Characterization of intrinsically harmonic forms

Journal of Topology 1 (2008) 643–650

c 2008 London Mathematical Society
doi:10.1112/jtopol/jtn014

Characterization of intrinsically harmonic forms Evgeny Volkov Abstract Let M be a closed oriented manifold of dimension n and ω a closed 1-form on M . We discuss the question of whether there exists a Riemannian metric for which ω is co-closed. For closed 1-forms with nondegenerate zeros, the question was answered completely by Calabi in 1969; (cf. Calabi, Global analysis, Papers in honour of K. Kodaira). The goal of this paper is to give an answer in the general case, that is, without making any assumptions on the zero set of ω.

1. Introduction and statement of results In this paper, we will be concerned with the characterization of intrinsically harmonic forms in terms of the topological and smooth structure of the underlying manifold. A k-form on a smooth oriented n-manifold (not necessarily closed) is called intrinsically harmonic if it is closed and there exists a Riemannian metric which makes the form coclosed, in which case it is common to say that the form is harmonic with respect to the Riemannian metric. The question is: given a closed k-form on a smooth closed oriented manifold, when is it intrinsically harmonic? Let us give a short historical overview. Only forms of degrees 1 and n − 1 have been considered seriously. The forms of degrees strictly between 1 and n − 1 seem to present considerable additional difficulties; see Section 4. The following classical theorem of Calabi from 1969 answers the question for forms of degree 1 with nondegenerate zeros. Theorem 1 (Calabi [4]). Let ω be a closed 1-form on a closed oriented manifold M . Assume that all the zeros of ω are nondegenerate. Then ω is intrinsically harmonic if and only if it is transitive. Transitivity for 1-forms will be discussed in detail later in Section 2, and for rank-2 2-forms on 4-manifolds in Section 4. For now suffice it to say that a closed 1-form is called transitive if there exists a closed transversal to its kernel foliation through every point which is not a zero of the form. Note that one can define the concept of transitivity for (n − 1)-forms repeating the above definition verbatim. In 1997, Honda gave a complete answer to the question for forms with nondegenerate zeros in degree n − 1. Having understood the concept of transitivity for both 1- and (n − 1)-forms, we can give a unified formulation of the theorems of Calabi and Honda. We abbreviate forms with nondegenerate zeros as ‘nondegenerate forms’. Theorem 2 (Calabi [4], Honda [7]). Let k ∈ {1, n − 1}. For a nondegenerate closed k-form α on a closed oriented connected n-manifold M to be intrinsically harmonic, it is necessary and sufficient that: (a) the form α is locally intrinsically harmonic; and (b) the form α is transitive.

Received 12 June 2007; published online 9 June 2008. 2000 Mathematics Subject Classification 57R30, 57R70.

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We say that a k-form is locally intrinsically harmonic if it becomes intrinsically harmonic when restricted to a suitable open neighborhood of its zero set. Note that this is stronger than saying that each point of the zero set of the form has such a neighborhood. Note also that for 1- and (n − 1)-forms, points of the manifold where the form does not vanish have the desired neighborhood automatically. For 1-forms with nondegenerate zeros, transitivity implies local intrinsic harmonicity and Theorem 2 simplifies to Theorem 1. For a discussion of local intrinsic harmonicity in the case of (n − 1)-forms, see the thesis of Honda [7]. All the theorems that we have discussed so far have assumed that the zeros of the k-form in question are nondegenerate. In 2006, Latschev managed to weaken the assumptions on the zero set of the form. Theorem 3 (Latschev [8]). Let k ∈ {1, n − 1}. Let α be a closed k-form on a closed oriented connected n-manifold. Assume that the zero set of α is a Euclidean neighborhood retract. Then α is intrinsically harmonic if and only if the following two conditions are satisfied: (a) the form α is locally intrinsically harmonic (if k = 1, assume in addition that the local metric which provides local harmonicity for α is real analytic); and (b) the form α is transitive. In this paper, we stick to the case of 1-forms and prove the characterization theorem in complete generality, that is, without assuming anything on the local structure of the zero set of the 1-form. We also do not assume any regularity higher than C ∞ for the local metric. Theorem 4. For a closed 1-form ω on a closed oriented manifold M to be intrinsically harmonic, it is necessary and sufficient that: (a) the form ω is locally intrinsically harmonic; and (b) the form ω is transitive. For general background we refer to ([6]).

2. Preliminaries We start by discussing the concept of transitivity in a little more detail, then recall the Poincar´e recurrence theorem, and then briefly discuss the Hodge-star operator, the codifferential and the Dirac operator with its basic properties. Definition 1. A closed 1-form ω is called transitive if for any point p ∈ M at which ω does not vanish, there is a closed strictly ω-positive smooth (embedded) path γ : S 1 −→ M through p. Here ‘strictly ω-positive’ means that ω(γ(t)) ˙ > 0 for all t ∈ S 1 = R/Z. That is to say, that there exists a closed transversal to the kernel foliation of ω through every point of our manifold which does not lie in the zero set of ω. Note that if there exists a smooth, not necessarily embedded, strictly ω-positive path through p, then the path is immersed and if n = dim M > 2, then we can achieve embeddedness by a small perturbation. If n = 2, then we perform the obvious modifications at double points. In the proof below, however, we get embeddedness automatically. To fix conventions from now on, an ‘ω-positive path’ means an ‘embedded strictly ω-positive path’. The following proposition is a classical result from dynamical systems — the Poincar´e recurrence theorem.

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Proposition 1. Let Ω be a set, Σ a σ-algebra on Ω, and µ a measure on Σ with finite µ(Ω). Let {φt }t∈R be a measure-preserving dynamical system on Ω. Assume that A is a σ-algebra element of positive measure. Then for any positive N, there exists n0 greater than N such that µ(A ∩ φn 0 (A)) > 0.

The following definition is the main piece of the background. Definition 2. For an oriented Euclidean n-dimensional vector space V and k ∈ {0, . . . , n}, the Hodge-star operator is defined as a unique operator ∗ : Λk V −−→ Λn −k V with the following property: for an oriented orthonormal basis e1 , . . . , en , the operator ∗ maps e1 ∧ . . . ∧ ek to ek +1 ∧ . . . ∧ en . For a smooth oriented n-dimensional Riemannian manifold (X, g), the Hodge-star operator ∗g : Λk T ∗ X −−→ Λn −k T ∗ X is defined fiberwise. Definition 3. For a smooth oriented n-dimensional Riemannian manifold (X, g), the codifferential d∗ : Ωk (X) −−→ Ωk +1 (X) is defined by the formula d∗ := (−1)n k +n +1 ∗ d∗; a form α ∈ Ω∗ (M ) satisfying d∗ α = 0 is called coclosed. The Dirac operator D : Ω∗ (X) −−→ Ω∗ (X) is defined by the formula D := d + d∗ .

The Dirac operator D is a first-order elliptic differential operator. It is well known that this operator possess the strong unique continuation property (any solution s0 to Ds = 0 that vanishes at a point up to infinite order must vanish identically). The standard way to see this is to consider the Laplace operator ∆ := D2 and use Aronszajn’s theorem (cf. [1]) to show the strong unique continuation property for ∆. See [2] for more details. Note that coclosedness of a form is equivalent to the Hodge dual of the form being closed.

3. Proof of Theorem 4 We let S denote the zero set of ω throughout this section and assume that S = M , since otherwise there is nothing to prove. For necessity, assume that there exists a Riemannian metric g that makes ω harmonic. Condition (a) is obviously satisfied. To show condition (b), we apply the Poincar´e recurrence theorem. We set Ω to be our manifold M , the σ-algebra Σ to be the usual Borelian σ-algebra, and µ to be the measure defined by a distinguished volume form dvol on M . Furthermore, let the vector field X be defined by the following equation: iX dvol = ∗g ω. Note that X is transverse to the kernel foliation of ω outside S. By the Cartan

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formula, we see that the Lie derivative LX dvol = 0. Let {φt }t∈R be the flow of X on M . In our setting, {φt }t∈R becomes a measure-preserving dynamical system on (Ω, Σ, µ). Now let p ¯ Φ) be a bifoliated closed chart around p; that is, ξ¯ is a closed be a given point in M \S. Let (ξ, subset of M containing an open neighborhood of p, and Φ : ξ¯ −→ B × I is a diffeomorphism, where B is a closed ball in Rn −1 and I = [0, 1] is a unit time interval. Moreover, under the diffeomorphism Φ, flowlines of {φt }t∈R correspond to the vertical leaves b × I, b ∈ B, and integral submanifolds of the kernel foliation of ω correspond to the horizontal leaves B × t, t ∈ I. In further considerations, we identify ξ¯ with its image under Φ. Since ξ¯ is compact, all points of ξ¯ will leave it by some time N , as we follow the flow {φt }t∈R . We set A := ξ¯ and apply Proposition 1 with the above choices of Ω, Σ, µ, A, and N . This gives us a trajectory of {φt }t∈R that leaves ξ¯ at some point (b1 , 1) and then enters it again for the first time at some point (b0 , 0). Let us denote the flowline between (b1 , 1) and (b0 , 0) by c˜. It is clear ¯ Now we close up this flowline artificially that except for its endpoints the path c˜ lies outside ξ. ¯ inside the bifoliated chart ξ, by connecting (b0 , 0) and (b1 , 1) with a smooth path cˆ through p, transverse to the horizontal leaves B × t, t ∈ I. Clearly, this can be done in such a way that the concatenation c of the paths c˜ and cˆ is smooth and embedded. So, as c is a smooth closed ω-positive path and the point p was arbitrary, we see that the form ω is transitive. This is condition (b). For sufficiency, assume that conditions (a) and (b) hold true. Let U be a neighborhood of S such that ω|U is coclosed with respect to some Riemannian metric gU on U . In order to see that we can choose U so small that the form ∗g U ω|U is exact (and not only closed), we need the following (key) technical result. Lemma 1. Let (X, g) be a smooth oriented n-dimensional Riemannian manifold without boundary. Let Z be the compact zero set of a 1-form γ on X, which is both closed and coclosed. Assume that Z = X; then there exists an open neighborhood U of Z, such that any closed (n − 1)-form on X becomes exact when restricted to U. Proof. Recall the Dirac operator D = d + d∗ from Section 2 and note that Dγ = 0. This allows us to draw a conclusion about the zero set of γ using the following result of B¨ ar (cf. [2]). Theorem 5. Let X be a smooth connected n-dimensional manifold and let L be a linear elliptic first-order differential operator acting on sections s of some vector bundle over X. Let the operator L satisfy the strong unique continuation property. Assume that s does not vanish identically and that it satisfies Ls = 0; then the zero set s−1 (0) of s is contained in a countable union of smooth (n − 2)-dimensional submanifolds. Theorem 5 applies to the operator D and the
1-form γ to give a sequence {Lk }k ∈N of submanifolds of X of codimension 2, with Z ⊂ k ∈N Lk . Since every submanifold Lk can be countably exhausted by compact ones (possibly with boundary), we may assume without loss of generality that each Lk is compact, possibly with boundary. Set Zk := Z ∩ Lk . Let ‘dim’ denote the covering dimension of a topological space. Then dim Zk
n − 2, since Lk is a compact manifold (possibly with boundary) of dimension n − 2 and Zk ⊂ Lk . We endow Z ⊂ X with the induced topology and note that Z is normal as a subspace of a metric space. Note also that each Zk is closed in Z, because each Lk is closed in X. We recall the following result from general topology, known as the countable sum theorem (cf. [5, Theorem 7.2.1, p. 394]).

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Theorem 6. Let X be
a normal topological space and {Xk }k ∈N a sequence of closed subsets of X . Assume that X = k ∈N Xk and that dim Xk
l for all k. Then dim X
l. This applies to Z =


k ∈N

Zk to give the estimate dim Z
n − 2. The latter implies that n −1 (Z) = 0, HCech ˇ

ˇ because Cech cohomology vanishes above the covering dimension of a compact metrizable space just as de Rham cohomology vanishes above the dimension of a manifold.  Take a sequence {Uj }j ∈N of open neighborhoods Uj of Z such that Uj +1 ⊂ Uj and j ∈N Uj = ˇ Z with U0 = X. The continuity property of Cech cohomology, (cf. [3, Section 14 ‘Continuity’, Theorem 14.4]) implies that n −1 (Uj ) = 0. lim HCech −→ ˇ ˇ cohomology is the same as de Rham cohomology Since Uj is a manifold, we find that its Cech and is finite-dimensional. The direct limit of the sequence of finite-dimensional vector spaces n −1 (Uj ) vanishes. This implies that for j large enough, the image of the 0th vector space HCech ˇ n −1 HCech (U0 ) of the sequence ˇ n −1 n −1 n −1 HCech (U0 ) −−→ HCech (U1 ) −−→ · · · −−→ HCech (Uj ) −−→ · · · ˇ ˇ ˇ n −1 (Uj ), vanishes. In other words, if i : Uj −→ X denotes the obvious in the jth one, HCech ˇ inclusion, then i∗ H n −1 (X) is the trivial subspace of H n −1 (Uj ). Take U := Uj .

We apply Lemma 1 with X := U , γ := ω|U , and Z := S to get the relevant neighborhood U ⊂ U of S and rename U := U. Now the form ∗g U ω|U is exact and we can pick a primitive (n − 2)-form α for ∗g U ω|U , that is, dα = ∗g U ω|U . This finishes the work near S and now we set things up away from S. Since the form ω is transitive, given any point m ∈ M \S, there exists a closed path γm : S 1 −→ M through m. By a standard ‘thickening of a transversal argument’ (see, for example, [6]), we obtain an open neighborhood Wm of γm (S 1 ), diffeomorphic to S 1 × B, where B is an open ball in Rn −1 centered at the origin. Moreover, when restricted to Wm , the form ω is a positive multiple of dθ, where θ denotes the coordinate along the S 1 direction. Let (x1 , . . . , xn −1 ) be the Cartesian coordinates on B ⊂ Rn −1 and assume without loss of generality that the top degree form dθ ∧ dx1 ∧ . . . ∧ dxn −1 orients Wm consistently with the given orientation on M . Let ρ : [0, 1] −→ R be a smooth cut-off function: ρ|[0,1/5] = 1, ρ|[4/5,1] = 0. Set   ψm = ρ x21 + . . . + x2n −1 dx1 ∧ . . . ∧ dxn −1 . Clearly, the (n − 1)-form ψm is closed and vanishes in a neighborhood of the boundary of Wm . The top degree form ω ∧ ψm is non-negative everywhere and is bounded away from zero in some open neighborhood Vm ⊂ Wm of γm (S 1 ). Vanishing of ψm near the boundary of Wm implies that ψm vanishes in some open neighborhood Um of S with Um ⊂ U . This construction almost literally follows that given by Calabi in [4]. Since M \U is compact, it can be covered by Vm 1 , . . . , Vm l for some natural number l, where m1 , . . . , ml ∈ M \U . Set U0 := Um 1 ∩ . . . ∩ Um l , V := Vm 1 ∪ . . . ∪ Vm l and


ψ := Σli=1 ψm i .


Note that U0 ⊂ M \V ⊂ U and ψ |U 0 = 0.

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We pause for a moment to summarize what we have. We have an open neighborhood U of S with an (n − 2)-form α on U such that dα = ∗g U ω; also, open sets U0 and V with U0 ⊂

M \V ⊂ U and an (n − 1)-form ψ with ω ∧ ψ bounded away from zero on V , non-negative everywhere and satisfying


ψ |U 0 = 0. This allows us to finish the proof with the standard gluing argument. We let φ be a smooth function with φ|M \V = 1 and φ|M \U = 0. Such a function φ exists, since both sets M \V and

M \U are closed and the first one is contained in the complement of the second. Set α = φα





and ψ = dα . Note that ψ |M \V = dα|M \V = ∗g U ω|M \V . Consider the closed form


ψ := Kψ + ψ





for a sufficiently large positive constant K. We claim that the form ψ has the following properties: (i) ψ|U 0 = ∗g U ω|U 0 ; and (ii) ω ∧ ψ > 0 everywhere on M \S.
Indeed, since ψ |U 0 = 0, we have ψ|U 0 = ψ  |U 0 = ∗g U ω|U 0 . This shows the first property. For the second one, consider









ψ|M \V = Kψ |M \V + ψ |M \V = Kψ |M \V + ∗g U |M \V ; multiplying by ω gives us


ω ∧ ψ|M \V = Kω ∧ ψ |M \V + ω ∧ ∗g U ω|M \V . The last expression is the sum of two non-negative terms, the second one being strictly positive
outside S. We are left with the expression ω ∧ ψ, restricted to V . Since ω ∧ ψ |V is bounded away from zero, we see that






ω ∧ ψ|V = Kω ∧ ψ |V + ω ∧ ψ |V > 0 for K sufficiently large. Now having the form ψ with the properties above, we construct the desired metric g by

gluing. Let φU , φV be a partition of unity, subordinate to the cover U, V . Let g be any metric on V , making ω and ψ orthogonal to each other (that is, the Hodge dual of one is proportional to the other; on M \S this is just the orthogonality of the respective kernel foliations). Consider

the metric g˜ := φU gU + φV g on M . It makes ω and ψ orthogonal everywhere on M and ∗g˜ ω|U 0 = ∗g U ω|U 0 = ψ|U 0 .

(1)

Consider the following g˜-orthogonal decomposition over M \S: g˜ = g˜|ker ω ⊕ g˜|ker ψ . There exists a unique smooth positive function f˜ on M \S such that for the metric g := f˜g˜|ker ω ⊕ g˜|ker ψ on M \S, we have ∗g ω|M \S = ψ|M \S . Note that f˜|U 0 \S = 1 because of equation (1); therefore g|U 0 \S = g˜|U 0 \S , and hence g can be C ∞ -regularly continued across points of S by just setting g|S := g˜|S . This means that the metric g is well defined everywhere on M . The equation ∗g ω = ψ holds on M \S, by the choice of f˜ and it also holds on U0 , by the first property of the form ψ, because g|U 0 = g˜|U 0 = gU |U 0 . Thus, since the form ψ is closed, we find that the form ω is harmonic with respect to the metric g. This completes the proof of Theorem 4.

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4. Concluding remarks As we mentioned in the introduction, the forms of degrees strictly between 1 and n − 1 present considerable additional difficulties. Indeed, the simplest case of such a form would be a 2-form on a 4-manifold. A generic 2-form on a 4-manifold has no zeros at all. So, let α be a nowhere zero closed 2-form on a closed 4-manifold. To simplify things even further, assume that α has constant rank. For dimension reasons, we have only two possibilities for the rank of α, namely 2 or 4. In the latter case, the form α is symplectic. Therefore, α coincides with its Hodge dual taken with respect to any Riemannian metric g compatible with α (here the ambient manifold is oriented by the volume form α ∧ α). This can be seen easily by choosing for every point p ∈ M a local frame for T M near p, that is both α-standard and g-orthonormal. It follows that α is harmonic with respect to g. The question of intrinsic harmonicity is answered trivially and positively in this case. So, the only potentially interesting case is when α has constant rank 2. In this case, we have the 2-dimensional kernel foliation of the closed 2-form α. The following definition of transitivity for such 2-forms seems to be a natural extension of that for 1-forms given in Section 2. Definition 4. Let α be a closed rank-2 2-form on a closed 4-manifold. We say that α is transitive if there exists an embedded surface Σ passing through every point of M transverse to the kernel foliation of α; that is, α restricts to Σ as a volume form. As the following example shows, transitivity is not sufficient for harmonicity. This illustrates the serious difficulties that we have when discussing forms of degrees strictly between 1 and n − 1. Example 1. Let M be the total space of a nontrivial orientable S 2 -bundle ξ = (S 2 −→ π M −→ S 2 ) over S 2 . Any such bundle can be obtained by gluing the two trivial S 2 -bundles over the two copies of the closed 2-disc along the common boundary via a gluing map h : S 1 −−→ Diff + (S 2 ), where Diff + (S 2 ) is the group of orientation-preserving diffeomorphisms of S 2 . The latter group retracts onto SO(3) (cf. [9]) and π1 (SO(3)) ∼ = Z2 . So, up to an isomorphism there are only two orientable S 2 -bundles over S 2 . One of them is trivial; the other is not. Therefore, the bundle ξ is well defined up to an isomorphism and the manifold M is well defined up to a diffeomorphism. The above picture of ξ in terms of the gluing map also shows that there exists a section of ξ through every point of M . Indeed, a constant map c from the closed 2-disc D to S 2 (the fiber of ξ) may be considered as a section of the restriction of ξ to this disc. After twisting the map c with the gluing map h on the boundary of D, it extends to S 2 \D ∼ = D, because the fundamental group of S 2 is trivial. Let dvolS 2 be a volume form on the base S 2 and set α := π ∗ dvolS 2 . The form α is a closed 2-form of constant rank 2 on the 4-dimensional manifold M , where the fibers of ξ are the leaves of the 2-dimensional kernel foliation of α. Sections of ξ provide closed 2-dimensional submanifolds of M to which α restricts as a volume form, so α is transitive. But α is not (!) intrinsically harmonic. Indeed, assume by contradiction, that there exists a Riemannian metric g on M such that the form ψ := ∗g α is closed. The form ψ has constant rank 2 and the leaves of the kernel foliation of ψ are transverse to those of α, that is, to the fibers of ξ. Take a leaf L of the kernel foliation of ψ. The restriction π|L : L −→ S 2 is a submersion and therefore, for dimensional reasons, a covering map. Since S 2 is simply connected, we see that π|L is a diffeomorphism, and hence L intersects every fiber of ξ exactly once. This is true for any leaf of the kernel foliation of ψ, so the total space M of ξ admits a foliation by closed leaves

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transverse to the fibers, with every leaf intersecting every fiber exactly once. This contradicts the nontriviality of ξ. Tautologically, one can say that a closed 2-form of rank 2 on a 4-manifold is intrinsically harmonic if and only if its kernel foliation ker α admits a complementary foliation F defined by a closed 2-form ψ. Indeed, given g which makes α harmonic, we can set ψ := ∗g α. Conversely, given a closed 2-form ψ defining a foliation F complementary to ker α, we can define a Riemannian metric g by requiring that ker α and F be orthogonal. Then ∗g α is proportional to ψ. By rescaling g on T ker α, we can achieve that ∗g α is equal to ψ, which is closed. The big problem with this characterization is that deciding the existence of a complementary foliation defined by a closed form is just as hard as deciding whether the given form is intrinsically harmonic. Finally, note that all this does not even begin to touch the case of nonconstant rank. Acknowledgements. The author thanks D. Kotschick for supervision, J. Latschev for many valuable discussions, and the referee for critical readings. This work was supported by a DFG grant.

References 1. N. Aronszajn, ‘A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order’, J. Math. Pures Appl. (9) 36 (1957) 235–249. ¨r, ‘Zero sets of solutions to semilinear elliptic systems of first order’, Invent. Math. 138 (1999) 2. C. B a 183–202. 3. G. Bredon, Sheaf theory (McGraw-Hill, New York, 1967). 4. E. Calabi, ‘An intrinsic characterization of harmonic one-forms’, Global analysis, Papers in honour of K. Kodaira (eds D. C. Spencer and S. Iyanaga; University of Tokyo Press, Tokyo, 1969) 101–107. 5. R. Engelking, General topology (Helderman, Berlin, 1989). 6. M. Farber, Topology of closed one-forms, Mathematical Surveys and Monographs 108 (American Mathematical Society, Providence, RI, 2004). 7. K. Honda, ‘On harmonic forms for generic metrics’, PhD Thesis, Princeton University, Princeton, 1997. 8. J. Latschev, ‘Closed forms transverse to singular foliations’, Manuscripta Math. 121 (2006) 293–315. 9. S. Smale, ‘Diffeomorphisms of the 2-sphere’, Proc. Amer. Math. Soc. 10 (1959) 621–626.

Evgeny Volkov Mathematics Institute Ludwig-Maximilians-Universit¨ at Theresienstr. 39 80333 M¨ unchen Germany [email protected]