = (p + q)(a + b)v + (a + b)2v˜ − p − q e(a+b)v + p + q . 2 (a + b)

7.4 Defining Systems of Type I

183

Now formulas (7.59), (7.92), (7.93) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A). Case (iv.e.3.2.2). Assume g + h = 0. Then we have

Ψ˜ =

  1 (c+d)v+(g+h)v˜ c+d e e(a+b)v − 1 . −1 − g+h (a + b)(g + h)

Now formulas (7.59), (7.92), (7.93) yield that the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of class (A).

7.4 Defining Systems of Type I As we showed in Section 4.2, if F(x) is the solution of a defining system of type I, the hypersurface x0 = F(x) near the origin can be written in the form (4.25). Thus, as in Sections 5.1, 6.4, our aim is to determine the functions G1 , G2 . We will now consider three cases corresponding to the signs of the eigenvalues of the matrices H1 and H2 (see (4.19), (4.23)). As in Section 6.4, here we do not go into computational details but only outline the argument by which we proceed. Calculations that need to be done here are completely elementary in their nature but rather long and involved technically. In all cases listed below the graph x0 = F(x) extends to a closed hypersurface in Rn+1 which is affinely equivalent to the base of a hypersurface of one of the classes (A), (B), (C), (D). Case (i). Assume that H1 is positive-definite, that is, H1 = IN . In this case we have 0 ≤ N ≤ n − 2. We will consider two situations. Case (i.a). Let N = 0. Then we have G1 = 1. Since χH˜ 2 = 2 (see (4.23)), the function G2 can be found as described in case (ii.c) of Section 6.4. Case (i.b). Let N ≥ 1. In this case χH˜ 1 = 1, and the function G1 can be found as described in cases (i)–(iv) of Section 5.1. Case (i.b.1). If N = n − 2, then χH˜ 2 = 1, and the function G2 also can be found as described in cases (i)–(iv) of Section 5.1. Case (i.b.2). If 1 ≤ N ≤ n − 3, then χH˜ 2 = 2, and the function G2 can be found as described in case (ii.c) of Section 6.4. Case (ii). Assume that −1 is an eigenvalue of each of H1 , H2 . In this case we have 1 ≤ N ≤ n − 1 and χH˜ 2 = 1. Then the function G2 can be found as described in cases (i)–(iv) of Section 5.1. We will consider three situations. ˜ 1 = −I2 , and G1 can be found as Case (ii.a). Let N = 1. In this case we have H described at the beginning of Section 5.1. Case (ii.b). Let N = 2. In this case we have χH˜ 1 = 1, and the function G1 can be found as described in cases (i)–(iv) of Section 5.1. Case (ii.c). Let N ≥ 3. In this case we have χH˜ 1 = 2, and the function G1 can be found as described in case (ii.c) of Section 6.4.

184

7 (n − 2, 2)-Spherical Tube Hypersurfaces

Case (iii). Assume that H2 is positive-definite. In this case we have 2 ≤ N ≤ n ˜ 2 = In−N+1 . Thus, either G2 = 1 (for N = n), or G2 can be found as described and H at the beginning of Section 5.1 (for N < n). We will consider four situations. ˜ 1 = −I3 , and G1 can be found as Case (iii.a). Let N = 2. In this case we have H described at the beginning of Section 5.1. Case (iii.b). Let N = 3. In this case we have χH˜ 1 = 1, and the function G1 can be found as described in cases (i)–(iv) of Section 5.1. Case (iii.c). Let N = 4. In this case we have χH˜ 1 = 2, and the function G1 can be found as described in case (ii.c) of Section 6.4. Case (iii.d). Let N ≥ 5. In this case we have χH˜ 1 = 3. By Proposition 4.2 and Remark 4.1 we obtain that for every α = 1, . . . , N there exists Cα ∈ GL(N + 1, R) such that either Cα−1 A1α Cα is diagonal or identity (5.5) holds, where Xα1 is a diagonal matrix and Xα2 is either: (1) a 6 × 6-matrix having only non-real eigenvalues whose Jordan normal form is diagonal, or (2) a 6 × 6-matrix having only non-real eigenvalues whose Jordan normal form contains exactly two cells of size 2 × 2, or (3) a 6×6-matrix having two non-real mutually conjugate eigenvalues whose Jordan normal form consists of two cells of size 3× 3, or (4) a matrix having non-real eigenvalues and a real eigenvalue, with the Jordan normal form corresponding to the nonreal eigenvalues being a diagonal 4 × 4-matrix and Jordan normal form corresponding to the real eigenvalue consisting of a single cell of size κ × κ for κ = 0, 2, 3, or (5) a matrix having exactly one pair of non-real mutually conjugate eigenvalues and a real eigenvalue, with the Jordan normal form corresponding to the non-real eigenvalues consisting of two cells of size 2 × 2 and Jordan normal form corresponding to the real eigenvalue consisting of a single cell of size κ × κ for κ = 0, 2, 3, or (6) a matrix having exactly two non-real mutually conjugate eigenvalues, each having multiplicity one, and a real eigenvalue, with the Jordan normal form corresponding to the real eigenvalue consisting of a single cell of size κ × κ for κ = 0, 2, 3, 4, 5, or (7) a matrix having exactly two non-real mutually conjugate eigenvalues, each having multiplicity one, and real eigenvalues, with the Jordan normal form corresponding to the real eigenvalues consisting of two cells, where the size of each cell is either 2 × 2 or 3 × 3, or (8) a matrix having a real eigenvalue whose Jordan normal form consists of a single cell of size κ × κ for κ = 2, 3, 4, 5, 6, 7, or (9) a matrix having only real eigenvalues whose Jordan normal form consists of two cells, with the cell sizes being k × k and m × m, where k = 2, 3, m = k, . . . , 5, or (10) a matrix having only real eigenvalues whose Jordan normal form consists of three cells, with the cell sizes being either 2 × 2 or 3 × 3. Hence, to determine the function G1 analogously to how it was done in Section 5.1, one needs to consider the following situations: either every matrix A1α is diagonalizable by means of a real transformation, or for some α0 the matrix A1α0 can be reduced to the form (5.5) with Xα20 of the kind described in (1)–(10) above.

Chapter 8

Number of Affine Equivalence Classes of (k, n − k)-Spherical Tube Hypersurfaces for k ≤ n−2

Abstract It follows from the results of Chapters 5, 6, 7 that the number of affine equivalence classes of closed (k, n − k)-spherical tube hypersurfaces in Cn+1 , with n ≤ 2k, is finite in the cases: (a) k = n, (b) k = n − 1, and (c) k = n − 2 with n ≤ 6. The first result of this short chapter states that this number is infinite (in fact uncountable) in the following situations: (i) k = n − 2 with n ≥ 7, (ii) k = n − 3 with n ≥ 7, and (iii) k ≤ n − 4. The question about the number of affine equivalence classes in the only remaining case k = 3, n = 6 had been open since 1989 until it was resolved by Fels and Kaup in 2009. They gave an example of a family of (3, 3)-spherical tube hypersurfaces in C7 that contains uncountably many pairwise affinely non-equivalent elements. The original approach due to Fels and Kaup is explained in Chapter 9. In this chapter we present the Fels-Kaup family but deal with it by different methods. Namely, we give a direct proof of the sphericity of the hypersurfaces in the family and use the j-invariant to show that the family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.

8.1 Statement of Results The first result of this chapter is the following theorem announced in [53] (cf. Theorem 2 therein). A detailed proof was given in recent paper [59]. Theorem 8.1. [53], [59] The number of affine equivalence classes of closed (k, n − k)-spherical tube hypersurfaces in Cn+1 , with n ≤ 2k, is infinite (in fact uncountable) in the following cases: (i) k = n − 2, n ≥ 7, (ii) k = n − 3, n ≥ 7, (iii) k ≤ n − 4. We prove Theorem 8.1 in Section 8.2. We explicitly present two 1-parameter families of algebraic (k, n − k)-spherical tube hypersurfaces that cover cases (i)–(iii) of A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 8, 

185

186

8 Number of Affine Equivalence Classes

Theorem 8.1 and show that all hypersurfaces in the families are pairwise affinely non-equivalent. Every hypersurface in either family is equivalent to the corresponding quadric by means of an explicitly given polynomial automorphism of Cn+1 . Theorem 8.1 does not cover the case k = 3, n = 6, and the question about the number of affine equivalence classes in this situation has remained open until recently. In [42] Fels and Kaup constructed a 1-parameter family of (3, 3)-spherical hypersurfaces in C7 and showed that it contains uncountably many pairwise affinely non-equivalent hypersurfaces. Let St be the hypersurface in C7 with the base given by the equation x0 = x1 x6 + x2 x5 + x3 x4 + x34 + x35 + x36 + tx4 x5 x6 ,

t ∈ R.

(8.1)

For every t ∈ R consider the following cubic of three real variables: ct (u1 , u2 , u3 ) := u31 + u32 + u33 + tu1 u2 u3 .

(8.2)

Fels and Kaup assign every cubic ct a certain real commutative associative unital algebra Ect with the property that two such algebras Ect1 , Ect2 are isomorphic if and only if the cubics ct1 , ct2 are linearly equivalent (i.e. can be mapped into each other by a transformation from GL(3, R)). On the other hand, Ect1 , Ect2 are isomorphic if and only if the hypersurfaces St1 , St2 are affinely equivalent. It is shown in [42] that the cubics ct are pairwise linearly non-equivalent, for example, for t lying in a certain interval I ⊂ R. Furthermore, the (3, 3)-sphericity of every hypersurface St is a consequence of the general algebraic-analytic approach of [41], [42] to classifying all local tube realizations of a given CR-manifold (a Levi non-degenerate quadric of CR-codimension one in this case). Details of this approach will be given in Section 9.2. Hence, for the exceptional case k = 3, n = 6 the following result holds. Theorem 8.2. [42] The hypersurfaces St are (3,3)-spherical for all t ∈ R and there exists an interval I ⊂ R such that St are pairwise affinely non-equivalent for t ∈ I . In particular, the number of affine equivalence classes of closed (3,3)-spherical tube hypersurfaces in C7 is infinite (in fact uncountable). In Section 8.3 below we give a direct (and mostly analytic) proof of Theorem 8.2, different from that in [42] (for an outline of the original proof due to Fels and Kaup see Remark 9.10). We follow the arguments of [59]. First of all, we show that every St is (3, 3)-spherical by explicitly presenting a polynomial automorphism of C7 that transforms St into the quadric. Next, we use the same general method as in the proof of Theorem 8.1 to show that the linear non-equivalence over R of two cubics ct1 , ct2 implies the affine non-equivalence of the corresponding hypersurfaces St1 , St2 . Finally, we establish the pairwise linear non-equivalence of ct for small |t| by comparing the values of the j-invariant for the elliptic curves defined as the zero loci of ct in CP2 . We note that both proofs of Theorem 8.2 (the one found in [42] and the one presented in Section 8.3 below) yield an alternative proof of Theorem 8.1 in cases (ii) and (iii) (see Remarks 8.2 and 9.11).

8.2 Proof of Theorem 8.1

187

8.2 Proof of Theorem 8.1 First, assume 5 ≤ k ≤ n − 2, n ≥ 7 and let Pt be the algebraic tube hypersurface with the base given by the following equation: x0 =

k−2

n

α =1

α =k+3

∑ x2α + xk−1xk + xk+1xk+2 − ∑

x2α +

 √ 1+t 2 2(1 + t)xk−4 xk−1 xk+1 + √ xk−3 x2k−1 + 2 3t xk−3 x2k+1 + 3t     −t 2 + 34t − 1 xk−2 x2k−1 + x2k−1 + x2k+1 x2k−1 + tx2k+1 , 3t

(8.3)

√ where 1 ≤ t ≤ 17 + 12 2 (cf. equations (5) in class (D) in Section 7.1). Every hypersurface Pt is (k, n − k)-spherical. Indeed, setting for any 0 ≤ m ≤ n   Qm,n−m :=

(z, w) ∈ Cn+1 : Im w =

m

n

α =1

α =m+1

∑ |zα |2 − ∑

|zα |2 ,

(8.4)

one can verify that Pt is mapped onto the quadric Qk,n−k by the following polynomial automorphism of Cn+1 (cf. map (7.1)): 1 z∗α = √ zα , α = 1, . . . , k − 5, k + 3, . . ., n, 2  1 1 ∗ zk−4 = √ zk−4 + 2(1 + t)zk−1 zk+1 , 2 2

 √ 1 1+t 2 3t 2 ∗ z zk−3 + √ zk−1 + zk−3 = √ , 2 k+1 4 3t 2 1 z∗k−2 = √ 2 z∗k−1



1 zk−2 + 4



 −t 2 + 34t − 1 2 zk−1 , 3t

  i 1+t 2zk−1 + zk + 2(1 + t)zk−4 zk+1 + √ zk−3 zk−1 + =− 4 3t   2 1+t −t + 34t − 1 3 2 zk−2 zk−1 + zk−1 + zk−1 zk+1 , 3t 2

z∗k = −

 √ i 2zk+1 + zk+2 + 2(1 + t)zk−4 zk−1 + 2 3t zk−3 zk+1 + 4 1+t 2 zk−1 zk+1 + tz3k+1 , 2

188

8 Number of Affine Equivalence Classes



 i 1+t z∗k+1 = − −2zk−1 + zk + 2(1 + t)zk−4 zk+1 + √ zk−3 zk−1 + 4 3t   2 −t + 34t − 1 1+t 3 2 zk−2 zk−1 + zk−1 + zk−1 zk+1 , 3t 2  √ i −2zk+1 + zk+2 + 2(1 + t)zk−4 zk−1 + 2 3t zk−3 zk+1 + 4 1+t 2 zk−1 zk+1 + tz3k+1 , 2

z∗k+2 = −



1 k−2 2 zk−1 zk zk+1 zk+2 1 n − + zα − ∑ z2α − 2 α∑ 2 2 2 =1 α =k+3  √ 2(1 + t) 1+t 3t 2 zk−4 zk−1 zk+1 − √ zk−3 zk−1 − zk−3 z2k+1 − 2 2 4 3t     1 −t 2 + 34t − 1 1 2 2 2 2 2 zk−2 zk−1 − zk−1 + zk+1 zk−1 + tzk+1 . 4 3t 8

w = i z0 −

The above formulas for the map are not a result of guessing. They arise when one attempts to reduce the equation of the hypersurface Pt to the Chern-Moser normal form by performing the transformations specified in Lemmas 3.2 and 3.3 of [24]. It turns out that these transformations are sufficient for normalizing the equation of Pt and no further (harder) steps of the Chern-Moser normalization process are necessary. This is the case for all known examples of algebraic spherical tube hypersurfaces and may be a manifestation of a general fact. We note that every algebraic (k, n − k)-spherical tube hypersurface is equivalent to Qk,n−k by means of a polynomial automorphism of Cn+1 (see [42]). Next, we show that the hypersurfaces Pt1 and Pt2 are affinely non-equivalent for t1 = t2 . To establish this, one can use the general method outlined in Section 5.3. We, however, apply a different technique that does not require utilizing the explicit formulas for the CR-isomorphisms between each of Pt1 , Pt2 and Qk,n−k presented above but relies on certain affine-geometric considerations instead.1 We need the following general lemma, which is also of independent interest. Lemma 8.1. [59] A hypersurface in Rn+1 given by an equation of the form x0 =

k−2

n

α =1

α =k+3

∑ x2α + xk−1xk + xk+1xk+2 − ∑

x2α + axk−4xk−1 xk+1 +

(8.5)

b xk−3 x2k+1 + cxk−3 x2k−1 + dxk−2x2k−1 + Q4 (xk−1 , xk+1 ),

where a, b, c ∈ R∗ , d ∈ R, 5 ≤ k ≤ n − 2 and Q4 is a homogeneous polynomial of order 4, is affinely homogeneous. 1

This line of proof has been suggested to us by M. G. Eastwood.

8.2 Proof of Theorem 8.1

189

Proof. It is sufficient to prove the lemma for k = 5, n = 7. Let S be a hypersurface of the form (8.5) in R8 . For p ∈ S let t p be the translation on R8 that maps p to the origin. Absorbing the linear terms into x0 , we turn the equation of t p (S) into an equation of the form x0 = x21 + x22 + x23 + x4 x5 + x6 x7 + L1(x1 , x2 , x3 , x4 , x6 )x4 + L2 (x1 , x2 , x6 )x6 + ax1 x4 x6 + bx2x26 + cx2 x24 + dx3x24 + Ax34 + Bx36 + Cx4x26 + Dx24 x6 + Q4 (x4 , x6 ), where L1 and L2 are linear functions and A, B,C, D ∈ R. Replacing x2 by x2 − (A/c) x4 − (B/b) x6 , we obtain the equation x0 = x21 + x22 + x23 + x4x5 + x6 x7 + L3(x1 , x2 , x3 , x4 , x6 )x4 + L4 (x1 , x2 , x6 )x6 + ax1 x4 x6 + bx2x26 + cx2 x24 + dx3 x24 + C x4 x26 + D x24 x6 + Q4 (x4 , x6 ), where L3 and L4 are linear functions and C , D ∈ R. Further, replacing x1 by x1 − (D /a) x4 − (C /a)x6 , we get x0 = x21 + x22 + x23 + x4x5 + x6 x7 + L5(x1 , x2 , x3 , x4 , x6 )x4 + L6 (x1 , x2 , x6 )x6 + ax1 x4 x6 + bx2x26 + cx2 x24 + dx3 x24 + Q4 (x4 , x6 ), where L5 and L6 are linear functions. Finally, absorbing L5 into x5 and L6 into x7 , we obtain the original equation of S. This argument shows that S is affinely homogeneous as required.

 Consider the bases Pt1 R and Pt2 R of Pt1 and Pt2 , respectively. Lemma 8.1 implies that the hypersurfaces Pt1 R and Pt2 R are affinely homogeneous, hence Pt1 R and Pt2 R are affinely equivalent if and only if they are linearly equivalent. It is immediate from (8.3) that the cubic terms in the equations of Pt1 R and Pt2 R are trace-free, where the trace is calculated with respect to the non-degenerate quadratic form n 2 2 ∑k−2 α =1 xα +xk−1 xk +xk+1 xk+2 − ∑α =k+3 xα . Therefore, Proposition 1 of [30] (see also [79]) yields that any linear equivalence between Pt1 R and Pt2 R has the form (4.16). It then follows that a sufficient condition for the linear non-equivalence of the hypersurfaces Pt1 R and Pt2 R is the linear non-equivalence over R of the fourth-order terms in their equations. Thus, to establish the pairwise affine non-equivalence of the hypersurfaces Pt it is sufficient to establish the pairwise linear non-equivalence of the quartics qt (ξ , η ) := (ξ 2 + η 2 )(ξ 2 + t η 2 ) over R. Suppose that there exists a non-degenerate linear map ϕ (ξ , η ) → (αξ + β η , γξ + δ η ),

α , β , γ , δ ∈ R,

that transforms qt1 into qt2 with t1 < t2 . We allow ξ and η to be complex and consider ϕ as a transformation of C2 . Then ϕ maps the zero locus of qt1 into the zero locus of q√ t2 (for any t the zero locus of qt consists of the complex lines {ξ = ±iη }, {ξ = ±i t η }). Let mϕ be the M¨obius transformation of CP1 arising from ϕ . Clearly, on the subset η = 0 of CP1 for ζ := ξ /η we have

190

8 Number of Affine Equivalence Classes

mϕ ( ζ ) =

αζ + β , γζ + δ

√ √ and mϕ maps the set σt1 := {±i, ±i t1 } onto the set σt2 := {±i, ±i t2 }. If t1 = 1, then σt1 is a two-point set and cannot be mapped onto the four-point set σt2 , hence such a map ϕ does not exist for t1 = 1. Now, assume t1 > 1. Then mϕ preserves the imaginary axis in the ζ -plane, which immediately implies αγ = 0, β δ = 0. Then we have either β = γ = 0 or α = δ = 0, thus mϕ is either a real dilation or the composition of a real dilation and 1/ζ . In either case mϕ cannot map σt1 onto σt2 since t1 < t2 . We have thus shown that qt1 and qt2 are not equivalent by means of a transformation from GL(2, R). Note that this statement no longer holds if transformations from GL(2, C) are allowed. Thus, the algebraic spherical tube hypersurfaces Pt defined in (8.3) are pairwise affinely non-equivalent. It then follows that the number of affine equivalence classes of (k, n − k)-spherical tube hypersurfaces is uncountable in the following situations: (a) k = n − 2 with n ≥ 7, (b) k = n − 3 with n ≥ 8, (c) k = n − 4 with n ≥ 9, and (d) k ≤ n − 5. It remains to prove the theorem for k = 4, n = 7 and for k = 4, n = 8. We present a 1-parameter family of hypersurfaces defined in a more general setting than these two remaining cases. Assume 4 ≤ k ≤ n − 3, n ≥ 7 and let P˜t be the algebraic tube hypersurface with the base given by the following equation: x0 =

k−2

n

α =1

α =k+4

∑ x2α − x2k−1 + xkxk+1 + xk+2xk+3 − ∑

x2α +

 √ 1+t 2 2(1 + t)xk−3 xk xk+2 + √ xk−2 x2k + 2 3t xk−2 x2k+2 + 3t     2 t − 34t + 1 xk−1 x2k + x2k + x2k+2 x2k + tx2k+2 , 3t √ where t ≥ 17 + 12 2. Every hypersurface P˜t is (k, n − k)-spherical. Indeed, P˜t is mapped onto Qk,n−k (see (8.4)) by the following polynomial automorphism of Cn+1 : 1 z∗α = √ zα , α = 1, . . . , k − 4, k + 4, . . ., n, 2  1 1 ∗ zk−3 = √ zk−3 + 2(1 + t)zk zk+2 , 2 2 

√ 1 1+t 2 3t 2 ∗ , z zk−2 + √ zk + zk−2 = √ 2 k+2 4 3t 2  √ i z∗k−1 = − 2zk+2 + zk+3 + 2(1 + t)zk−3 zk + 2 3t zk−2 zk+2 + 4 1+t 2 zk zk+2 + tz3k+2 , 2

8.2 Proof of Theorem 8.1

191

  i 1+t ∗ zk = − 2zk + zk+1 + 2(1 + t)zk−3 zk+2 + √ zk−2 zk + 4 3t   2 t − 34t + 1 1+t 2 zk−1 zk + z3k + zk zk+2 , 3t 2 z∗k+1

1 =√ 2



1 zk−1 − 4



 t 2 − 34t + 1 2 zk , 3t

  i 1+t z∗k+2 = − −2zk + zk+1 + 2(1 + t)zk−3 zk+2 + √ zk−2 zk + 4 3t   2 1+t 2 t − 34t + 1 zk−1 zk + z3k + zk zk+2 , 3t 2  √ i −2zk+2 + zk+3 + 2(1 + t)zk−3 zk + 2 3t zk−2 zk+2 + 4 1+t 2 3 z zk+2 + tzk+2 , 2 k

z∗k+3 = −



1 k−2 2 z2k−1 zk zk+1 zk+2 zk+3 1 n zα + − − + ∑ z2α − 2 α∑ 2 2 2 2 α =k+4 =1  √ 2(1 + t) 1+t 3t 2 zk−3 zk zk+2 − √ zk−2 zk − zk−2 z2k+2 − 2 2 4 3t     1 2 1 t 2 − 34t + 1 2 2 2 2 zk−1 zk − zk + zk+2 zk + tzk+2 . 4 3t 8

w = i z0 −

A proof completely analogous to that for the hypersurfaces Pt above shows that the hypersurfaces P˜t are affinely homogeneous and pairwise affinely non-equivalent. It then follows that the number of affine equivalence classes of (k, n − k)-spherical tube hypersurfaces is uncountable in the cases k = 4, n = 7 and k = 4, n = 8 as well.

 √ Remark 8.1. For t = 17 + 12 2 one can write the families of hypersurfaces Pt and P˜t in a unified form. Namely, let Psτ be the tube hypersurface in Cn+1 for n ≥ 7 with the base given by the equation x0 = 4x1 x7 + 4x2 x6 − τ x23 + 2x24 − τ x25 + 4x3 x5 +

s+7

n

α =8

α =s+8

∑ x2α − ∑

x2α −

τ τ 2τ x21 x3 − 2τ x22 x5 + 4x21 x5 + 4x22 x3 + 8x1 x2 x4 − x41 + 4x21 x22 − x42 , 3 3 where τ ∈ R, τ = ±2 and 0 ≤ s ≤ n − 7. Note that every hypersurface Psτ is affinely equivalent to a hypersurface Psτ for which τ ∈ [−6, −2) ∪ (−2, 2). The Levi form of Psτ has signature (s + 5, n − (s + 5)) for τ ∈ [−6, −2) and signature (s + 4, n − (s + 4)) for τ ∈ (−2, 2) up to sign. Using Lemmas 3.2 and 3.3 of [24] as

192

8 Number of Affine Equivalence Classes

in the proof of Theorem 8.1, one can show that Psτ is CR-equivalent to Qs+5,n−(s+5) for τ ∈ [−6, −2) and to Qs+4,n−(s+4) for τ ∈ (−2, 2) by means of a polynomial automorphism of Cn+1 . Next, observe that the function √ 12 t Λ : t → − t+1 √ √ is a bijection from [1, 17 + 12 2) onto [−6, −2) and from (17 + 12 2, ∞) onto (−2, 0). It is now not hard to see that Pk−5 is affinely equivalent to PΛ −1 (τ ) for τ k−4 τ ∈ [−6, −2), 5 ≤ k ≤ n − 2, and that Pτ is affinely equivalent to P˜Λ −1 (τ ) for k−4 for τ ∈ (−2, 2), τ ∈ (−2, 0), 4 ≤ k ≤ n − 3. In particular, the family √ Pτ 4 ≤ k ≤ n − 3 extends the family P˜t with t = 17 + 12 2. For algebraic structures behind the family Psτ and the Fels-Kaup family St defined in (8.1) we refer the reader to Remark 9.12.

8.3 Proof of Theorem 8.2 First of all, we observe that every hypersurface St defined in (8.1) is (3, 3)-spherical. Indeed, St is mapped onto the quadric

1 7 (z, w) ∈ C : Im w = Re (z1 z6 + z2 z5 + z3 z4 ) 2 by the following polynomial automorphism of C7 : 3 t z∗1 = z1 + z26 + z4 z5 , 2 2

3 t z∗2 = z2 + z25 + z4 z6 , 2 2

3 t z∗3 = z3 + z24 + z5 z6 , z∗α = zα , α = 4, 5, 6, 2 2   1 1 w = i z0 − (z1 z6 + z2 z5 + z3 z4 ) − z34 + z35 + z36 + tz4 z5 z6 . 2 4 Next, we observe that the base St R of every hypersurface St is affinely homogeneous. Indeed, for p ∈ St R let us apply to St R the translation t p on R7 that maps p to the origin. Absorbing the linear terms into x0 , we turn the equation of t p (St R ) into x0 = Q2 + L1(x4 , x5 , x6 )x4 + L2 (x5 , x6 )x5 + L3 (x6 )x6 + Q3 , where Q j , j = 2, 3, are the terms of order j in the right-hand side of formula (8.1) and L1 , L2 , L3 are linear functions. Absorbing L1 into x3 , L2 into x2 , and L3 into x1 , we obtain the original equation of St R . This proves that St R is affinely homogeneous. Thus, St1 R and St2 R are affinely equivalent if and only if they are linearly equivalent.

8.3 Proof of Theorem 8.2

193

Further, it is straightforward to see that for every t the cubic terms in the equation of St R are trace-free, where the trace is calculated with respect to the non-degenerate quadratic form Q2 . We now argue as in Section 8.2 and use Proposition 1 of [30] to conclude that two hypersurfaces St1 R and St2 R are affinely non-equivalent if the corresponding cubics ct1 , ct2 (see (8.2)) are linearly non-equivalent over R. To see when two cubics are linearly non-equivalent, we think of them as functions of three complex variables and find sufficient conditions for their zero loci, viewed as curves in CP2 , to be projectively non-equivalent. Namely, for every t ∈ R define   Zt := (ζ0 : ζ1 : ζ2 ) ∈ CP2 : ct (ζ0 , ζ1 , ζ2 ) = 0 . (8.6) If t = −3, the set Zt is a non-singular elliptic curve, whereas for t = −3 it has singularities at the points (1 : q : q2 ) with q3 = 1. We only consider small values of t and show that for t lying in some interval around 0 the elliptic curves Zt are pairwise projectively non-equivalent, which implies that for such t the cubics ct are pairwise linearly non-equivalent over R as required. The projective equivalence class of an elliptic curve (which coincides with its biholomorphic equivalence class) is completely determined by the value of the j-invariant for the curve (see, e.g. [72], pp. 56–67 or [104]). Saito’s calculation for simple elliptic singularities of type E˜6 in [92] gives a formula for the value of the j-invariant for the curve Zt (see also [21], [29]). Furthermore, it is explained in [29] (see also [21]) how one can recover this value directly from the corresponding moduli algebra (see Remark 9.12). For completeness of our exposition we will compute the value of the j-invariant for Zt below. In order to apply well-known formulas, we first transform Zt for t = −3 into an elliptic curve given by an equation in the Weierstrass form (see, e.g. formula (1) in [104]). We perform the following projective transformation:  t t  (ζ0 : ζ1 : ζ2 ) → ζ0 + ζ1 − ζ2 : cζ2 : −ζ1 + ζ2 , 3 3   t3 3 1 +1 . c := − 3 27

where

This transformation takes Zt into the elliptic curve

ζ0 ζ22 −

t t2 1 ζ0 ζ1 ζ2 + ζ02 ζ2 = ζ13 − 2 ζ0 ζ12 − ζ03 . 3c 9c 3

One can now compute the value of the j-invariant for this curve as shown in [72], [104]. Using standard notation, we have a1 = − hence

t t2 1 , a2 = − 2 , a3 = 1, a4 = 0, a6 = − , 3c 9c 3

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8 Number of Affine Equivalence Classes

t 3 1 (t + 72c3 ), Δ = 3 . 9c4 c Therefore, the value of the j-invariant for Zt is c4 =

j(Zt ) = −t 3

(t 3 − 216)3 . (t 3 + 27)3

Setting s := t 3 we obtain j(Zt ) = Φ (s) := −s

(s − 216)3 . (s + 27)3

The function Φ is strictly increasing for small |s| since Φ (0) > 0. This shows that the values j(Zt ) are pairwise distinct for t lying in a sufficiently small interval around 0. The proof of the theorem is complete.

 Remark 8.2. The proof of Theorem 8.2 given above in fact yields an alternative proof of Theorem 8.1 in cases (ii) and (iii). Indeed, for n ≥ 6 consider the family of hypersurfaces in Cn+1 with the bases given by the following equations: x0 = x1 x6 + x2 x5 + x3 x4 +

s+6

n

α =7

α =s+7

∑ x2α − ∑

x2α + x34 + x35 + x36 + tx4 x5 x6 ,

(8.7)

where t ∈ R and 0 ≤ s ≤ n − 6, n ≤ 2s + 6. As in the proof of Theorem 8.2, one can show that for a given s every hypersurface defined in (8.7) is CR-equivalent to the quadric Qs+3,n−(s+3) by means of a polynomial automorphism of Cn+1 , its base is affinely homogeneous, and such hypersurfaces are pairwise affinely non-equivalent for small |t|. This shows that the number of affine equivalence classes in cases (ii) and (iii) of Theorem 8.1 is uncountable as required. We note that by considering family (8.7) one can also derive a proof of Theorem 8.1 in cases (ii) and (iii) from the original proof of Theorem 8.2 given in [42] (see Remark 9.11). In this chapter we showed that the bases of certain algebraic spherical hypersurfaces are affinely homogeneous. This is the case for many other examples of algebraic spherical tube hypersurfaces. Until recently, it has been an open question whether algebraic spherical hypersurfaces whose bases are not affinely homogeneous exist. First examples of this kind appeared in [43] where proofs relied on computer algebra (see Remark 9.8).

Chapter 9

Further Results

Abstract In the first section of this chapter we consider tube hypersurfaces in Cn+1 locally CR-equivalent to a quadric Qg where the Hermitian form g is degenerate. For g ≡ 0 we show that every tube hypersurface of this kind is real-analytic and extends to a closed non-singular real-analytic tube hypersurface in Cn+1 represented as the direct sum of a complex linear subspace of Cn+1 and a closed spherical tube hypersurface lying in a complementary complex subspace. For g ≡ 0 such a hypersurface is an open subset of a real affine hyperplane in Cn+1 . Thus, the study of tube hypersurfaces locally CR-equivalent to Levi-degenerate quadrics reduces to the study of spherical tube hypersurfaces. In the second section we briefly describe the approach to the problem of affine classification of spherical tube hypersurfaces recently proposed by Fels and Kaup in [41], [42]. Many results of this chapter apply to CR-manifolds of arbitrary CR-codimension.

9.1 Tube Hypersurfaces with Degenerate Levi Form We start with general facts on complex foliations on arbitrary CR-manifolds. More details can be found in Chapter 5 of survey article [25].

9.1.1 Complex Foliations on CR-Manifolds Let M be a CR-manifold of dimension d and {Mλ }λ ∈Λ a family of disjoint immersed connected submanifolds of M such that ∪λ ∈Λ Mλ = M. The family {Mλ }λ ∈Λ is called a (smooth) -dimensional complex foliation on M if: (i) for every λ ∈ Λ and every p ∈ Mλ the tangent space Tp (Mλ ) is a complex subspace of Tpc (M) of complex dimension  (in particular, every Mλ is an almost complex manifold; since the CR-structure on M is integrable, the almost complex structure on every Mλ is integrable as well and therefore by the Newlander-Nirenberg theorem turns Mλ into A. Isaev, Spherical Tube Hypersurfaces, Lecture Notes in Mathematics 2020, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19783-3 9, 

195

196

9 Further Results

a complex manifold), and (ii) for every p ∈ M there exist a neighborhood U of p, a domain U ⊂ C , a domain V ⊂ Rd−2 , and a diffeomorphism ϕ : U × V → U such that for every v ∈ V one can find λ ∈ Λ for which the restriction of ϕ to U × {v} is a biholomorphic map onto Mλ ∩ U . If M is real-analytic and ϕ can be chosen to be real-analytic, the foliation {Mλ }λ ∈Λ is called real-analytic. The submanifolds Mλ are called the leaves of the foliation {Mλ }λ ∈Λ . Further, we say that an -dimensional complex foliation is locally CR-straightenable if for every p ∈ M there exist a neighborhood U of p, a domain U ⊂ C , a CR-manifold M0 of dimension d − 2, and a CR-isomorphism ϕ : U × M0 → U such that for every v ∈ M0 one can find λ ∈ Λ for which ϕ maps U × {v} onto Mλ ∩ U (note that the restriction of ϕ to U × {v} is necessarily biholomorphic). When M is a locally closed submanifold of a complex manifold N with induced CR-structure, one can also speak of locally holomorphically straightenable foliations on M by taking in the above definition the neighborhood U to be the intersection U ∩ M for a sufficiently small neighborhood U of p in N, the CR-manifold M0 to be a CR-manifold with induced CR-structure contained in a domain V of Cl with l := dim N − , and the map ϕ to be the restriction to U × M0 of a biholomorphic map ψ : U × V → U. Sufficient conditions for the local holomorphic straightenability of a real-analytic complex foliation were given in [45]. These conditions are also easily seen to be necessary for the local CR-straightenability of a smooth complex foliation. [We note that many results of [45] apply to real submanifolds M ⊂ N for which dim Tp (M) is not necessarily constant on M (see (1.1)).] Suppose now that M is an immersed submanifold of CK with induced CRstructure. In this situation an elegant necessary and sufficient condition for the local CR-straightenability of a complex foliation was found in [87]. In order to state this condition, consider the map

Φ : M → GC (, K) ,

p → Tp (Mλ ),

(9.1)

with GC (, K) being the Grassmannian of all -dimensional complex linear subspaces in CK and Mλ the (unique) submanifold in the family {Mλ }λ ∈Λ passing through p, where Tp (Mλ ) is regarded in a natural way as a point in GC (, K). The result of [87] can now be stated as follows: {Mλ }λ ∈Λ is locally CR-straightenable if and only if Φ is a CR-map. Furthermore, if M is locally closed in CK and the foliation {Mλ }λ ∈Λ is real-analytic, this condition is also sufficient for the local holomorphic straightenability of {Mλ }λ ∈Λ . Examples of non–CR-straightenable foliations can be found, e.g. in Section 5.2 of [25]. We are interested in the complex foliation arising from the kernel of the Levi form. With the Levi form given by formula (1.3), we define its kernel kerLM (p) at a point p ∈ M as   (1,0) (1,0) ker LM (p) := Z ∈ Tp (M) : LM (p)(Z, Z ) = 0 for all Z ∈ Tp (M) . Suppose that the complex dimension of ker LM (p) is constant and equal to r on M. In this case the linear subspaces ker LM (p) form a subbundle of the bundle

9.1 Tube Hypersurfaces with Degenerate Levi Form

197

T (1,0) (M). Let kerc LM be the corresponding tangent distribution on M, which is (1,0) a subbundle of T c (M). [Recall that for every p ∈ M and every Z ∈ Tp (M) there c exists a unique X ∈ Tp (M) such that Z = X − iJp X (see (1.2)). The fiber kerc LM (p) of kerc LM over p is the image of ker LM (p) under the map Z → X.] The integrability of the CR-structure on M implies that the distribution kerc LM is involutive (see, e.g. Proposition 5.3 in [25]). The foliation of M formed by the integral manifolds of kerc LM is an r-dimensional complex foliation called the Levi foliation. If the Levi form of M is everywhere zero (in this case r = CRdim M and M is called Levi-flat), then the Levi foliation is CR-straightenable (see, e.g. Proposition 5.1 in [25]). Furthermore, any CRdim M-dimensional complex foliation on M coincides with the Levi foliation and hence is CR-straightenable. Note that the Levi foliation need not be CR-straightenable for r < CRdim M (see examples in Section 5.2 of [25]).

9.1.2 Levi Foliation on Tube Manifold We introduce tube manifolds in the complex vector space CK by generalizing the notion of tube hypersurface to the case of an arbitrary codimension. Fix a totally real K-dimensional linear subspace V ⊂ CK . A tube manifold in CK is an immersed submanifold of CK of the form M = MR + iV, where MR is an immersed submanifold  of V called the base ofM. Choosing coordinates z1 , . . . , zK in CK such that V = Im z j = 0, j = 1, . . . , K , we identify V with RK by means of the coordinates x j := Re z j , j = 1, . . . , K. Below we always regard MR as a submanifold of RK and represent the tube manifold M as M = MR + iRK . Observe that for the tube manifold M the dimension of Tp (M) is constant (see (1.1)), hence CK induces a CR-structure on M. Clearly, for the induced structure we have CRdim M = dim MR and CRcodim M = K − dim MR . Note that for Z := (z1 , . . . , zK ) any transformation of CK of the form Z → Z + ib,

b ∈ RK ,

(9.2)

is a CR-automorphism of M. We need the following proposition (cf. [41]). Proposition 9.1. [56] Let M be a locally closed tube manifold in CK . Suppose that M is not Levi-flat and dimC ker LM (p) ≡ r. Assume further that the Levi foliation on M is locally CR-straightenable. Then there exist linear subspaces L1 , L2 of RK , with RK = L1 ⊕ L2 , dim L2 = r, and a locally closed submanifold S of L1

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9 Further Results

with non-degenerate second fundamental form such that for every p ∈ MR one can find a domain S p in S and a domain Vp in L2 satisfying the following conditions: (i) p ∈ S p + Vp , and (ii) M contains the tube manifold (S p + iL1 ) + (Vp + iL2 ) as an open subset. Every leaf of the Levi foliation on M is a connected component of a set of the form (s + (L2 + iL2 )) ∩ M for some s ∈ S + iL1 . Proof. 1 By the result of [87], the local CR-straightenability of the Levi foliation implies that the map Φ associated to the foliation is a CR-map (see (9.1)). The Levi foliation is invariant under all translations of the form (9.2), hence Φ is constant on every real affine subspace p + iRK with p ∈ MR . Since Φ is a CR-map, this implies that Φ is in fact a constant map. For p ∈ MR let T p (MR ) be the kernel of the second fundamental form of MR at p and consider the map

ΦR : MR → GR (r, K) ,

p → T p (MR ),

where GR (r, K) is the Grassmannian of all r-dimensional linear subspaces in RK and T p (MR ) is regarded in a natural way as a point in GR (r, K). Observe that for every p ∈ MR and every q ∈ p + iRK we have Φ (q) = ΦR (p) + iΦR (p), where for a subspace L ∈ GR (r, K) its complexification L + iL is regarded as a point in GC (r, K). Therefore, the constancy of the map Φ implies that ΦR is constant as well. We now let L2 be the (unique) linear subspace of RK lying in the range of ΦR . Choose any complementary subspace L1 in RK and let S be image of MR under the projection to L1 along L2 . Clearly, every leaf of the Levi foliation on M is a connected component of a set of the form (s + (L2 + iL2 )) ∩ M for some s ∈ S + iL1 . It is now straightforward to see that for every p ∈ MR there exist a domain S p in S and a domain Vp in L2 such that conditions (i) and (ii) of the proposition hold.

Remark 9.1. The above proof also works if M is Levi-flat, in which case the submanifold S is a single point. In this situation one obtains the well-known result that a Levi-flat tube manifold is an open subset of a real affine subspace. We now return to the case of CR-hypersurfaces. Generalizing the definition of sphericity given at the end of Section 1.2, we say that a CR-hypersurface M of CR-dimension n ≥ 1 is (k, m − k, n − m)-spherical if M is locally CR-equivalent to the quadric Qg in Cn+1 , where g is a Hermitian form on Cn with k positive, m − k negative, and n − m zero eigenvalues, where 0 ≤ k ≤ m, m ≤ 2k, m ≤ n. If m < n, the Levi form of a (k, m − k, n − m)-spherical CR-hypersurface M is everywhere degenerate and dimC ker LM (p) ≡ n − m. Furthermore, the above definition of (k, m− k, n − m)-sphericity immediately yields that the Levi foliation on M is locally CR-straightenable. We will now obtain the main result of Section 9.1. It implies that the study of Levi degenerate spherical tube hypersurfaces reduces to that of Levi non-degenerate ones. By Remark 9.1 every Levi-flat tube hypersurface is an open subset of a real affine hyperplane in Cn+1 , thus below we assume m ≥ 1. 1

This proof has been suggested to us by N. Kruzhilin and is similar to that of Corollary 5.5 in [25]. Alternatively, Proposition 9.1 can be derived from the results of [45] (see [55]).

9.2 Approach of G. Fels and W. Kaup

199

Theorem 9.1. [56] Let M be a (k, m − k, n − m)-spherical tube hypersurface in Cn+1 with 1 ≤ m < n. Then the following holds: (i) M is real-analytic, and (ii) M extends to a closed non-singular real-analytic hypersurface M ext in Cn+1 that has the form M ext = (S + iL1 ) + (L2 + iL2 ), where L1 and L2 are linear subspaces of Rn+1 , with Rn+1 = L1 ⊕ L2 , dim L2 = n − m, and S is a closed non-singular real-analytic hypersurface in L1 such that S + iL1 is a (k, m − k)-spherical tube hypersurface in the complex linear (m+ 1)-dimensional subspace L1 + iL1 of Cn+1 . Proof. Fix p ∈ MR and let M be a neighborhood of p in M which is a locally closed tube hypersurface in Cn+1 . Since the Levi foliation on M is locally CR-straightenable, by Proposition 9.1 there exist an (m+ 1)-dimensional linear subspace L1 ⊂ Rn+1 , a complementary (n − m)-dimensional subspace L2 ⊂ Rn+1 , a locally closed submanifold S ⊂ L1 with non-degenerate second fundamental form, and a domain V in L2 such that p ∈ S + V and M contains the tube manifold T := (S + iL1 ) + (V + iL2 ) as an open subset. Clearly, the leaves of the Levi foliation on T have the form s + (V + iL2 ) for s ∈ S + iL1 . For every point q ∈ T there exist a neighborhood U of q in T, a domain U ⊂ Cn−m , a domain Ω ⊂ Qg with g being a non-degenerate Hermitian form on Cm with signature (k, m − k), and a CR-isomorphism ϕ : U × Ω → U such that for every ω ∈ Ω the isomorphism ϕ maps the set U × {ω } biholomorphically onto the intersection of a leaf of the Levi foliation on T with U . Hence,   for every σ ∈ V + iL2 the inverse image Rσ := ϕ −1 [(S + iL1 ) + σ ] ∩ U is a CR-submanifold of U × Ω represented as the graph of a map defined on an open subset Ωσ of Ω with values in U. Clearly, Rσ is CR-equivalent to Ωσ for all σ , which shows that the tube hypersurface S + iL1 in L1 + iL1 is (k, m − k)-spherical. By Proposition 3.1 this hypersurface is real-analytic, hence T is real-analytic. Since the above arguments are applicable to any point in MR , it follows that M is realanalytic, thus we have obtained statement (i) of the theorem. By Theorems 4.1 and 4.2 the hypersurface S extends to a closed non-singular real-analytic hypersurface S in L1 such that S + iL1 is a (k, m − k)-spherical tube hypersurface in L1 + iL1 . Since M is real-analytic and contains T as an open subset, it lies in the closed non-singular hypersurface (S + iL1 ) + (L2 + iL2 ) as an open subset. We have now obtained statement (ii) of the theorem, and the proof is complete.

9.2 Approach of G. Fels and W. Kaup In this section we give a brief overview of recent papers by Fels and Kaup [41], [42]. Paper [41] is concerned with the question of describing all possible (local) tube realizations of a real-analytic CR-manifold up to affine equivalence. In [42] the authors further develop the techniques of [41] aiming at determining all tube realizations of a Levi non-degenerate quadric of CR-codimension one. The approach of [41],

200

9 Further Results

[42] allows one to recover the affine classifications of closed (n, 0)-spherical and (n − 1, 1)-spherical tube hypersurfaces in Cn+1 presented in Chapters 5, 6, without utilizing defining systems (see Remark 9.9). Furthermore, Fels and Kaup give alternative proofs of Proposition 3.1 (see Remark 9.4) as well as Theorems 4.1 and 4.2 (see Theorem 9.3). They also show that the number of affine equivalence classes of (3, 3)-spherical tube hypersurfaces is infinite (see Theorem 8.2 and Remark 9.10), thus resolving the only case not covered by Theorem 8.1. Furthermore, their proof of Theorem 8.2 provides an alternative proof of Theorem 8.1 in cases (ii), (iii) (see Remark 9.11). We now turn to details of the approach proposed in [41], [42]. Let Q be a locally closed real-analytic submanifold of a complex manifold N with induced CR-structure. We assume that Q is generic in N, that is, for every q ∈ Q one has Tq (N) = Tq (Q) + JqN Tq (Q), where JqN is the operator of complex structure on Tq (N). Observe that any tube manifold in CK is generic in CK . We denote by hol(Q) the real Lie algebra of all real-analytic infinitesimal CR-automorphisms of Q.2 A vector field on Q lies in hol(Q) if and only if it extends to a holomorphic vector field defined on a neighborhood U of Q in N. [We think of holomorphic vector fields on U as holomorphic sections over U of the tangent bundle T (U). In particular if N = CK , a holomorphic vector field on U is just a holomorphic map f : U → CK . To indicate that we think of f as a vector field we write f (Z) ∂∂Z .] Further, for q ∈ Q we denote by hol(Q, q) the real Lie algebra of all germs at q of vector fields in hol(V ) with V running over all neighborhoods of q in Q. Clearly, hol(Q, q) is a real Lie subalgebra of the complex Lie algebra hol(N, q).3 For the germ (Q, q) of Q at a point q ∈ Q we define Aut(Q, q) to be the group of all real-analytic CR-automorphisms of (Q, q), i.e. the group of germs of local real-analytic CR-automorphisms of Q defined near the point q and preserving it. Clearly, the group Aut(Q, q) acts on hol(Q, q) by means of the linear representation f → f∗ , where f∗ denotes the push-forward map of vector field germs arising from f ∈ Aut(Q, q). We say that two Lie subalgebras v, v ⊂ hol(Q, q) are Aut(Q, q)-conjugate if there exists f ∈ Aut(Q, q) such that v = f∗ (v). Next, we say that the germ (M , a) of a real-analytic tube manifold M ⊂ CK at a point a ∈ M is a tube realization of (Q, q) if the germs (Q, q) and (M , a) are CR-equivalent by means of the germ of a real-analytic CR-isomorphism between a neighborhood of q in Q and a neighborhood of a in M . Observe that we necessarily have K = CRdim Q + CRcodim Q = dim N and dim MR = CRdim Q. Two germs (M , a) and (M , a ) of real-analytic tube manifolds M , M ⊂ CK are called affinely equivalent if (M , a) and (M , a ) are equivalent by means of the germ of an affine transformation of the form Z → CZ + b with C ∈ GL(K, R), b ∈ RK . (9.3) Observe that if (M , a) is a tube realization of (Q, q) and ϕ : (Q, q) → (M , a) the germ of some real-analytic CR-isomorphism, then hol(M , a) contains the commu  tative Lie subalgebra iv ∂∂Z , v ∈ RK , and its pull-back 2 3

We allow Lie algebras to be infinite-dimensional. For a holomorphic vector field ξ on U ⊂ N the vector field iξ is defined as (iξ )(p) := J pN ξ (p).

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201



 ∂ ∗ K v(M , a) := ϕ iv ,v∈R ∂Z

(9.4)

under ϕ is a commutative Lie subalgebra of hol(Q, q). We are interested in the class of commutative Lie subalgebras v ⊂ hol(Q, q) satisfying the following conditions: (i) v is totally real in hol(N, q), and (ii) the evaluation map εq at q from vC := v ⊕ iv ⊂ hol(N, q) to Tq (N) is surjective.

(9.5)

It is straightforward to verify that v(M , a) satisfies conditions (9.5). We now state a proposition that lies at the foundation of the Fels-Kaup approach. Proposition 9.2. [41] The map (M , a) → v(M , a) induces a one-to-one correspondence between the affine equivalence classes of the tube realizations of the germ (Q, q) and the Aut(Q, q)-conjugacy classes of commutative Lie subalgebras v ⊂ hol(Q, q) satisfying conditions (9.5). Remark 9.2. It is not hard to construct a tube realization of (Q, q) from a Lie subalgebra v ⊂ hol(Q, q) satisfying conditions (9.5). First of all, by [41] the evaluation map εq is a complex-linear isomorphism between vC and Tq (N), so we identify vC with CK and v with iRK , where K := dim N. For a sufficiently small neighborhood U of 0 in CK let Ψ : U → U be the biholomorphic map onto a neighborhood U of q in N defined by Ψ (ξ ) := exp(ξ )(q), where exp(ξ )(q) denotes the point that corresponds to parameter value 1 on the integral curve of the vector field ξ originating at q. Then for M = Ψ −1 (Q ∩ U ) the germ (M , 0) isa tube of(Q, q), realization

∂ −1 ∗ and taking ϕ to be the germ of Ψ at q, we have v = ϕ iv ∂ Z , v ∈ RK . Remark 9.3. Commutative Lie subalgebras satisfying conditions similar to (9.5) were considered in [4] in the context of CR-manifolds more general than tubes. It was shown in Theorem I.2 in [4] that every such Lie algebra yields a realization of (Q, q) of a certain form that contains a tube realization as a special case. The results of [41], [42], however, aim at describing all local tube realizations of a given manifold up to affine equivalence. Remark 9.4. Although the proof of Proposition 9.2 is essentially straightforward, this proposition is extremely useful. For example, coupled with Theorem 3.1 of [3], it almost immediately implies the real-analyticity result of Proposition 3.1 (see [41]). It then follows that every germ of a spherical tube hypersurface in Cn+1 is a tube realization of any germ of the corresponding quadric Qg in the sense of the definition given above (recall that this definition requires M to be real-analytic). We will now restate Proposition 9.2 in different terms. A map f : M1 → M2 between two CR-manifolds is called an anti–CR-map if for every p ∈ M1 one has: (a) the differential d f (p) of f at p maps Tpc (M1 ) into T fc(p) (M2 ), and (b) d f (p) is

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9 Further Results

anti–complex-linear on Tpc (M1 ). Note that for every tube manifold in CK the map Z → Z is an anti–CR-map fixing every point in its base. This observation motivates the following definition. A real-analytic self-map τ : M → M of a real-analytic CR-manifold M is called an involution of M if τ is an anti–CR-map and satisfies τ 2 = id. If, in addition, τ (p) = p for some p ∈ M, we call τ an involution of the germ (M, p). For example, for a real-analytic tube manifold M in CK and a ∈ M the map τa : Z → Z + 2i Im a is an involution of (M , a). Further, if (M , a) is a tube realization of (Q, q) with ϕ : (Q, q) → (M , a) being the germ of a real-analytic CR-isomorphism ϕˆ , then the map

τ(M , a) := ϕˆ −1 ◦ τa ◦ ϕˆ (considered on an τ(M , a) -invariant neighborhood of q in Q) is an involution of (Q, q). For an involution τ of (Q, q) its differential d τ (q) at q is a linear isomorphism of Tq (Q). We denote by Tq (Q)τ and Tq (Q)−τ the +1-eigenspace and −1-eigenspace of d τ (q) in Tq (Q), respectively. Further, the Lie algebra hol(Q, q) is preserved by the push-forward map of vector fields τ∗ arising from τ . We denote by hol(Q, q)τ and hol(Q, q)−τ the corresponding +1-eigenspace and −1-eigenspace of τ∗ in hol(Q, q). Observe that the commutative Lie algebra v(M , a) defined in (9.4) lies in hol(Q, q)−τ(M , a) . Next, we say that two involutions τ and τ of the germ (Q, q) are equivalent if on a suitable neighborhood of q in Q one has τ = fˆ ◦ τ ◦ fˆ−1 for some fˆ representing an element f ∈ Aut(Q, q). The following result is essentially a reformulation of Proposition 9.2. Proposition 9.3. [41] The map (M , a) → v(M , a) gives rise to a one-to-one correspondence between the affine equivalence classes of the tube realizations of the germ (Q, q) and the Aut(Q, q)-conjugacy classes of commutative Lie subalgebras v ⊂ hol(Q, q) satisfying condition (ii) of (9.5) and the following condition:

there exists an involution τ of (Q, q) with v ⊂ hol(Q, q)−τ .

(9.6)

The involution τ in (9.6) is determined by v uniquely, namely if v = v(M , a) , then τ = τ(M , a) . The involution τ satisfies  τ dim Tq (Q)/Tqc (Q) = 0, (9.7) and for every f ∈ Aut(Q, q) the involution τ corresponding to v := f∗ (v) is equivalent to τ, namely τ = fˆ ◦ τ ◦ fˆ−1 on a suitable neighborhood of q in Q for some fˆ representing f . Remark 9.5. By Proposition 9.2 the explicit determination of all tube realizations of the germ (Q, q) up to affine equivalence requires finding all, up to conjugation by elements of Aut(Q, q), commutative Lie subalgebras v ⊂ hol(Q, q) that satisfy conditions (9.5). Proposition 9.3 allows us to modify this task as follows: determine first, up to equivalence, all involutions of (Q, q) that satisfy (9.7) and then for every such involution τ search for all suitable Lie subalgebras v ⊂ hol(Q, q)−τ . We use

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203

this modification below for the case where Q is a Levi non-degenerate quadric of CR-codimension one (cf. Remark 9.7). The definition of affine equivalence for germs of tube hypersurfaces given in this section is too restrictive for the purposes of the affine equivalence problem for spherical tube hypersurfaces that we are interested in. Indeed, if M0 is the closed strongly pseudoconvex spherical tube hypersurface in C2 with the base given by the equation sin x0 = ex1 , 0 < x0 < π (cf. equation (2) in Theorem 5.1), then any germ (M0 , a) is a tube realization of any germ of the sphere S3 ⊂ C2 , but two germs (M0 , a) and (M0 , a ) are affinely equivalent only if the x1 -coordinates of the points a and a coincide. For our purposes, however, it would be natural to treat any two germs of M0 as affinely equivalent. Motivated by examples of this kind, Fels and Kaup introduce a weaker notion of affine equivalence for germs of tube hypersurfaces than the one given above. For a fixed K let T be the set of all germs (M , a) of all real-analytic tube manifolds M in CK . We introduce on T the coarsest topology with respect to which the set [M ] := {(M , a) : a ∈ M } is open in T for every M . Endowed with this topology,  T is a Hausdorff topological space. Next, we define a map π : T → CK by π (M , a) := a and introduce on every connected component of T the structure of a CR-manifold with respect to which the restriction π |[M ] : [M ] → M is a CR-isomorphism for every M . Further, every affine transformation A of the form (9.3) gives rise to the homeomorphism of T defined as Aˆ : (M , a) → (A (M ), A (a)). If Aˆ maps a connected component T1 of T onto a connected component T2 , then Aˆ is in fact a CR-isomorphism between T1 and T2 . Further, for every M we denote by M˜ the connected component of T containing [M ] and call the pair (M˜, π ) the abstract globalization of M and also the abstract globalization of (M , a) for any a ∈ M . We now say that two germs (M , a) and (M , a ) of real-analytic tube manifolds M , M ⊂ CK are globally affinely equivalent if there exists an affine transformation A of the form (9.3) such that Aˆ(M˜) = M˜ . Let Autω (Q) ⊂ Aut(Q) be the subgroup of all real-analytic CR-automorphisms of Q and Glob(Q, q) the subgroup of the group Aut(hol(Q, q)) of Lie algebra automorphisms of hol(Q, q) generated by the group of inner automorphisms of hol(Q, q) and by { f ∗ : f ∈ Aut(Q, q)}. We now state a result on the global affine equivalence of two tube realizations of the germ (Q, q). Theorem 9.2. [41] Let N be compact and Q closed in N . Assume further that Q is Autω (Q)-homogeneous and has the property that every real-analytic CR-isomorphism between any two germs (Q, q1 ) and (Q, q2 ), with q1 , q2 ∈ Q, extends to a holomorphic automorphism of N . Let (M , a) and (M , a ) be two tube realizations of a germ (Q, q). Then (M , a) and (M , a ) are globally affinely equivalent if and only if v(M , a) can be mapped into v(M , a ) by an element of Glob(Q, q). Remark 9.6. For a non-degenerate C-valued Hermitian form g on Cn we consider, as in Chapter 1, the quadric Qg associated to g and its closure Qg in CPn+1 (see (1.60)). As explained in Section 1.2, the pair Q = Qg , N = CPn+1 satisfies

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9 Further Results

the assumptions of Theorem 9.2. Next, for every q ∈ Qg any element of hol(Qg , q) extends to an element of the Lie algebra hol(Qg ) = suH g , which is the Lie algebra of the group Autω (Qg ) = Aut(Qg ) = Bir(Qg ) = PSU± H g (see the part of Section 1.2 that follows formula (1.59)). Furthermore, upon identification of hol(Qg , q) with   hol(Qg ), we have Glob(Qg , q) = f∗ : f ∈ Aut(Qg ) = Ad(Aut(Qg ))  Aut(Qg ). We also note that hol(Qg , q) is a real form of hol(CPn+1 ) = sl(n + 2, C), which is the Lie algebra of the group Aut(CPn+1 ) = PSL(n + 2, C). Hence, condition (i) of (9.5) is satisfied for any Lie subalgebra v ⊂ hol(Qg , q). Suppose now that for a real-analytic tube manifold M ⊂ CK and its abstract globalization (M˜, π ) the image π (M˜) is an immersed submanifold of CK . In this case we set Mˆ := π (M˜) and call Mˆ the globalization of M and also the globalization of (M , a) for any a ∈ M . Clearly, the globalization Mˆ contains M as an open subset and is maximal with respect to this property. In particular, every real-analytic tube manifold M ⊂ CK which is closed as a submanifold of CK is the globalization of every germ of M . We now state a general globalization result which implies our globalization Theorems 4.1 and 4.2. Theorem 9.3. [41] Let Q be closed in N and suppose that for a point q ∈ Q and every Lie subalgebra v ⊂ hol(Q, q) satisfying conditions (9.5), every element of vC extends to a complete vector field in hol(N). Then every tube realization (M , a) of the germ (Q, q) has a globalization Mˆ ⊂ CK which is closed as a submanifold of CK and for which there exists a locally biholomorphic map ψ : CK → N with ψ (Mˆ) ⊂ Q. Application of Theorem 9.3 to Q = Qg , N = CPn+1 yields Theorems 4.1 and 4.2. With Z = (ζ0 , ζ1 , . . . , ζn+1 ), we now reduce the Hermitian form H g (Z, Z) on Cn+2 defined in (1.48) to the diagonal form h(Z, Z) :=

k

n+1

α =0

α =k+1

∑ |ζα |2 − ∑

|ζα |2

(9.8)

for some 0 ≤ k ≤ n. Here the signature of g is (k, n − k) and the signature of H g is (k + 1, n − k + 1). Without loss of generality we assume n ≤ 2k and let r := k + 1, s := n − k + 1 with r ≥ s. Consider the hypersurface   Q := Z ∈ CPn+1 : h(Z, Z) = 0 , which is CR-equivalent to Qg (here, as before, Z = (ζ0 : ζ1 : . . . : ζn+1 )). We now set Q = Q and N = CPn+1 (in particular, K = n + 1 for every tube realization of any germ of Q). Define   G := L ∈ SL(n + 2, C) : h(LZ, LZ) = ±h(Z, Z), Z ∈ Cn+2 , SU(h) := G◦ ,   g := su(h) := L ∈ sl(n + 2, C) : Re h(LZ, Z) = 0, Z ∈ Cn+2 .

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205

Clearly, we have Aut(Q) = G/(center). For future use we denote by σ the anti– complex-linear real Lie algebra automorphism of sl(n + 2, C)R for which the fixed point set sl(n + 2, C)σ coincides with g (whenever the symbol R is used as a superscript for complex Lie groups and algebras, it is meant that on such occasions they are regarded as real Lie groups and algebras). Remark 9.7. For every germ (Q, q) all involutions τ satisfying condition (9.7) with Q = Q can be found explicitly (see [41]). Namely, every such τ extends to an involution of CPn+1 (which we denote by the same symbol) such that there exists f ∈ Aut(Q)◦ with τ = f ◦ τ0 ◦ f −1 , where τ0 is the standard involution of CPn+1 arising from the complex conjugation Z → Z on Cn+2 . It then follows that any two involutions τ and τ of (Q, q) are equivalent: there exists an element f ∈ Aut(Q)◦ fixing the point q such that τ = f ◦ τ ◦ f −1 . Further, Proposition 9.2, Theorem 9.2 and Remark 9.6 yield that in order to obtain an affine classification of closed tube hypersurfaces in Cn+1 locally CRequivalent to Q, one needs to determine all Ad(G)-conjugacy classes of commutative Lie subalgebras v ⊂ g satisfying the following condition: the Lie algebra vC ⊂ hol(CPn+1 ) has an open orbit in CPn+1 , that is, for some q ∈ CPn+1 the evaluation map εq at q from vC to Tq (CPn+1 ) is surjective.

(9.9)

Note that if v satisfies condition (9.9), then such a point q can be chosen to lie in Q. Further, utilizing Proposition 9.3, Remark 9.7 and certain results of [41], one observes that condition (9.9) implies the following conditions: (i) v is a maximal commutative Lie subalgebra in g, and (ii) Ad(L)(v) ⊂ g−τ0 for some L ∈ G◦ .

(9.10)

We call a maximal commutative Lie subalgebra in g a MASA (Maximal Abelian SubAlgebra) in g, and any MASA that satisfies condition (ii) of (9.10) a qualifying MASA. Instead of classifying, up to Ad(G)-conjugation, Lie subalgebras of g satisfying (9.9), we will classify, up to Ad(G)-conjugation, all qualifying MASAs. It will turn out a posteriori (see Theorems 9.4 and 9.5) that every qualifying MASA in fact satisfies (9.9). Below we briefly describe the approach to the classification problem for qualifying MASAs proposed in [42]. In what follows, for a Lie algebra a and a subset b ⊂ a we denote by Ca (b) the centralizer of b in a and by Z (a) the center of a. If a is reductive, we denote by ass the (uniquely defined) semi-simple part of a. Also, if a is a Lie subalgebra of gl(, C), we call a maximal commutative Lie subalgebra of a that consists of nilpotent elements of End(C ) a MANSA (Maximal Abelian Nilpotent SubAlgebra) in a. The approach of [42] is based on the observation that every MASA v ⊂ g ⊂ gl(n + 2, C) has a unique decomposition into its toral and nilpotent parts v = vred ⊕ vnil , where the Lie subalgebras vred and vnil consist of all semi-simple and nilpotent elements of v in End(Cn+2 ), respectively. Suppose that v lies in g−τ0 .

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9 Further Results

Then the centralizer Cg (vred ) is a τ0 -stable real reductive Lie subalgebra of g, and the maximality of v implies vred = Z (Cg (vred )). Further, vnil is a MANSA in Cg (vred ) con−τ  tained in (Cg (vred ))ss 0 . These observations reduce the task of classifying, up to Ad(G)-conjugation, all qualifying MASAs in g to the following two tasks: (R) classify up to Ad(G)-conjugation all reductive Lie subalgebras r ⊂ g that are the centralizers of the toral parts of qualifying MASAs in g, and

(9.11)

(N) given a τ0 -stable reductive Lie subalgebra r = Z (r) ⊕ r ⊂ g as above, classify all MANSAs in r contained in (rss )−τ0 . ss

We start with task (R). Let v be a qualifying MASA with vred ⊂ g−τ0 . It is wellknown that Cn+2 can be decomposed into the direct sum of complex linear subspaces V j , with j ranging in a finite index set J , such that the following holds: Cgl(n+2,C) (vred ) = Cg+ig (vred ) =



gl(V j , C).

(9.12)

j∈J

Since vred is τ0 -stable, there is an involution j → j of J with V j = Vj . Furthermore, the restriction of the Hermitian form h defined in (9.8) to V j + V j (which we denote by h j ) can be shown to be non-degenerate for every j ∈ J . We now split the index set J as follows: J = K L L with K := { j ∈ J : j = j}. For every j ∈ K the involution σ preserves the Lie subalgebra sl(V j , C) ⊂ sl(n + 2, C), and we have sl(V j , C)σ = su(h j ). For every j ∈ L the Hermitian form h j has signature (m j , m j ), where m j := dimV j , we have σ (sl(V j , C)) = sl(V j , C), and the fixed point set (sl(V j , C) ⊕ sl(V j , C))σ is isomorphic to sl(V j , C)R . The involution σ acts on sl(V j , C) ⊕ sl(V j , C) as follows. Choose a basis of the form e1 , . . . , em j , e1 , . . . , em j in V j ⊕ V j , where e1 , . . . , em j is a basis in V j . In such a basis, h j is represented by a matrix of the form

0H , H0 where H is a symmetric non-degenerate m j × m j matrix. Then for an element of sl(V j , C) ⊕ sl(V j , C) represented by a pair of matrices (A, B) one has

σ ((A, B)) = (−H−1 BT H, −H−1AT H). Equivalently, σ ((A, B)) = (−BV , −AV j ), where j

AV j is the adjoint of the operator A ∈ End(V j ) with respect to the non-degenerate bilinear form on V j defined as

β j (v1 , v2 ) := h(v1 , v2 ),

(9.13)

and BV is the adjoint of the operator B ∈ End(V j ) with respect to the non-degenerate j bilinear form on V j defined analogously. Observe that in the basis e1 , . . . , em j the bilinear form β j is given by the matrix H. We now state a solution to task (R) of (9.11).

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Proposition 9.4. [42] Let v be a qualifying MASA with vred ⊂ g−τ0 . Then we have  

  red v = iR idV j +V j ⊕ R idV j − idV j j∈K ∪L

and (Cg (vred ))ss =

trace = 0



su(h j ) ⊕

j∈K



j∈L

(sl(V j , C) ⊕ sl(V j , C))σ .

(9.14)

j∈L

Furthermore, there exist only finitely many Ad(G)-conjugacy classes of the toral parts of qualifying MASAs in g. The last statement of Proposition 9.4 follows by considering Cartan subalgebras of g. One can show that every Ad(G◦ )-conjugacy class of Cartan subalgebras in g (there are exactly s + 1 such classes) has a representative in g−τ0 and such a representative can be constructed explicitly as the Lie subalgebra of all elements in g diagonal with respect to a special basis in Cn+2 . [For the purposes of this construction, considering the form h defined in (9.8) is more convenient than the form H g defined in (1.48).] For every qualifying MASA v the conjugation by a suitable element L ∈ Ad(G◦ ) of its toral part vred lies in one of these representatives, say h. Then for the Lie algebra Ad(L)(v) each of the subspaces V j in Proposition 9.4 is spanned by some of the vectors of the special basis corresponding to h, and the result on the finiteness of the number of Ad(G)-conjugacy classes of the toral parts of qualifying MASAs in g follows. We now turn to task (N) of (9.11). It follows from (9.14) that for r = Cg (vred ) with red v ⊂ g−τ0 and for every MANSA n in rss one has the corresponding decomposition 

n=

n j,

(9.15)

j∈K ∪L

with

 n j := n ∩ s j for s j :=

su(h j ) (sl(V j , C) ⊕ sl(V j

if j ∈ K , , C))σ

if j ∈ L .

(9.16)

Every n j is a commutative Lie subalgebra of s j that consists of nilpotent elements of End(V j +V j ). For j ∈ K the Lie algebra n j has dimension r j + s j − 1, where (r j , s j ) is the signature of the form h j (see Theorem 9.4). Hence, r j = 0 is only possible for s j = 1, and s j = 0 is only possible for r j = 1. For j ∈ L the Lie algebra n j has dimension 2m j − 2 (see Theorem 9.5). Before approaching task (N), we extract a certain combinatorial invariant from the decompositions of Proposition 9.4. For every set A we denote by F (A) the free commutative monoid over A, i.e. the set of all linear combinations of the form ∑a∈A na · a with na ∈ Z+ and ∑a∈A na < ∞. Define K := {(t, u) ∈ Z2+ : (tu = 0) ⇒ (t + u = 1)}, J := K ∪ L,

D := F (J) = F (K) + F (L).

L := N

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9 Further Results

We write a point in D as D = ∑ j∈J n j · j. The permutation of J defined as (t, u) → (u,t) on K and as the identity on L induces an involution D → Dopp of D. Further, for every r, s ∈ Z+ we let Dr,s ⊂ D be the subset of all elements D such that ∑ n(t,u)t + ∑ n j j = r, ∑ n(t,u) u + ∑ n j j = s. (t,u)∈K

(t,u)∈K

j∈L

j∈L

We now introduce an invariant, which we call the D-invariant. It takes values in D and for a qualifying MASA v ⊂ g its value D(v) is D(v) :=

∑

j∈K

1 · (r j , s j ) + ∑ 1 · m j , j∈L

where r j , s j , with j ∈ K , and m j , with j ∈ L , are the corresponding numbers arising as explained above from decomposition (9.12) for the Lie algebra Ad(L)(v), where L ∈ G◦ is any element for which Ad(L)(vred ) ⊂ g−τ0 . The relevance of the D-invariant to the classification problem for qualifying MASAs in g is demonstrated by the following proposition. Proposition 9.5. [42] An element D ∈ D is the D-invariant of a qualifying MASA in g if and only if D ∈ Dr,s . Furthermore, for two qualifying MASAs v1 , v2 in g the red ◦ Lie subalgebras vred 1 , v2 are Ad(G )-conjugate in g if and only if D(v1 ) = D(v2 ). red For r = s the Lie subalgebras v1 , vred 2 are Ad(G)-conjugate in g if and only if either D(v1 ) = D(v2 ), or D(v1 ) = D(v2 )opp . By Proposition 9.4 there exist only finitely many Ad(G)-conjugacy classes of the toral parts of qualifying MASAs in g. Using the D-invariant, one can enumerate these conjugacy classes by the elements of Dr,s as explained in Proposition 9.5. red red red Observe that vred 1 , v2 are Ad(G)-conjugate if and only if Cg (v1 ), Cg (v2 ) are Ad(G)conjugate. Therefore, there are only finitely many Ad(G)-conjugacy classes of Lie subalgebras r in g that need to be considered in task (N) of (9.11), and these classes are enumerated by the elements of Dr,s . Thus, we now assume that r lies in the Ad(G)-conjugacy class in g given by a particular element of Dr,s . Next, it follows from (9.15), (9.16) that the problem of determining all MANSAs in rss contained in (rss )−τ0 reduces to the problem of determining all MANSAs in su(h j ) contained in su(h j )−τ0 for j ∈ K and all MANSAs in  − τ (sl(V j , C) ⊕ sl(V j , C))σ  sl(V j , C)R contained in sl(V j , C)R 0 for j ∈ L . Here

for j ∈ L the involution τ0 acts on (sl(V j , C))R as τ0 (A) = −AV j , where, as before, AV j is the adjoint of the operator A ∈ End(V j ) with respect to the non-degenerate bilinear form β j on V j defined in (9.13). These two cases will be dealt with separately. Firstly, MANSAs in su(h j ) for j ∈ K are described in the following theorem.

Theorem 9.4. [42] The following statements hold: (1) For every MANSA n in su(h j ) contained in su(h j )−τ0 the subset N := in|V τ0 of j

End(V jτ0 ), where V jτ0 is the fixed point set of τ0 in V j , is a real nilpotent commutative

9.2 Approach of G. Fels and W. Kaup

209

associative subalgebra of dimension r j + s j − 1 in End(V jτ0 ) having a 1-dimensional annihilator. (2) For two MANSAs n and n˜ in su(h j ) contained in su(h j )−τ0 the following statements are equivalent: (i) N := in|V τ0 and N˜ := i˜n|V τ0 are isomorphic as real associative algebras, j

j

(ii) n and n˜ are conjugate by an element of Ad(SO(h j )), where SO(h j ) is the closed connected subgroup of SU(h j ) with Lie algebra su(h j )τ0 . (3) Every real nilpotent commutative associative algebra N of dimension r j + s j − 1 having a 1-dimensional annihilator A can be realized (by means of restricting the left regular representation of its unital extension N 0 := R ⊕ N  V jτ0 to N ) as a τ subalgebra in End(V j 0 ) such that N = in|V τ0 , with n being a MANSA in su(h j ) j

contained in su(h j )−τ0 for which nC has an open orbit in P(V j ). Here h j is the non-degenerate Hermitian form on V j obtained by continuing from V jτ0  N 0 the non-degenerate bilinear form b : N 0 ×N 0 → A  R given by b(v1 , v2 ) := π (v1 v2 ), with π ∈ End(N 0 ) being any projection with range A satisfying π (1) = 0. (4) Any MANSA n in su(h j ) contained in su(h j )−τ0 is related to some N as described in (3). Secondly, we describe MANSAs in sl(V j , C)R for j ∈ L . Theorem 9.5. [42] The following statements hold: (1) Every MANSA n in sl(V j , C)R contained in (sl(V j , C)R )−τ0 , regarded as a subset of End(V j ), is a complex nilpotent commutative associative subalgebra N of dimension m j − 1 in End(V j ) having a 1-dimensional annihilator. Every element of N is symmetric with respect to the bilinear form β j defined in (9.13). (2) For two MANSAs n and n˜ in sl(V j , C)R contained in (sl(V j , C)R )−τ0 the following statements are equivalent: (i) n and n˜ , regarded as complex associative algebras N and N˜ , respectively, are isomorphic, (ii) n and n˜ are conjugate by an element of Ad(SO(V j , C)R ), where SO(V j , C)R is the closed connected subgroup of SL(V j , C)R with Lie algebra (sl(V j , C)R )τ0 . (3) Every complex nilpotent commutative associative algebra N of dimension m j − 1 having a 1-dimensional annihilator A can be realized (by means of restricting the left regular representation of its unital extension N 0 := C ⊕ N  V j to N ) as a subalgebra in End(V j ) consisting of trace-free transformations such that, regarded as a subset of sl(V j , C)R , it is a MANSA n in sl(V j , C)R contained in (sl(V j , C)R )−τ . Here τ acts on sl(V j , C)R as τ (A) = −A , where A is the adjoint of the operator A ∈ End(V j ) with respect to the non-degenerate bilinear form b : N 0 × N 0 → A  C given by b(v1 , v2 ) := π (v1 v2 ), with π ∈ End(N 0 ) being any projection with range A satisfying π (1) = 0. Furthermore, the sum C(idV j , − idV j )⊕ (n× n) has an open orbit in P(V j ⊕V j ), where the elements of n× n

210

9 Further Results

are represented by pairs of matrices (A, A) in a basis of the form e1 , . . ., em j , e1 , . . .,em j in V j ⊕ V j , with A representing an element of n in the basis e1 , . . . , em j in V j . (4) Any MANSA n in sl(V j , C)R contained in (sl(V j , C)R )−τ0 is related to some N as described in (3).

Theorems 9.4 and 9.5 reduce task (N) of (9.11) to the problem of classifying, up to isomorphism, all finite-dimensional real and complex nilpotent commutative associative algebras with 1-dimensional annihilator. For brevity we call such algebras admissible. In fact, admissible algebras are exactly the maximal ideals of real and complex Gorenstein algebras of finite dimension greater than 1 (recall that a local commutative associative algebra of finite dimension greater than 1 is a Gorenstein algebra if and only if the annihilator of its maximal ideal is 1-dimensional – see, e.g. [5], [51]). Furthermore, following the procedure outlined in Remark 9.2 above, Fels and Kaup explain in [42] how one can explicitly find an equation of the closed spherical tube hypersurface corresponding to any qualifying MASA in g. In particular, there exists an algorithm (which we call Algorithm (A)) associating a spherical tube hypersurface to every complex admissible algebra and to every real admissible algebra of dimension greater than 1. The hypersurfaces associated to two admissible algebras are affinely equivalent if and only if the algebras are isomorphic. Remark 9.8. We note that application of Algorithm (A) to a real admissible algebra produces a spherical tube hypersurface affinely equivalent to a hypersurface of the form x0 = F(x), F is a polynomial, (9.17) and all spherical tube hypersurfaces of the form (9.17) are obtained in this way. Similarly, one can construct a certain algebraic hypersurface in the maximal ideal of every Gorenstein algebra of finite dimension greater than 1 over any field of characteristic zero. Such hypersurfaces and their applications to problems in algebra and geometry were studied in [40], [43], [60], [61]. In particular, until recently it has been an open question whether such hypersurfaces are always affinely homogeneous. First examples that fail to have the affine homogeneity property appeared in [43] where proofs relied on computer algebra. Such examples can be constructed over R, which yields the existence of spherical tube hypersurfaces whose bases are not affinely homogeneous (cf. Chapter 8). We note that the affine homogeneity does take place over any field if the corresponding Gorenstein algebra is Z+ -graded (see [40], [42], [43], [61]). Remark 9.9. While it seems impossible to obtain a reasonable classification of admissible algebras in full generality, article [42] contains such a classification for s = 1, 2. As a consequence, the affine classifications of closed (k, n − k) spherical tube hypersurfaces in Cn+1 for k = n and k = n − 1 presented in Chapters 5, 6 above, were recovered in [42]. We will now give an example of a special family of real admissible algebras. Let E = E0 ⊕ E1 ⊕ E2 ⊕ E3 be a real vector space, with E0 = R, dim E3 = 1, and dim E1 = dim E2 = d ≥ 1. Choose a basis {ξ0 , . . . , ξd , η0 , . . . , ηd } in E such that

9.2 Approach of G. Fels and W. Kaup

211

ξ0 = 1 ∈ E0 , ξ1 , . . . , ξd ∈ E1 , η1 , . . . , ηd ∈ E2 , η0 ∈ E3 . Define a bilinear form h on E by the relations h(ξi , η j ) = δi j ,

h(ξi , ξ j ) = h(ηi , η j ) = 0,

i, j = 0, . . . , d.

We are interested in commutative associative algebra structures on E with unit ξ0 satisfying (i) E j Ek ⊂ E j+k , j, k = 0, 1, 2, 3, (9.18) (ii) ab = h(a, b)η0 for all a ∈ E1 , b ∈ E2 , where E j := {0} for j ≥ 4. For every such structure on E the (unique) maximal ideal N := E1 ⊕ E2 ⊕ E3 is a real admissible algebra with annihilator E3 . Clearly, E is the unital extension of N . All such structures on E turn out to be given by cubics as stated in the following proposition. Proposition 9.6. [42] For every real cubic c on E1 there exists a unique commutative associative algebra structure Ec on E satisfying (9.18) and such that a3 = c(a)η0 for all a ∈ E1 . Furthermore, every commutative associative algebra structure on E satisfying (9.18) is obtained in this way. Let Nc be the maximal ideal of Ec . The next proposition describes the equivalence classes of the admissible algebras Nc . Proposition 9.7. [42] For two cubics c and c on E1 the algebras Ec , Ec (and hence their maximal ideals Nc , Nc ) are isomorphic if and only if c and c are linearly equivalent. Remark 9.10. Applying Algorithm (A) to Nc , one obtains a tube hypersurface in C2d+1 affinely equivalent to the tube hypersurface with the base defined by the equation x0 =

d

∑ xα x2d−α +1 + c(xd+1, . . . , x2d ).

(9.19)

α =1

For d = 3 and c(x4 , x5 , x6 ) = ct (x4 , x5 , x6 ) = x34 + x35 + x36 +tx4 x5 x6 , with t ∈ R, equation (9.19) gives the base of the hypersurface St defined in (8.1). It follows from the results of this section that every hypersurface St is (3,3)-spherical, and two such hypersurfaces St1 , St2 are affinely equivalent if and only if the corresponding cubics ct1 , ct2 are linearly equivalent. For sufficiently small |t| the pairwise linear nonequivalence of the cubics ct is not hard to establish. This was done in [42] (cf. the alternative argument utilizing the j-invariant given in Section 8.3). Thus, the results of this section yield a proof of Theorem 8.2, which is in fact the original proof found in [42]. Remark 9.11. It is clear from techniques described above that the tube hypersurfaces defined in (8.7) are spherical and pairwise affinely non-equivalent for small |t|. By considering these hypersurfaces Fels and Kaup obtain an alternative proof of Theorem 8.1 in cases (ii) and (iii) (cf. Remark 8.2).

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9 Further Results

Remark 9.12. Observe that one source of admissible algebras is isolated hypersurface singularities. Suppose that W is a complex hypersurface germ at the origin in C given as the zero locus of a holomorphic function germ f . We denote by O the algebra of holomorphic function germs at the origin in C and assume that f is a generator of the ideal I (W ) ⊂ O that consists of all elements of O vanishing on W . Suppose that the origin is an isolated singularity of W and consider the Tjurina algebra, or moduli algebra, of W A (W ) := O /I( f ), where I( f ) is the ideal in O generated by f and all its first-order partial derivatives. It is well-known that A (W ) is a commutative associative algebra of finite positive dimension independent of the choice of f and the coordinate system near the origin (see, e.g. [50], Section 2.1). Nakayama’s lemma implies that the (unique) maximal ideal N (W ) of A (W ) is a complex nilpotent associative algebra. Assume now that the origin is a quasi-homogeneous singularity (see [91]), that is, if for some (hence for any) generator f of I (W ) there exist positive integers p1 , . . . , pm , q such that, modulo a biholomorphic change of coordinates, f is the germ of a polynomial Q satisfying Q(s p1 z1 , . . . , s pm zm ) ≡ sq Q(z1 , . . . , zm ) for all s ∈ C. By a theorem due to Saito (see [91]), the singularity of W is quasi-homogeneous if and only if I( f ) = J( f ), where J( f ) is the Jacobian ideal of f , that is, the ideal in O generated by all first-order partial derivatives of f . Hence, for a quasi-homogeneous singularity, A (W ) coincides with the Milnor algebra O /J( f ) for any generator f of I (W ). Further, the singularity of W is quasi-homogeneous if and only if N (W ) is a complex admissible algebra, provided N (W ) is positive-dimensional (see [5], [75], [82], [92]). Thus, if N (W ) is non-trivial, by applying Algorithm (A) one can construct a spherical tube hypersurface from N (W ). Furthermore, if dim N (W ) ≥ 2 and N (W ) has real forms, one can construct a spherical tube hypersurface from every real form of N (W ). For example, for n = 7 and every τ ∈ [−6, −2) ∪ (−2, 2) the hypersurface P0τ introduced in Remark 8.1 arises from a real form of the maximal ideal of the moduli algebra of a simple elliptic hypersurface singularity of type E˜ 7 . Similarly, for every t = 0, 6 the hypersurface St defined in (8.1) arises from a real form of the maximal ideal of the moduli algebra of a simple elliptic hypersurface singularity of type E˜6 (for details concerning simple elliptic singularities see [92]). This real form is isomorphic to the algebra Nc−18/t , and the corresponding moduli algebra is isomorphic to Ec−18/t . It is interesting to observe that if, in addition, t = −3, then the values of the j-invariant for the elliptic curves Zt and Z−18/t defined in (8.6) are reciprocal (see [29]).

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•

Index

(k, m − k, n − m)-spherical CR-hypersurface, 198 (k, n − k)-spherical CR-hypersurface, 23 G-structure, 2 G-structures reducible to absolute parallelisms, 3 χQ , 59 absolute parallelism, 3 abstract globalization of a germ of a tube manifold, 203 abstract globalization of a tube manifold, 203 admissible algebra, 210 affinely equivalent tube hypersurfaces, 42 affinely equivalent germs of tube manifolds, 200 anti–CR-map, 201 base of a tube hypersurface, 41 base of a tube manifold, 197 Bianchi identities, 20 Cartan connection, 19 complex foliation on a CR-manifold, 195 complex tangent space, 1 CR-automorphism, 2 CR-codimension, 1 CR-curvature form, 18 CR-dimension, 1 CR-equivalence, 2 CR-equivalent CR-manifolds, 2 CR-flat CR-hypersurface, 23 CR-function, 3 CR-hypersurface, 1 CR-isomorphism, 2 CR-manifold, 1 CR-map, 2

CR-structure, 1 CR-submanifold, 2 curvature form, 7, 18 defining system, 50 defining systems of types I, II, III, 62 equivalent G-structures, 2 equivalent Hermitian forms, 4 equivalent involutions of a germ of a CR-manifold, 202 generic CR-manifold in a complex manifold, 200 globalization of a germ of a tube manifold, 204 globalization of a tube manifold, 204 globally affinely equivalent germs of tube manifolds, 203 Gorenstein algebra, 210 induced CR-structure, 2 infinitesimal CR-automorphism, 2 integrable CR-structure, 3 involution of a CR-manifold, 202 involution of a germ of a CR-manifold, 202 isomorphism of G-structures, 2 leaf of a foliation, 196 Levi foliation, 197 Levi form, 4 Levi non-degenerate CR-manifold, 4 Levi-flat CR-manifold, 197 local CR-automorphism, 2 locally CR-straightenable foliation on a CR-manifold, 196 locally holomorphically straightenable foliation on a CR-manifold, 196

219

220 MANSA, 205 MASA, 205 matrix symmetric with respect to a bilinear form, 46 Maurer-Cartan equation, 22 Maurer-Cartan form, 22 Milnor algebra of a function germ, 212 moduli algebra of an isolated hypersurface singularity, 212 non-degenerate Hermitian form, 4 quadric associated to a Hermitian form, 5 quasi-homogeneous isolated hypersurface singularity, 212 real hypersurface in a complex manifold, 2 rigid hypersurface, 35 rigid polynomial hypersurface, 39 rigid representation, 35

Index signature of a non-degenerate Hermitian form, 4 signature of the Levi form, 4 spherical CR-hypersurface, 23, 198 standard representation of a tube hypersurface, 42 strongly pseudoconvex CR-hypersurface, 43 strongly uniform CR-manifold, 4 Tjurina algebra of an isolated hypersurface singularity, 212 torsion of the curvature form, 18 tube hypersurface, 41 tube manifold, 197 tube neighborhood, 42 tube realization, 200 umbilic point, 31 weakly uniform CR-manifold, 6

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