A RIGIDITY RESULT FOR KÄHLERIAN MANIFOLDS ENDOWED WITH CLOSED CONFORMAL VECTOR FIELDS

We show that if a connected compact Kählerian surface M with nonpositive Gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We also consider the case of a connected complete Kählerian manifod M of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is well known that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M . Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete Kählerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of ξ.


Introduction
A conformal vector field ξ on a semi-Riemannian manifold M is closed if its metrically dual 1-form is closed. In this case, if ∇ stands for the Levi-Civita connection of M , one has ∇ X ξ = ψX for some smooth function ψ on M (the conformal factor of ξ) and every X ∈ X(M ).
The geometry of Riemannian submanifolds of Lorentzian and Riemannian manifolds in which either the submanifold or the ambient space is endowed with a closed conformal vector field has been the object of intense research in recent years (see, for instance, [1,3,6,7] and the references therein). The presence of such a (nontrivial) vector field also imposes strong restrictions on the structure of the ambient manifold M itself. For example, it is a well known fact (cf. [4] and [7]) that, in a neighborhood of each nonsingular point, M is isometric to a warped product structure.
In what concerns Kählerian manifolds, a canonical class of examples possessing closed conformal vector fields is that of the warped products M n = I × t N 2n−1 (n standing for the complex dimension of M and t for the I-variable), where N is a (2n − 1)-dimensional Sasakian manifold; in this case, ξ = t∂ t is closed conformal and ψ = 1 is its conformal factor. Another one is that of the Riemannian products M n = N n−1 × T, where N is a Kählerian manifold of complex dimension n − 1 and T is a flat torus of complex dimension 1; in this case, the conformal vector field ξ is the lift, to M , of a parallel vector field in T, and ψ = 0 is its conformal factor. In both of these classes of examples, the holomorphic sectional curvature of M in the direction of ξ vanishes identically and, if J stands for the quasi-complex structure of M , then the leaves of the distribution generated by ξ and Jξ are totally geodesic in M . However, in the first class such leaves are noncompact, whereas in the second class they are compact.
On the other hand, if (M n , J, g) is a compact complex symmetric space of complex dimension n, positive scalar curvature, vanishing first De Rham cohomology group and not isometric to S 2n , then [8,Theorem 1] implies that M does not possess a nontrivial closed conformal vector field. A particular instance of this situation is that of CP n endowed with the Fubini-Study metric. Therefore, covering space theory shows that if (M n , J, g) is a compact connected Kählerian manifold of positive constant holomorphic sectional curvature (n being its complex dimension), then M does not possess a nontrivial closed conformal vector field.
The purpose of this paper is to show that, under a reasonable set of conditions on the closed conformal vector field ξ, the second class of examples presented in the third paragraph is essentially the only one. More precisely, we first show that if a connected compact Riemann surface M , endowed with a Kählerian metric of nonpositive Gaussian curvature, possesses a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We then consider the case of a connected complete Kählerian manifod M n , of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is a well known fact (cf. [7]) that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M . Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori, and also by a family of totally geodesic, isometric complete Kählerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R 2 as a factor.
Our approach is based on a certain kind of deformation of the original Kählerian metric of M , which is interesting in itself and generalizes the way the metric of C n deforms into that of CH n .

Deforming Kählerian metrics
In the sequel, we let (M n , J, g) be an n-dimensional Hermitian manifold, where n stands for its complex dimension (hence, M has real dimension 2n). We also let ω ∈ Ω 2 (M ) denote the corresponding Kählerian form, so that ω(X, Y ) = JX, Y for all X, Y ∈ X(M ). It is a standard fact that M is a Kählerian manifold if and only if J is parallel with respect to the Levi-Civita connection ∇ of g.
Whenever convenient, we write g = ·, · and let | · | denote the corresponding norm. Also, for X ∈ X(M ), we let θ X denote the 1-form metrically dual to X, so that θ X (Y ) = X, Y for Y ∈ X(M ); we also let θ 2 X denote the symmetrization of θ X ⊗ θ X .
We recall that a conformal vector field ξ on (M, g, ∇) is said to be closed conformal if θ ξ is a closed 1-form. These conditions are readily seen to be equivalent to the existence of a smooth function ψ on M (called the conformal factor of ξ) such that ∇ X ξ = ψX for all X ∈ X(M ).
If (M, J, g, ∇) is endowed with a nontrivial closed conformal vector field, the following result presents a simple way to construct, out of g, a new Kählerian metric on (M, J). Proof. We briefly sketch the (simple) proof. The 2-tensorg is clearly positive definite, and thus defines a Riemannian metric on M . On the other hand, for X, Y ∈ X(M ), the Hermitian character of ·, · with respect to J readily implies that ofg (also with respect to J). Next, if ω andω stand for the Kählerian forms of (M, J, g) and (M, J,g), respectively, then another straightforward computation givesω = µω + µ 2 θ ξ ∧ θ Jξ , and hence Letting ψ be the conformal factor of ξ and X ∈ X(M ), one also computes dµ = 2ψµ 2 θ ξ and dθ Jξ = 2ψω. Therefore, Our next result gives a set of conditions under which (M, J,g) is a complete Riemannian manifold.
Now, choose K ⊂ M to be a compact set such that ψ = 0 on K c , and t 0 > 0 such that γ(t) / ∈ K for t > t 0 . Letting sup M |ψ| = α < +∞, we estimatẽ Let ǫ > 0 be given. Since |ξ| 2 is proper, |ξ| 2 < c and sup M |ξ| 2 = c, there exists a compact subset Hence, for t > t 0 , t ǫ , the above computations givel Remark 2.3. The previous result continues to hold if we assume that (in the notations of the proof) ψ −1 (0) ∩ K c is a set of isolated points. It suffices to split the trace of γ into pieces along each of which ψ = 0, estimate the length of each such piece as we did above and add the results.

Example 2.4.
In the complex Euclidean n-space C n , let J be the standard quasicomplex structure, g = ·, · the standard metric and B n = {z ∈ C n ; |z| < 1}. Since the vector field ξ(p) = p is closed and conformal, the previous construction endows B n with a Kählerian metricg, such that An immediate application of the previous result establishes the completeness of (B n ,g). Therefore, the formula of Lemma 2.7 for the holomorphic sectional curvature of (B n , J,g), together with the Hawley-Igusa theorem, shows that (B n , J,g) is nothing but the complex hyperbolic space CH n . Example 2.5. Let (N n−1 , J N , g N ) be a Kählerian manifold (of complex dimension n − 1) and T be a flat torus with its standard quasi-complex structure. If M n = N ×T is endowed with the product quasi-complex structure and the product metric, then M n is a Kählerian manifold. Moreover, if T is the quotient of the lattice L in R 2 and Z stands for the canonical vector field along one of the directions of the lattice, then Z induces a nontrivial parallel smooth vector field on T, which can be lifted to a corresponding one on M .
Henceforth, we let (M n , J, g) be a Kählerian manifold, ξ ∈ X(M ) be a nontrivial closed conformal vector field with conformal factor ψ and such that |ξ| 2 < c, andg be the Kählerian metric on (M, J) given as in Proposition 2.1. We want to relate the holomorphic sectional curvatures of (M, J, g) and (M, J,g), and to this end we start by relating their Levi-Civita connections. Lemma 2.6. If ∇ and∇ stand for the Levi-Civita connections of g andg, respectively, theñ Proof. The proof is a somewhat lengthy, though straightforward computation. On the one hand, for X, Y, Z ∈ X(M ) we have On the other, Koszul's formula gives By computing each summand at the right hand side of the last expression above we obtain, after some cancellations, Setting W =∇ X Y − ∇ X Y and comparing the two expressions for 2g(∇ X Y, Z), we arrive at Taking the inner product of (2.3) with ξ and Jξ, respectively, and recalling that However, Jξ, ξ = 0 gives F (X, Y ), ξ = 2ψµc( ξ, X ξ, Y − Jξ, X Jξ, Y ) and F (X, Y ), Jξ J = 2ψµc( ξ, X Jξ, Y + ξ, Y Jξ, X ). Substituting these formulas in the right hand side of the expression of W , we arrive at (2.2).

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A. CAMINHA Before we can proceed to relate the holomorphic sectional curvatures of g = ·, · andg, we need a few more preliminaries. Firstly, the closed conformal condition on ξ ∈ X(M n ) (recall that n stands for the complex dimension of M ) readily gives and Lemma 1 of [7] shows that where ∇(div ξ) stands for the gradient of the divergence of ξ with respect to g and Ric(ξ) for the normalized Ricci curvature of (M, g) in the direction of ξ. In particular, at each point where ξ = 0, we get
Proof. We first perform the computations at a point p ∈ M such that ξ(p) = 0. LettingR denote the curvature tensor of (M,g), we havẽ Extend X to a neighborhood of p, with (∇ v X)(p) = 0 and (∇ v JX)(p) = 0 for all v ∈ T p M (with e 1 (p) = X p , the parallelism of J allows us to take a Hermitian geodesic frame (e 1 , Je 1 , . . . , e n , Je n ) around p); hence, [X, JX] = 0 at p. Setting α = 2ψµ X, ξ and β = 2ψµ X, Jξ , relation (2.2) gives, at the point p and after some straightforward computations, Computing each of I , II and III at p, we get Substituting these expressions in (2.6), we obtaiñ If ξ(p) = 0, take a sequence (p j ) j≥1 in M , converging to p and such that ξ(p j ) = 0 (such a sequence does exist, for we are assuming that the zeros of ξ are isolated).
The corollary below extends, to a general deformation g →g as above, the phenomenon of holomorphic sectional curvature decay that takes place when we pass from C n to CH n . Proof. If X⊥ξ, Jξ, theng(X, X) = µ. Therefore, our previous result gives For a general X ∈ T p M unitary, let A = X, ξ 2 + X, Jξ 2 and writẽ For the first summand, note that For the second summand, substitutingg(X, X) = µ + µ 2 A we get where y = µA. It now suffices to observe that 1 µ ≤ c and (since y ≥ 0) 3y + 2y 2 1 + 2y + y 2 = 2 − 1 y + 1 − 1 (y + 1) 2 < 2.

Two rigidity results for Kählerian manifolds
We now use the metric deformation discussed in the previous section to study the structure of a connected complete Kählerian manifold endowed with a nontrivial closed conformal vector field. We start by looking at the compact case, for which we need the following result.
We have finally arrived at our first main result.

Theorem 3.2. Let (M 1 , g, J) be a connected, compact Kählerian surface with
Gaussian curvature K ≤ 0. If M possesses a closed conformal vector field ξ ∈ X(M ) whose zeros are all isolated, then K ≡ 0, ξ is parallel, and M is isometric to a flat torus.
Proof. As before, let ψ be the conformal factor of ξ, choose a real number c > 0 such that c > max M |ξ| 2 , and letg be defined as in (2.1).
Since ξ −1 (0) is a set of isolated points and M is compact, we conclude that ξ −1 (0) is finite. Therefore, ifK stands for the Gaussian curvature of (M,g), then, at every point of M \ ξ −1 (0) and in the notations of Lemma 2.7, we have K = K(ξ) andK =K(ξ). That result also furnishes Since ξ , Jξ = 0 andg(ξ,ξ) = cµ 2 , we obtain, after some simple algebra, By continuity, this last formula relating K andK holds in all of M . We now apply Gauss-Bonnet theorem twice, with the aid of Lemma 3.1: Thus, the inequality above must be an equality, which implies K ≡ 0 and ψ ≡ 0 and, in turn, X (M ) = 0. This means that M is diffeomorphic to a torus and ξ is parallel. Also, since ∇J = 0, we get that Jξ is also parallel. Therefore, |ξ| = |Jξ| are constant on M , and since ξ is nontrivial, neither of these vectors does vanish on M .
Since M is diffeomorphic to a torus, a theorem of Cartan assures the existence of closed geodesics γ 1 and γ 2 in M , representing the free homotopy classes of a set of generators of π 1 (M ). Letting the flow of ξ act (by isometries, since ξ is parallel) on γ 1 and γ 2 , we can assume that both of them start and end at the same point p of M , so that π 1 (M ; p) is generated by [γ 1 ] and [γ 2 ]. If γ ′ 1 (0) = αξ + βJξ, for some α, β > 0, then the parallelism of ξ and Jξ give that γ ′ 1 = αξ + βJξ along γ 1 , so that γ 1 is a geodesic loop based at p. Accordingly, so is γ 2 (see Figure 1).
Hence, ψ is constant along α and, since ψ(q) = 0, we get that ψ ≡ 0 along α. Since X was arbitrarily chosen subjected to the condition X⊥ξ, Jξ, we conclude that ψ ≡ 0 in a neighborhood of p in M \ ξ −1 (0). The discussion on [5, Section 1] assures that ψ and ξ are uniquely determined by the values of ψ, ∇ψ, ξ, and ∇ξ at a single point of M . Therefore, since ψ vanishes on an open subset of M \ ξ −1 (0), and (as we have observed above, for n > 1) such a set is connected, we conclude that ψ ≡ 0 on M \ ξ −1 (0). However, since ξ −1 (0) is a set of isolated points, we actually have that ψ ≡ 0 on M . In turn, this shows that both ξ and Jξ (since ∇J = 0) are parallel on M .
Finally, for X ∈ X(M ) we have X ξ, ξ = 2 ψX, ξ = 0, so that |ξ| 2 is constant on M . Since ξ = 0 at most at a set of isolated points, this implies that |Jξ| = |ξ| is constant and positive on M .
For what follows, we recall that, given a connected submanifold N of a Riemannian manifold M , p ∈ N , and a closed and piecewise differentiable curve α : [0, 1] → N such that α(0) = p, parallel translation along α defines a linear operator P α : is the parallel transport of v along α. It is immediate to check that the set of such linear operators, endowed with the product P α · P β = P α·β , form a closed subgroup of O(T p N ⊥ ), called the normal holonomy subgroup of N at p and denoted by Hol ⊥ p (N ) (for more details, see [2,Chapter 4]). If q ∈ N and δ is a piecewise smooth curve in N joining p and q, we have Hol ⊥ p (N ) ≃ Hol ⊥ q (N ) via P α → P δ −1 ·α·δ . Therefore, from now on we shall refer to the normal holonomy group of N , which will henceforth be denoted by Hol ⊥ (N ).
We then have our second main result. Proof. Lemma 3.4 assures that Σ is totally geodesic in M , so that its Gaussian curvature K Σ coincides with the holomorphic sectional curvature of M along Σ. Therefore, K Σ ≤ 0, and we can apply Theorem 3.2 to conclude that Σ is isometric to a flat torus (hence, K Σ ≡ 0) and the conformal factor ψ of ξ vanishes along it.  ; q). In particular, it is a well known fact that γ ′ (0)⊥T p Σ, so that γ ′ (0), ξ p = γ ′ (0), Jξ p = 0. Now, the parallelism of ξ assures that, along γ, Then, γ ′ , ξ is constant along γ, so that γ ′ , ξ q = γ ′ , ξ p = 0. Analogously, γ ′ , Jξ q = 0. On the other hand, if Σ q is the leaf of D passing through q, then T q Σ q is generated by ξ q and Jξ q , so that γ ′ (l)⊥T q Σ q (cf. Figure 2). Figure 2. Comparing Σ to Σ q .
By Lemma 3.4, Σ q is totally geodesic in M . For v ∈ T q Σ q , the maximal geodesic of Σ q departing from q with velocity v coincides with that of M , which is complete. Hence, Σ q is also complete. Let K Σq stand for the Gaussian curvature of Σ q and K(ξ, Jξ) for the holomorphic sectional curvature of M along the planes generated by ξ and Jξ. Letting R denote the curvature operator of M , the parallelism of ξ and Jξ give R(Jξ, ξ)ξ = 0, so that K(ξ, Jξ) ≡ 0. However, since Σ q is totally geodesic in M , we conclude that K Σq = K(ξ, Jξ) |Σq = 0. Therefore, being a connected, complete flat surface, Σ q is isometric to a torus, a plane or a cylinder over a plane curve. In what comes next, we shall show that is is isometric to a torus.
We now turn to (c). If X and Y are smooth vector fields in D ⊥ , then the parallelism of ξ and Jξ give ∇ X Y, ξ = 0 and ∇ X Y, Jξ = 0. In particular, [X, Y ] ∈ D ⊥ , so that D ⊥ is integrable. Letting N denote a leaf of D ⊥ and α its second fundamental form, we have α(X, Y ) = ∇ X Y, ξ ξ |ξ| 2 + ∇ X Y, Jξ Jξ |ξ| 2 = 0, and N is totally geodesic in M . The completeness of N (in the induced metric) now follows from that of M , together with the fact that geodesics in N are also geodesics in M .
It is immediate to check that X ∈ X(N ) ⇒ JX ∈ X(N ). Therefore, J is an almost complex structure on N , and the fact that the Levi-Civita connection of N is the restriction of that of M guarantees that J is parallel on N . Finally, since the Nijenhuis tensor of N is the restriction of that of M , which vanishes identically, we conclude that N is a Kählerian manifold in the induced metric.
For the last part, we argue pretty much as in (i). To this end, let N 1 and N 2 be two distinct leaves of D ⊥ , and take p 1 ∈ N 1 ∩ Σ and p 2 ∈ N 2 ∩ Σ. Let δ : [0, a] → Σ be a geodesic of Σ joining p 1 to p 2 , and δ ′ (0) = aξ p1 + bJξ p1 for some a, b ∈ R. The parallelism of ξ and Jξ assure that δ ′ is the restriction of the parallel (hence, complete) vector field η = aξ + bJξ to δ. If Φ : R × M → M denotes the flow of η, then Φ a : M → M is an isometry such that Φ a (p 1 ) = p 2 and (dΦ a ) p1 (T p1 N 1 ) = T p2 N 2 . Since N 1 and N 2 are connected, complete and totally geodesic in M , an argument pretty much like the one presented in the proof of (b) guarantees that Φ a applies geodesics in N 1 to geodesics in N 2 . Hence, Φ a (N 1 ) ⊂ N 2 and, likewise, Φ −a (N 2 ) ⊂ N 1 . Thus, Φ a (N 1 ) = N 2 . Corollary 3.8. Let n > 1 be an integer and (M n , g, J, ∇) be a connected, complete Kählerian manifold satisfying the hypotheses of the previous result. IfM stands for the universal covering of M , endowed with the covering metric, thenM is isometric to a Riemannian productÑ × R 2 , whereÑ is a connected, simply connected, complete Kählerian manifold.
Proof. Lettingg denote the covering metric, the covering map π :M → M turns into a local isometry, so thatM is naturally a Kählerian manifold. Moreover, the orthogonal foliations on M lift to two orthogonal foliations ofM with totally geodesic leaves, and one of which has leaves isometric to R 2 . SinceM is simply connected, it now suffices to apply the complex version of the De Rham decomposition theorem.