3-nets with maximal family of two-dimensional subnets

Abh. Math. Sem. Univ. Hamburg 61 (1991), 203-211

3-nets with Maximal Family of Two-dimensional Subnets

By PETERT. NAGY

1 Introduction Plane 3-webs with the hexagonality closure condition were investigated by W. BLASCHKEand his school in the early period of the history of web geometry [5]. The study of the fundamental properties of multidimensional hexagonal 3-webs was initiated by S.S. Cr~ERN in 1936 [7] and systematically proved by M.A. A~:Ivls [2]. He introduced the notion of transversal geodesic 3-web, and characterized the transversal geodesic and hexagonal properties of 3-webs using curvature tensor identities for the associated Chern connection. Moreover, he obtained as necessary and sufficient condition for these classes of 3-webs the existence of local one-parameter subloops or subgroups, respectively, in each tangential direction at the identity in local coordinate loops. These results can be formulated in the way that the classes of transversal geodesic and hexagonal 3-webs are characterized by the existence of maximal family of two-dimensional subwebs, or group-subwebs, respectively (cf. [3]). The early results of web geometry, especially its coordinatization theory gave a motivation for the development of the abstract theory of nets, quasigroups and loops [6], [13]. In the global differential geometric investigation of web structures the notions of 3-webs and 3-nets have to be distinguished. The first one is defined by tensorfields such that three tangent distributions determined by these tensorfields are integrable and the second one means a special 3-web which is a differential geometric version of an incidence structure studied in the foundation of geometry and combinatorial geometry (cf. [1], [4], [8]). The purpose of this paper is a geometric investigation of differentiable 3-nets which are globalized versions of transversal geodesic and hexagonal 3-webs and to discuss the Akivis' conditions in this context. We will use the notation and terminology introduced in our paper [10], [11].

2

3-webs and 3-nets

Definition 2.1. A differentiable 3-web on a manifold M is a triple of foliations {21,22,23} such that the tangent spaces of the leaves of any two different foliations 2~, 2/~(~ ~ fl) through any point of M are complementary subspaces

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of the tangent vector space of M. The leaves of the foliations 21,22,23 are called horizontal, vertical or transversal, respectively.

Definition 2.2. A 3-web on a manifold M is called a differentiable 3-net if the leaves of different foliations of the 3-web intersect exactly in one point. The leaves of a 3-net are called (horizontal, vertical and transversal) lines of M. The projection operators of the tangent bundle T M onto the horizontal, vertical and transversal tangent distributions T(h)M, T(~)M and TIt)M with respect to the decomposition T M = T(h)M ~ T~O)M determine the (1,1)tensorfields H and V on M satisfying H 2 = H, V 2 = V and H + V = Id. With help of these tensorfields we can define the (1,1)-tensorfield J satisfying j 2 ~__ Id, J o H = V o J and T(t)M = KerV o (J + Id).

Definition 2.3. An {H, J}-structure on the manifold M is defined to be a pair of (1,1)-tensorfields H and J satisfying H2=H,

J2 = Id,

HoJ+J•H=J.

We write V := I d - H and say that H and V are horizontal and vertical

projection operators on TM. For any {H,J)-structure on the manifold M there exists a unique affine connection

Vx Y = H{JiHX, JHY] + [VX, HY]} + V{J[VX, J V Y ] + [HX, VY]} satisfying VH --- VJ = 0 and whose torsion tensorfield T has the property T(HX, VY) = 0 for all vectorfields X and Y on M. If the {H,J}-structure is associated to a 3-web or 3-net then this connection V is called the Chern

connection.

3

Subwebs and Transversally Geodesic 3-webs

Definition 3.1. Suppose that a 3-web structure is given on the submanifold S of the 3-web manifold M. S is called a subweb of the 3-web M if its leaves are intersections of S with the leaves of the 3-web M. The 3-net K is a subnet of a 3-net manifold M if it is a subweb of M.

Definition 3.2. A 2-plane through 0 in a tangent space TpM of a web M is called transversal 2-plane if it is invariant with respect to the Jp : TpM ~ TeM. A 2-surface r : S --* M immersed into the web M is called transversally geodesic if the images of its tangent planes T(o(q)M are transversal 2-planes.

manifold operator manifold ~b. TqS

Proposition 3.3. A 2-submanifold of a 3-web manifold M has an induced subweb structure if and only if it is a transversally geodesic 2-surface.

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Proof If the immersed 2-surface ~b : S ~ M is transversally geodesic then it is clear that the tensorfields H, V, J associated to the 3-web on M can be restricted to ~,TqS, consequently the transversally geodesic 2-surface S has an induced 3-web structure. On the other hand if S is an immersed two-dimensional subweb and p ~ S is a point on S then its horizontal, vertical and transversal tangent vectors are contained in the tangent plane TpS that is this plane is invariant with respect to the operator Jp. It follows that S is transversally geodesic. [] In the following for the simplicity we shall consider the immersed transversally geodesic surfaces as subsets in the manifold M.

Proposition 3.4. The transversally geodesic 2-surfaces of a web manifold M are autoparallel submanifolds with respect to the Chern connection V. Proof Let U and neighbourhood of tangent vectorfield /~ are functions on

J U be horizontal and vertical vectorfields defined on a a transversally geodesic 2-surface S c M. An arbitrary on S can be written in the form eU + flJU, where c~ and S. One can calculate

V~u+~Ju()~U -4- #JU) = = [~U)~-k- flJU2]U -b [c~U#-t- flJU#]JU -k + ct2VuU + ~t#Vu(JU) + fl2VsuU + fl#Vju(JU). We have

VvU = HJ[U, JU], VjvU = H[JU, U],

Vv(JU) = V[U, JU], Vjv = VJ[JU, U].

Since U and JU span an integrable 2-plane field we get from Frobenius theorem [U, JU] = pU + aJU, that is V~v+~jv(2U + #JU) is contained in the tangent 2-plane TpS. []

Proposition 3.5. The leaves of the subweb induced on a transversally geodesic 2-surface S of 3-web manifod M are geodesics with respect to the Chern connection. Proof The assertion follows from the fact that the leaves of a 3-web are autoparalM submanifolds with respect to the Chern connection (cf. [10] Theorem 3.6).

Definition 3.6. A 3-web on the manifold M is called transversally geodesic if for any horizontal vectors Up ~ T(h)M,p E M there exists a transversally geodesic 2-surface tangent to the 2-plane spanned by the vectors Up, JUp.

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Remark. Since the transversally geodesic 2-surfaces are totally geodesic, they are (locally) uniquely determined by their tangent plane at a given point p. It follows that the transversally geodesic 3-webs have a maximal family of subwebs. Theorem 3.7. A 3-web on the manifold M is transversally geodesic if and only if there exists a (2, O)-tensorfield g(X, Y) on M satisfying a{g(X, Y)Z = R(X, J Y ) Z } for all horizontal vectors X, Y, Z E T (h), p E M, where a denotes the summation for all permutations of the variables X, Y, Z. Proof Let S be a transversally geodesic 2-surface in M. We suppose that the horizontal vectorfield U on S is normalized by the condition Vu U = 0. Then its integral curves are affinelly parametrized horizontal geodesics on S. If we consider a vertical geodesic on S then with help of its parameter we can describe a 1-parameter family of horizontal geodesics which is a geodesic variation on M. The corresponding infinitesimal variation vectorfield W satisfies the Jacobi differential equation VuVuW + V u ( T ( W , U ) ) + R(W, U)U = 0 along a horizontal geodesic (cf. KOBAYASHI-NOMIZU[9] Theorem 1.2 in Chapter VIII). We denote H W = ~U, V W = flJU where ~ and fl ~ 0 are real functions, then the Jacobi equation can be written in the form VuVu(~U + flJU) = flR(U, JU)U, since T(U, U) = T(JU, U) = 0 and R(~U + flJU, U)U = flR(U, JU)U. Using Vu U = VuJU = 0 we obtain the equivalent equation (U(U~))U + (U(Ufl))JU = flR(U, JU)U. Since the right hand side of this equation is horizontal it can be written as (U(U~))U -=-flR(U, JU)U, u ( u f l ) = o.

It is easy to see that if the functions ~ and fl satisfy these equations then ~U + flJU is a Jacobi vectorfield along the integral horizontal geodesic curve of the vectorfield U. If the 3-web is transversally geodesic then it follows from the preceding equations that for all horizontal vectors Up E T~ph)M,p C M there exist a scalar f(Up) satisfying f(Ue)U e = Rp(Up, JUe)U e.

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Since the right hand side of this equation is a cubic form in Up the function f is a quadratic form in Up. Hence it can be written in the form f ( U ) = g(U, U), where g is a (2, 0)-tensorfield on M. Thus for transversally geodesic 3-webs we obtain the identity R(U, J U ) U = g(U, U)U

for horizontal vectorfields U on M, which is equivalent to the identity in the Theorem. Now, we suppose that this identity is satisfied. Let be given p ~ M, Uv ~ T~h)M and let 7(w), w ~ I c R be the vertical geodesic with the initial value 7(0) = p, y'(O) = JpUp. For a given w ~ I we consider the horizontal geodesic ~w(u) satisfying ~w(0) = y(w) and ~'(0) = JT(w)~1(O). The geodesics ~w(U) define in the neighbourhood (u, w) ~ I x I c IR: a 2-surface ~b(u,w) tangent to the transversal 2-plane spanned by the vectors Up and JUp. The tangent vectorfields U(u, w) = ~b. ~ and W(u, w) = c~. satisfy v v u = 0,

[u, w ] = 0,

and along the vertical geodesic y(w) V w W = O,

V w U = V w J W = J V w W = O.

We fix a Wo E I. The vectorfield W is a Jacobi vectorfield along the horizontal geodesic ~wo(u). Hence it satisfies V u V u W + V v ( T ( W , U)) + R ( W , U)U = 0

with the initial values Wr. ~ = (JU)rwo and ( v v w)rw ~ = ( T ( U , W) + VwU + [U, W])~w,, = 0,

since (T(U, W))v~~ = Tr.o (J~" o, ~ o ) = O.

From the other hand if a(u) is a real function satisfying the differential equation d2a Tu2 (u) = g(UT~o, U%,) with initial values a(0) = 0, a~r du"

= 0 then the vectorfield

W~wo(~) = ~(u)U~o + (Ju)~o is a Jacobi vectorfield along the horizontal geodesic %o(u). Since the tangent plane of the 2-surface ~b(u,w) at the point (u, w) is spanned by the horizontal vector U(u, w) and by the value of the Jacobi vectorfield W along the horizontal geodesic through ~b(u,w) E M and the Jacobi vectorfield W is a linear combination of the vectorfields U and J U the 2-surface ok(u, v) is transversally geodesic. It follows from this construction that the 3-web on M is transversally geodesic. []

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1-parameterSubloops in Coordinate Loops

Since the transversally geodesic 2-surfaces of a 3-net manifold have an induced subweb structure, we are interested on the question wether this 3-web structure could be continued to an immersed 3-net structure on a maximal transversally geodesic 2-surface. For the investigation of this question we consider the canonical 3-net structure on the product manifold M = L x L for a given loop L.

Proposition 4,1. The local coordinate loop of the induced 3-web structure on a transversally geodesic 2-surface S is a 1-parameter local subloop of the coordinate loop L of the 3-net on M. Proof Let 7 be a horizontal geodesic on the transversally geodesic 2-surface S through the point p E M. The construction of the coordinate loop L corresponding to the initial point P uses only the intersection properties of the leaves. Since the leaves of the induced 3-web on S are the intersections of the leaves of M with S the construction of the coordinate loop multiplication x o y gives the same point on 7 for the 3-web structure on S or on M. []

Proposition 4.2. Any local l-parameter subloop 7 of a differentiable loop L can be extended to an injectively immersed 1-parameter subloop. Proof The subloop F c L generated by the local 1-parameter subloop 7 c L is an injective immersion of a 1-manifold into L since every x E F has a neighbourhood LxN where Lx : L ~ L is the left multiplication mapping and N is a neighbourhood of e E L such that 7 C N and Lx7 ~ LxN is an imbedded 1-submanifold. []

Theorem 4.3. Let be given a 3-net structure on the manifold M. The following conditions are equivalent: (i) For every point p E M and horizontal tangent vector Up E T(h) M there -p exists an injectiveIy immersed transversally geodesic 2-surface S ~angent to Up with an induced 3-net structure, that is the 3-net manifold M has a maximal family of injectively immersed two-dimensional subnets. (ii) The curvature tensorfield R of the Chern connection of M satisfies tr{g(X, Y)Z = R(X, J Y ) Z }

for all horizontal vectors X, Y, Z E T(h), p E M, where a denotes the summation for all permutations of the variables X, Y , Z and g(X, Y) is a (2,0)-tensorfield on M. (iii) The coordinate loops of the 3-net structure on M have l-parameter subloops in every tangent direction at the identity element.

Proof The assertion follows from the preceding Theorem and Propositions.D

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209

g-hexagonal 3-nets

Definition 5.1. Let M be a 3-net manifold. We say that the points p, q E M are (horizontally, vertically or transversally) collinear if they are connected with a (horizontal, vertical or transversal) line o f M, (respectively).

Definition 5.2. Let M be a 3-net manifold. The ordered set of points po, pl,p2,p3,P4,ps, p6 is called a hexagonal configuration of M with center Po and initial vertex Pl if the triples o f points Pl,po, P4; p2,po, P5; p3,Po, P6 are horizontally, transversally and vertically collinear, respectively, and the pairs o f points Pi, pi+l, (i = 1, 2, 3, 4, 5) are collinear. The hexagonal configuration po, pl,p2, p3,p4, P5, P6 on M is called closed if the points Pl and p6 are collinear.

Definition 5.3. A 3-net manifold M is called hexagonal if all hexagonal configurations on M are closed. A 3-net manifold M is called g-hexagonal if all hexagonal configurations whose center and initial vertex are connected with a horizontal geodesic are closed. It is well k n o w n that a 3-net is hexagonal if and only if its coordinate loops satisfy the identity x o (x o x) = (x o x) o x, (cf. [1]). Now, we can formulate a characterization of g-hexagonal 3-nets. Here we will use the fact that a two-dimensional differentiable group 3-net is the canonical 3-net on IR x IR or on S 1 x S I.

Definition 5.4. The canonical 3-net structures on the manifolds R • IR or on S 1 x S 1 are called parallelizable 3-nets.

Theorem 5.5. Let be given a 3-net manifold M. The following conditons are equivalent: (i) The 3-net structure on M is g-hexagonal. (ii) Any coordinate loop L of the 3-net M satisfies the equation x o (x o x) = (x o x) o x for all x E L belonging to the image of the exponential mapping exPe : TeL ~ L with respect to the left canonical connection of L. (iii) For any point p c M and any horizontal tangent vector Up E T:h) M there

exists an injectively immersed parallelizable subnet S tangent to Up, that is the 3-net manifold M has a maximal family of injectively immersed parallelizable subnets. (iv) The curvature tensorfield R of the Chern connection of M satisfies

cr{R(X, J Y ) Z = 0} .for all horizontal vectors X, Y, Z E T(ph), p E M, where ~ denotes the summation for all permutations of the variables X, Y, Z.

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(v) The coordinate loops of the 3-net structure on M have 1-parameter subgroups in every tangent direction at the identity element.

Proof The condition (i) and (ii) are equivalent since the Chern connection of a 3-net induces the left-canonical connection on its coordinate loops (Theorem 6.4 in [10]). If the identity (x o x) o x = x o (x o x) is satisfied in a neighbourhood of e then its third derivative at e results the equation tr{R(X, J Y ) Z = 0} (cf. [3] p.20). Thus from (ii) follows (iv). From the other hand if (iv) is satisfied then the 3-net is transversally geodesic by our preceding theorem. Hence for any point p E M and any horizontal tangent vector Up E T~ph)Mthere exists a two-dimensional subnet S tangent to Up. Since this submanifold S is autoparallel its curvature tensor is induced by the tensorfield R. It follows that the curvature tensorfield on this submanifold S vanishes which means that this two-dimensional subnet is coordinatized by a 1-parameter (commutative) subgroup of the coordinate loop L (cf.[5] p.165). Thus the property (iv) implies (iii). Clearly the conditions (iii) and (v) are equivalent. We suppose now that (iii) or (v) is satisfied. If the center and the initial point of a hexagonal configuration are connected with a horizontal geodesic then the vertices of the hexagonal configuration are contained in the twodimensional subnet determined by the 1-parameter subgroup corresponding to the connecting geodesic. Since this subnet is coordinatized by a commutative group it is hexagonal. It follows that the given hexagonal configuration is closed that is the 3-net is g-hexagonal which is the condition (i). []

6

Subwebs of complete Moufang 3-webs

Definition 6.1. A 3-web structure on the manifold M is called complete if the associated Chern connection is complete. Since the universal covering manifold of a complete Moufang 3-web is a Moufang 3-net (cf. [12]) we can apply our preceding results to the description of the system of 2-dimensional subwebs of this class of webs.

Definition 6.2. The two-dimensional complete 3-webs which are the factorwebs of the canonical 3-net on the plane R x R by a free action of a properly discontinuous group of direction preserVing collineation are called toral 3webs.

Theorem 6.3. Let M be a complete Moufang 3-web. Then for any point p E M and any horizontal tangent vector Up E T(h)M there exists an immersed toral subweb S tangent to Up, that is the Moufang 3-web manifold M has a maximal family of immersed two-dimensional toral subwebs.

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References [1] J. ACZ~L, Quasigroups, Nets and Nomograms, Adv. in Math. 1 (1965), 383-450. [2] M. A. AKIVlS, Three Webs of Multidimensional Surfaces, Trudy Geom. Sem. 2 (1969), 7 31. [3] M. A. AKivls and A. M. SHELEKI-tOV,Foundations of the Theory of Webs, Kalinin Gos. Univ., Kalinin (1981). [4] A. BARLO3-rIand K. STRAMBACH,The Geometry of Binary Systems, Adv. in Math. 49 (1983), 1-105. [5] W. BLASCHKEand G. BOL,Geometrie der Gewebe, Springer-Verlag, Berlin (1938). [6] R. H. BRUCK,A Survey of Binary Systems, third printing, Springer-Verlag, Berlin (1971). [7] S. S. CHERN, Eine Invariantentheorie der Dreigewebe aus r-dimensionalen Mannigfaltigkeiten in R2r. Abh. Math. Sem. Univ. Hamburg 11 (1936), 335-358. [8] J. D/~NESand A. D. KEEDWELL,Latin Squares and Their Applications, Akadbmiai Kiad6, Budapest (1974). [9] S. KOBAYASmand K. NOMIZtJ, Foundations of Differential Geometry, Vol. II, Interscience, New York (t969). [10] P. T. NAGV, Invariant Tensorfields and the Canonical Connection of a 3-web, Aequationes Math, 35 (1988), 31-44. [11] P. T. NAGY,Complete Group 3-webs and 3-nets, Arch. Math. 53 (1989), 411-413. [12] P. T. NAGV,Extension of Local Loop Isomorphisms, to appear. [13] G. PtCKERT,Projektive Ebenen, zweite Auftage, Springer-Verlag, Berlin (1975).

Eingegangen am: 10.01.1991

Author's address: P&er T. Nagy, Bolyai Institute, Szeged University, Aradi v6rtanfik tere 1., H-6720 Szeged, Hungary.