Conification construction for Kaehler manifolds and its application in c-projective geometry
CCONIFICATION CONSTRUCTION FOR K ¨AHLER MANIFOLDS AND ITSAPPLICATION IN C-PROJECTIVE GEOMETRY
VLADIMIR S. MATVEEV AND STEFAN ROSEMANN
Abstract.
Two K¨ahler metrics on one complex manifold are said to be c-projectively equiv-alent if their J -planar curves, i.e., curves defined by the property that their acceleration iscomplex proportional to their velocity, coincide. The degree of mobility of a K¨ahler metric isthe dimension of the space of metrics that are c-projectively equivalent to it. We give the listof all possible values of the degree of mobility of simply connected n -dimensional RiemannianK¨ahler manifolds. We also describe all such values under the additional assumption that themetric is Einstein. As an application, we describe all possible dimensions of the space of essen-tial c-projective vector fields of K¨ahler and K¨ahler-Einstein Riemannian metrics. We also showthat two c-projectively equivalent K¨ahler Einstein metrics (of arbitrary signature) on a closedmanifold have constant holomorphic curvature or are affinely equivalent. Introduction
The conification construction will be recalled in §3. It is a special case of a local constructionfrom [1] that given a n -dimensional K¨ahler manifold ( M, g, J ) (of arbitrary signature) producesa (2 n + 2) -dimensional K¨ahler manifold. If we apply this construction to the Fubini-Study metric g F S , we obtain the flat metric.There are many similar and more general constructions that were described and successfullyapplied in K¨ahler geometry before, for example the Calabi construction or the interplay betweenSasakian and K¨ahler manifolds.The main results of the paper are applications of the construction in the theory of c-projectivelyequivalent metrics, let us explain what this theory is about.Let ( M, g, J ) be a K¨ahler manifold of real dimension n ≥ with Levi-Civita connection ∇ . Aregular curve γ : I → M is called J -planar for ( g, J ) if ( ∇ ˙ γ ˙ γ ) ∧ ˙ γ ∧ J ˙ γ = 0 (1)holds at each point of the curve. The condition (1) can equivalently be rewritten as follows: thereexist smooth functions α ( t ) , β ( t ) such that ∇ ˙ γ ( t ) ˙ γ ( t ) = α ( t ) · ˙ γ ( t ) + β ( t ) · J ˙ γ ( t ) . Every geodesic is evidently a J -planar curve. The set of J -planar curves is geometrically a muchbigger set of curves than the set of geodesics. For example in every point and in every directionthere exist infinitely many geometrically different J -planar curves, see figure 1. p X Figure 1.
Given p ∈ M , X ∈ T p M , there are infinitely many J -planar curves γ such that γ (0) = p, ˙ γ (0) = X . a r X i v : . [ m a t h . DG ] J u l VLADIMIR S. MATVEEV AND STEFAN ROSEMANN
Two metrics g, ˜ g on the complex manifold ( M, J ) which are K¨ahler w.r.t. the complex structure J are called c-projectively equivalent if their J -planar curves coincide. A trivial example of c-projectively equivalent metrics is when the metric ˜ g is proportional to g with a constant coefficient.Another trivial (in the sense it is relatively easy to treat it at least in the Riemannian, i.e.,positively-definite case) example is when the metrics are affinely equivalent , that is when theirLevi-Civita connections coincide. If these metrics are K¨ahler w.r.t. the same complex structure,they are of course c-projectively equivalent since the equation (1) defining J -planar curves involvesthe connection and the complex structure only.The theory of c-projectively equivalent K¨ahler metrics is a classical one. It was started in the50th in Otsuki et al [32] and for a certain period of time was one of the main research directionsof the japanese (Obata, Yano) and soviet (Sinjukov, Mikes) differential geometric schools, see thesurvey [29] or the books [39, 34] for an overview of the classical results. In the recent time, thetheory of c-projectively equivalent metrics has a revival. A number of new methods appearedwithin or were applied in the c-projective setting and classical conjectures were solved. Moreover,c-projectively equivalent metrics independently appeared under the name Hamiltonian -forms,see [3, 4, 5, 6], which are also closely related to conformal Killing or twistor -forms studiedin [31, 33]. These relations will be explained in more detail in Section 1.3. The c-projectivelyequivalent metrics also play a role in the theory of (finitely-dimensional) integrable systems [19]where they are closely related to the so called K¨ahler-Liouville metrics, see [20]. Remark . Most classical sources use the name “h-projective” or “holomorphically-projective” forwhat we call “c-projective” in our paper. We also used “h-projective” in our previous publications[14, 27]. Recently a group of geometers studying c-projective geometry from different viewpointsdecided to change the name from h-projective to c-projective, since a c-projective change of con-nections, though being complex in the natural sense, is generically not holomorphic. The prefix“c-” is chosen to be reminiscent of “complex-” but is not supposed to be pronounced nor regardedas such since “complex projective” is already used differently in the literature.As we recall in Section 2, the set of metrics c-projectively equivalent to a given one (say, g ) isin one-to-one correspondence with the set of nondegenerate hermitian symmetric (0 , -tensors A satisfying the equation ( ∇ Z A )( X, Y ) = g ( Z, X ) λ ( Y ) + g ( Z, Y ) λ ( X ) + ω ( Z, X ) λ ( JY ) + ω ( Z, Y ) λ ( JX ) (2)for all X, Y, Z ∈ T M . Here λ is a -form which is easily seen to be equal to λ = d trace A (here, A is viewed as a (1 , -tensor by “raising one index” with the metric). Since this equation is linear,the space of its solutions is a linear vector space. Its dimension is called the degree of mobility of ( g, J ) and will be denoted by D ( g, J ) . Locally, D ( g, J ) coincides with the dimension of the set(equipped with its natural topology) of metrics c-projectively equivalent to g .Our main result is the list of all possible degrees of mobility of Riemannian K¨ahler metrics onsimply connected manifolds (in what follows we always assume that simply connectedness impliesconnectedness). The degree of mobility is always ≥ since the metrics of the form const · g providea one-parameter family of metrics c-projectively equivalent to g . One can show that for a genericmetric the degree of mobility is precisely . This statement, though known in folklore, is up toour knowledge nowhere published. Let us mention therefore that by [29] irreducible symmetricRiemannian K¨ahler spaces of nonconstant holomorphic curvature have degree of mobility precisely . Using this result one could show, similar to [26, §3.1], that for any (local) K¨ahler metric g andevery (cid:15) > there exists a K¨ahler metric g (cid:48) that is ε − close to g in the C ∞ topology such that thereexists (cid:15) (cid:48) > such that for every metric g (cid:48)(cid:48) that is ε (cid:48) − close to g (cid:48) in the C ∞ topology the degree ofmobility of g (cid:48)(cid:48) is precisely . Theorem 1.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler manifold of real dimension n ≥ . Suppose at least one metric c-projectively equivalent to g is not affinely equivalent to it.Then, the degree of mobility D ( g, J ) belongs to the following list: • , • k + (cid:96) , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 3 C D Figure 2.
The possible values for the degree of mobility D for ≤ dim M ≤ . • ( n + 1) ,Moreover, every value from this list that is greater than or equal to is the degree of mobilityof a certain simply connected n -dimensional K¨ahler manifold ( M, g, J ) such that there exists ac-projectively but not affinely equivalent metric ˜ g . In the theorem above, (cid:2) . (cid:3) denotes the integer part. The condition D ( g, J ) ≥ is due tothe assumption that there exists a metric that is c-projectively but not affinely equivalent (andtherefore not proportional) to g .If all metrics that are c-projectively equivalent to g are affinely equivalent to g , then it is not abig deal to obtain in the Riemannian situation the list of all possible degrees of mobility of suchmetrics on simply connected manifolds by using the same circle of ideas as in the proof of Theorem1 (see also Section 1.4 below): it is { k + (cid:96) | k ≤ n − , ≤ (cid:96) ≤ n − k } (cid:91) { n } . Special cases of Theorem 1 were known before. It is a classical result (see e.g. [3, Proposition4] or [34, Chap. V, §3]) that the maximum value D ( g, J ) = ( n + 1) implies that the metrichas constant holomorphic curvature and is attained on simply connected manifolds of constantholomorphic curvature. It was also previously known ([4, Proposition 10] and [14, Lemma 6])that in the case when the dimension is n = 4 the degree of mobility (on a simply connectedmanifold) takes the values , , only. We also see that the submaximal degree of mobility is ( n − + 1 = n − n + 2 . This value was also known before, see [29, §1.2], though we did notfind a place where this statement was proved.Under the additional assumption that the manifold M is closed, the analog of Theorem 1 isalso essentially known and the list of possible degrees of mobility is { , , ( n + 1) } . Indeed, as itwas shown in [14], if D ( g, J ) ≥ , ( M, g, J ) must be equal to ( C P ( n ) , const · g F S , J standard ) andtherefore its degree of mobility is ( n + 1) . On the other hand, there are many examples (con-structed in [4, 5, 6] or [20]) of closed K¨ahler manifolds different from ( C P ( n ) , const · g F S , J standard ) admitting c-projectively equivalent but not affinely equivalent Riemannian K¨ahler metrics. Thus,on closed K¨ahler manifolds (and here the assumption that the manifold is simply connected is notimportant), D ( g, J ) takes the values , and ( n + 1) only.1.1. The dimension of the space of essential c-projective vector fields.
A (possibly, local)diffeomorphism f : M → M of a K¨ahler manifold ( M, g, J ) is called c-projective transformation ifit is holomorphic (i.e., preserves J ) and if it sends J -planar curves to J -planar curves. A clearlyequivalent requirement is that the pullback f ∗ g is c-projectively equivalent to g . A c-projective VLADIMIR S. MATVEEV AND STEFAN ROSEMANN transformation is called essential , if it is not an isometry. A vector field is called a c-projectivevector field if its (locally defined) flow consists of c-projective transformations; we call it essentialif it is not a Killing vector field.For a given K¨ahler structure ( g, J ) , let c ( g, J ) and i ( g, J ) denote the Lie algebras of c-projectiveand holomorphic Killing vector fields respectively. Both are linear vector spaces and c ( g, J ) ⊇ i ( g, J ) . The quotient vector space c ( g, J ) / i ( g, J ) will be called the space of essential c-projectivevector fields . Its dimension is dim( c ( g, J )) − dim( i ( g, J )) . In the proof of Theorem 2 it willbe clear that under the assumption that the degree of mobility is ≥ this vector space couldbe (canonically) viewed as a subspace of c ( g, J ) . It is not a subalgebra though: typically thecommutator of two vector fields from this space is a nontrivial Killing vector field. Let us alsomention that the number dim( c ( g, J )) − dim( i ( g, J )) remains the same if we replace g by a c-projectively equivalent metric ˜ g . Theorem 2.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler manifold of real dimension n ≥ . Suppose at least one metric c-projectively equivalent to g is not affinely equivalent to it.Then, the dimension of the space c ( g, J ) / i ( g, J ) is • , , or • k + (cid:96) − , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) , or • ( n + 1) − .Moreover, each of the values of the above list is equal to the number dim( c ( g, J ) / i ( g, J )) fora certain n -dimensional simply connected Riemanian K¨ahler manifold ( M, g, J ) such that thereexists a metric that is c-projectively but not affinely equivalent to g . Under the assumption that the manifold is closed, the analog of Theorem 2 is again knownand is due to [27] where the classical Yano-Obata conjecture is proved; this conjecture impliesthat on any closed Riemannian K¨ahler manifold dim( c ( g, J )) = dim( i ( g, J )) unless the manifoldis ( C P ( n ) , const · g F S , J standard ) .1.2. Einstein metrics.
Our next group of results concerns K¨ahler-Einstein metrics. Note thatEinstein metrics play a special important role in the c-projective geometry since they are closely re-lated to the normal sections of the so-called prolongation connection of the metrizability equation,see [13].The analog of Theorem 1 under the additional assumption that the metric is Einstein is
Theorem 3.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler-Einstein manifold of real di-mension n ≥ . Assume at least one metric c-projectively equivalent to g is not affinely equivalentto it. Then, the degree of mobility D ( g, J ) is equal to one of the following numbers: • , • k + (cid:96) , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) , • ( n + 1) .Conversely, each of the numbers of the above list that is greater than or equal to is the degree ofmobility of a certain simply connected Riemannian K¨ahler-Einstein manifold ( M n , g, J ) admittinga metric ˜ g that is c-projectively equivalent but not affinely equivalent to g . The analog of Theorem 2 under the additional assumption that the metric is Einstein looks asfollows:
Theorem 4.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler-Einstein manifold of real di-mension n ≥ . Suppose at least one metric c-projectively equivalent to g is not affinely equivalentto it.Then, the dimension of the space c ( g, J ) / i ( g, J ) is • , , or • k + (cid:96) − , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) , or • ( n + 1) − . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 5
Moreover, each of the values of the above list is equal to dim( c ( g, J ) / i ( g, J )) for a certain n ≥ -dimensional simply connected Riemannian K¨ahler-Einstein manifold ( M, g, J ) such thatthere exists a metric that is c-projectively but not affinely equivalent to g . We see that the main difference in the lists of Theorems 1 and 3 respectively Theorems 2and 4 is that in the general case we divide n + 1 − k by and in the Einstein case we divide n + 1 − k by . The additional difference is that k goes up to n − in the general case and upto n − in the Einstein case. As a by-product we also obtain that in dimension n = 4 , twoc-projectively equivalent Riemannian Einstein metrics that are not affinely equivalent must be ofconstant holomorphic curvature; this result was known before and is in [21].Under the assumption that the manifold is closed, the list of the degrees of mobility is againmuch more simple as the next theorem shows: Theorem 5.
Suppose g and ˜ g are c-projectively equivalent K¨ahler-Einstein metrics of arbitrarysignature on a closed connected complex manifold ( M, J ) of real dimension n ≥ . Then, g and ˜ g are affinely equivalent unless ( M, g, J ) is ( C P ( n ) , const · g F S , J standard ) . Note that as examples constructed in [4] show, the assumption that the second metric is alsoEinstein is important for Theorem 5.As a by-product, in the proof of Theorems 4 and 5 we obtain the following
Theorem 6.
Assume two nonproportional K¨ahler-Einstein metrics (of arbitrary signature) ona complex n ≥ -dimensional manifold ( M, J ) are c-projectively equivalent. Then, any K¨ahlermetric that is c-projectively equivalent to them is also Einstein. Modulo Theorem 6 and under the additional assumption that g is Riemannian, Theorem 5follows from known global results in K¨ahler-Einstein geometry . More precisely, if the first Chernclass c ( M ) is nonpositive, Theorem 5 follows from [35, Theorem 1]. The proof of [35] is a standardapplication of the Weitzenb¨ock formula which shows nonexistence of a Hamiltonian Killing vectorfield, see also [8, page 89] or [30, page 77] (it is known that nonaffine c-projective equivalenceimplies the existence of a Hamiltonian Killing vector field, see e.g. [3, 20]).If c ( M ) > , then by Bando and Mabuchi [7] (see also [8, Addendum D]), for any twoK¨ahler-Einstein metrics g, ˜ g on a closed connected complex manifold ( M, J ) , there exists a bi-holomorphism f : M → M , contained in the connected component of the group of bi-holomorphictransformations of ( M, J ) , such that f ∗ g = const · ˜ g . This bi-holomorphism is of course ac-projective transformation of g . Now, by Theorem 6, we have a one-parameter family of c-projectively equivalent Einstein metrics which gives us a one-parameter family of such bi-holomorphisms.By the standard rigidity argument, we obtain then the existence a nontrivial (i.e., containing notonly isometries) connected Lie group of c-projective transformations and [27, Theorem 1] impliesthat the manifold is ( C P ( n ) , const · g F S , J standard ) .As a direct corollary of Theorem 5 we obtain Corollary 1 (Generalization of the Yano-Obata conjecture to metrics of arbitrary siganture underthe additional assumption that the metrics are Einstein) . Let ( M, g, J ) be a closed K¨ahler-Einsteinmanifold of arbitrary signature of real dimension n ≥ . Then, every c-projective vector field is anaffine (i.e., connection-preserving) vector field unless ( M, g, J ) = ( C P ( n ) , const · g F S , J standard ) . Relation to hamiltonian and conformal Killing -forms. Let ( M, g, J ) be a K¨ahlermanifold of real dimension n with K¨ahler form ω = g ( ., J. ) . A hermitian -form φ is called hamiltonian -form if ∇ X φ = X (cid:91) ∧ Jλ + ( JX ) (cid:91) ∧ λ (3)for a certain λ ∈ Ω ( M ) , where we define Jλ = λ ◦ J . It is straight-forward to see that λ = d trace ω φ , where trace ω φ = (cid:80) ni =1 φ ( Je i , e i ) for an orthonormal frame e , ..., e n . Hamiltonian -forms where introduced in [2] and have been studied further in [3, 4, 5, 6]. Among other interesting We are grateful to D. Calderbank and C. T ø nnesen-Friedman for pointing this out to us. VLADIMIR S. MATVEEV AND STEFAN ROSEMANN results and applications in K¨ahler geometry [5, 6], a complete local [3] and global [4] classificationof Riemannian K¨ahler manifolds admitting hamiltonian -forms have been obtained.Let φ be a hamiltonian -form and consider the corresponding symmetric hermitian (0 , -tensor A = φ ( J., . ) . It is easy to see that this correspondence sends solutions of (3) to solutionsof (2) and vice versa, thus hamiltonian -forms and hermitian symmetric solutions of (2), i.e.c-projectively equivalent K¨ahler metrics, are essentially the same objects. We immediately obtainfrom Theorems 1 and 3 the next two corollaries. Corollary 2.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler manifold of real dimension n ≥ . Suppose there exist at least one hamiltonian -form on M that is not parallel.Then, the dimension of the space of hamiltonian -forms belongs to the following list: • , • k + (cid:96) , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) . • ( n + 1) ,Moreover, every value from this list that is greater than or equal to is the dimension ofthe space of hamiltonian -forms of a certain simply connected n -dimensional K¨ahler manifold ( M, g, J ) that admits a non-parallel hamiltonian -form. Corollary 3.
Let ( M, g, J ) be a simply connected Riemannian K¨ahler-Einstein manifold of realdimension n ≥ . Assume that there exists at least one hamiltonian -form on M that is notparallel. Then, the dimension of the space of hamiltonian -forms is equal to one of the followingnumbers: • , • k + (cid:96) , where k = 0 , ..., n − and (cid:96) = 1 , ..., (cid:2) n +1 − k (cid:3) , • ( n + 1) .Conversely, each of the numbers of the above list that is greater than or equal to is thedimension of the space of hamiltonian -forms of a certain simply connected Riemannian K¨ahler-Einstein manifold ( M n , g, J ) that admits a nonparallel hamiltonian -form. Following [3, Appendix A] and [31], we briefly recall the relation between hamiltonian -formsand conformal Killing or twistor -forms studied in [31, 33]. On a m -dimensional Riemannianmanifold ( M, g ) a conformal Killing or twistor -form is a -form ψ satisfying ∇ X ψ = X (cid:91) ∧ α + i X β (4)for certain α ∈ Ω ( M ) and β ∈ Ω ( M ) . It is straight-forward to see that β = 13 dψ and α = − n − δψ, where ( δψ )( X ) = (cid:80) mi =1 ( ∇ e i ψ )( X, e i ) for an orthonormal frame e , ..., e m .Now assume that ( M, g, J ) is a K¨ahler manifold of real dimension m = 2 n , with K¨ahler form ω = g ( ., J. ) . Then, as shown in [3, Appendix A] and [31, Lemma 3.11], the defining equation (4)for a conformal Killing -form which in addition is assumed to be hermitian, equivalently reads ∇ X ψ = X (cid:91) ∧ α − ( JX ) (cid:91) ∧ Jα + Jα ( X ) ω. (5)On the other hand, setting α (cid:48) = Jλ , (3) becomes ∇ X φ = X (cid:91) ∧ α (cid:48) − ( JX ) (cid:91) ∧ Jα (cid:48) (6)Thus, if φ is a hamiltonian -form we set f λ = trace ω φ and see that ψ = φ − f λ ω (7)is a conformal Killing -form with α = α (cid:48) . Conversely, let ψ be a conformal Killing -form. In thecase n > , we have Jα = df α , where f α = n − trace ω ψ , see [3, Appendix A] and [31, Lemma3.8]. Then, φ = ψ − f α ω (8) ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 7 is a hamiltonian -form with α (cid:48) = α , and the linear mappings in (7) and (8) are inverse to eachother. Thus, when n > , hamiltonian -forms and conformal (hermitian) Killing -forms areessentially the same objects and Corollaries 2 and 3 also descibe the possible dimensions of thespace of hermitian conformal Killing -forms for simply-connected Riemannian K¨ahler respectivelyRiemannian K¨ahler-Einstein manifolds of dimension n > that admit a non-parallel conformalKilling -form.1.4. Relation to parallel (0 , -tensors on the conification and circles of ideas used inthe proof. As we mentioned above and will recall later, the system of equations (2) (such thatthe dimension of the space of solutions is the degree of mobility) is linear and overdetermined.In theory, there exist algorithmic (sometimes called prolongation-projection or Cartan-K¨ahler)methods to understand the dimension of the space of solutions of such a system. In practice, thesemethods are effective in small dimensions only, or return the maximal and submaximal valuesfor the possible dimension of the space of solutions. Actually, the previously known results areobtained by versions of these methods.A key observation that allowed us to solve the problem in its full generality is that the problemcan be reduced to the following geometric one:
Find all possible dimensions of the spaces of parallel symmetric hermitian (0,2)-tensors onconifications of K¨ahler manifolds.
The reduction goes as follows. First of all, at least for the proof of Theorem 1, we may assumethat the degree of mobility of ( M n , g, J ) is ≥ (since otherwise it is and this case is clear).Then, as we recall in Theorem 7, the solutions of the system (2) are in one-one correspondencewith the solutions of the system (20).Since the construction of the conification ( ˆ M n +2 , ˆ g, ˆ J ) of ( M n , g, J ) in general only workslocally (we explain this in detail in Section 3 and Theorem 8) we apply it assuming that ourmanifold is diffeomorphic to a disc. The extension of the results to simply connected manifoldsis then a standard application of the theorem of Ambrose-Singer and will be explained in Section5.2.Consider now the number B from (20). If B = − , then the solutions of (20) are essentiallyparallel symmetric hermitian (0 , -tensors on ( ˆ M , ˆ g, ˆ J ) , see Theorem 9. If B (cid:54) = − but B (cid:54) = 0 ,one can always make B = − by an appropriate scaling of the metric g . If the initial constant B was positive, then ˆ g has signature (2 , n ) .The case B = 0 requires the following additional work: we show that on any simply connectedneighborhood such that its closure is compact there exists a c-projectively equivalent Riemannianmetric with B (cid:54) = 0 , see Section 4. For this new metric the above reduction works. Evidently, c-projectively equivalent metrics have the same degrees of mobility. Thus, also in this case, insteadof solving the initial problem, we study the possible dimensions of the space of parallel symmetrichermitian (0 , -tensors on the conification of a K¨ahler manifold.To do this, let us first assume, for simplicity and since the ideas will be already visible in thissetting, that B is negative so the conification we will work with is a Riemannian manifold. Inthe final proof we will assume that the signature is (2 , n ) which poses additional difficulties. Inorder to calculate the dimension of the space of parallel symmetric hermitian (0 , -tensors on theconification ( ˆ M , ˆ g, ˆ J ) , let us consider the (maximal orthogonal) holonomy decomposition of thetangent space at a certain point of ˆ M : T p ˆ M = T ⊕ T ⊕ ... ⊕ T (cid:96) . (9)where T is flat (in the sense that the holonomy group acts trivially on it) and all other T i areirreducible. Clearly, each T i is ˆ J -invariant so it has even dimension k i .It is well known, at least since de Rham [12], that parallel symmetric (0 , -tensor fields on aRiemannian manifold ˆ M are in one-one correspondence with the (0 , -tensors on T p ˆ M of the form(10) k (cid:88) i,j =1 c ij τ i ⊗ τ j + C g + ... + C (cid:96) g (cid:96) , VLADIMIR S. MATVEEV AND STEFAN ROSEMANN where { τ i } is a basis in T ∗ and g i is the restriction of ˆ g to T i . The assumption that the tensoris symmetric and hermitian implies that the k × k -matrix c ij is symmetric and is hermitianw.r.t. to the restriction of ˆ J to T . This gives us a k -dimensional space of such tensors. If ˆ M isflat, i.e. if the initial metric g has constant holomorphic curvature, we have k = n + 1 (=half ofthe dimension of the conification of our n -dimensional M ) which gives us the number ( n + 1) .Suppose now that our manifold is not flat so (cid:96) ≥ . We show that the dimension of each T i for i ≥ is ≥ (i.e., that k i ≥ ). The key observation that is used here is that each T i (or,more precisely, the restriction of ˆ g to the integral leafs of the integrable distribution T i ) is a conemanifold. Since it is not flat, it has dimension > and since the dimension is even it must be ≥ .Then, k = 0 , ..., n − and (cid:96) is at most (cid:2) n +1 − k (cid:3) , because the sum of the dimensions of all T i with i ≥ is n + 1) − k and each T i “takes” at least four dimensions, see (44).Now, in the case our manifold is Einstein, we show that the dimension of each T i is ≥ .This statement is due to the nonexistence of Ricci-flat but nonflat cones of dimension . Then, k = 0 , ..., n − and (cid:96) is at most (cid:2) n +1 − k (cid:3) , which gives us the list from Theorem 3.In the case the conification of the manifold has signature (2 , n ) , we use recent results of [15,Theorem 5], where an analog of (9) was proven under the additional assumption that our manifoldof signature (2 , n ) is a cone manifold (and conifications of K¨ahler manifolds of dimension n areindeed cone manifolds over n + 1 -dimensional manifolds). The additional work is required thoughsince the number k in this case is not necessary dim( T )2 but could also be dim( T )2 + 1 . In the lattercase we show that one of T i with i ≥ must necessary have dimension ≥ (resp. ≥ in theEinstein case) and the list of possible dimensions of the space of parallel symmetric hermitian (0 , -tensors remains the same.Let us now touch the proof of Theorems 2 and 4. The restriction dim( c ( g, J ) / i ( g, J )) ≤ D ( g, J ) − is straightforward. Now, in the case D ( g, J ) ≥ , under the additional assumption that theconstant B (cid:54) = 0 , we actually have(11) dim( c ( g, J ) / i ( g, J )) = D ( g, J ) − . Indeed, for every solution of (2) we canonically construct a c-projective vector field. Twosolutions of (2) give the same c-projective vector field if and only if their difference is a multipleof g . The formula (11) gives us the lists from Theorems 2 and 4. The case when B = 0 can bereduced to the case B (cid:54) = 0 by the same trick as in the proofs of Theorems 1 and 3.1.5. Organization of the paper.
In Section 2 we recall basic statements in c-projective geometrythat were proved before and that will be used in the proof.In Section 3, we describe the construction of the conification ( ˆ
M , ˆ g, ˆ J ) of a K¨ahler manifold ( M, g, J ) and will show that solutions to the system (20) with B = − on M correspond to parallelhermitian symmetric (0 , -tensors on ˆ M .Section 4 is a technical one, its first goal is to explain that w.l.o.g. we can assume that B in(20) is equal to − . The only nontrivial step here is in Section 4.1 and is as follows: if for theinitial metric B = 0 , we change the metric to a c-projectively equivalent one such that B (cid:54) = 0 for the new metric. The second goal of Section 4 is to prove the additional statement concerningthe case of an Einstein metric. Roughly speaking, one goal is to show that when we change themetric to make B (cid:54) = 0 , the metric we obtain is still Einstein. Another goal is to show that theconification construction applied to K¨ahler-Einstein metrics gives a Ricci-flat metric.In Section 5 we prove the Theorems 1 and 3. In Section 5.1 we will essentially calculate thepossible dimensions of the space of parallel symmetric hermitian (0 , -tensors on K¨ahler manifoldsthat arise from the conification construction. This will complete the proofs of the Theorems 1and 3 in the local situation. In Section 5.2 we will extend our local results to the global situation.In Section 5.3 we complete the proof of Theorem 1 respectively Theorem 3 and show that eachof the values in these theorems is the degree of mobility of a certain K¨ahler metric respectivelyK¨ahler-Einstein metric.In Section 6 we prove Theorems 2 and 4 and in the final Section 7 we prove Theorem 5. ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 9 Basic facts in the theory of c-projectively equivalent metrics
C-projective equivalence of K¨ahler metrics as a system of PDE.
Let g and ˜ g be twoK¨ahler metrics on the complex manifold ( M, J ) of real dimension n ≥ .It is well-known, see [37, equation (1.7)], that g and ˜ g are c-projectively equivalent if and onlyif for a certain -form Φ , the Levi-Civita connections ∇ , ˜ ∇ of g, ˜ g respectively satisfy ˜ ∇ X Y − ∇ X Y = Φ( X ) Y + Φ( Y ) X − Φ( JX ) JY − Φ( JY ) JX (12)for all vector fields X, Y . In the tensor index notation, (12) reads(13) ˜Γ ijk − Γ ijk = δ ij Φ k + δ ik Φ j − J ij Φ s J sk − J ik Φ s J sj . Actually, the -form Φ is exact, i.e., it is the differential of a function, and the function isexplicitly given in terms of g and ˜ g . Indeed, contracting (13) w.r.t. i and k , we obtain ∂ i ln (cid:18) det ˜ g det g (cid:19) = 2( n + 1)Φ i so Φ i = φ ,i = dφ for the function φ given by(14) φ := n +1) ln (cid:18) det ˜ g det g (cid:19) . The equation (13) allows us to reformulate the condition that the metrics g and ˜ g are c-projectively equivalent as a system of PDE on the components of ˜ g whose coefficients depend on g (and its derivatives). Indeed, in view of (12), the condition ˜ ∇ ˜ g = 0 which is the defining equationfor ˜ ∇ reads(15) ∇ Z ˜ g ( X, Y ) − Φ( X )˜ g ( Z, Y ) − Φ( Y )˜ g ( Z, X ) − Φ( JX )˜ ω ( Z, Y ) − Φ( JY )˜ ω ( Z, X ) − Z )˜ g ( X, Y ) = 0 , where we denote by ˜ ω the K¨ahler -form ˜ ω = ˜ g ( ., J. ) We will view this condition as a system of PDE on the entries of ˜ g whose coefficients depend onthe entries of g . This system of equations is nonlinear (since the entries of Φ depend algebraicallyon the entries ˜ g ). A remarkable observation by Mikes et al [28] is that one can make this systemlinear by a clever substitution. For this, consider the symmetric hermitian (0 , -tensor A and the -form λ given by A = A ( g, ˜ g ) = (cid:18) det ˜ g det g (cid:19) n +1) g ˜ g − g. (16) λ = − Φ g − A. (17)Here g − is viewed as an isomomorphism g − : T ∗ M → T M given by the condition g ( g − ψ, Y ) := ψ ( Y ) for all Y and A is viewed as a mapping A : T M → T ∗ M given by the condition AX ( Y ) = A ( X, Y ) . If we view (0 , -tensors as matrices and (0 , -tensors as n -tuples, the matrix of A (up to multiplying it with the scalar expression involving the determinants of the metrics) is theproduct of the matrices g , ˜ g − , and again g . The 1-form λ in the matrix-notation is minus theproduct Φ , g − and A . Using index notation, A ij = (cid:18) det ˜ g det g (cid:19) n +1) g is ˜ g sr g rj and λ i = − Φ s g sr A ri , where ˜ g ij is the inverse (2 , -tensor to ˜ g ij .Straightforward calculations show that the condition (15) is equivalent to the formula (2). Inindex notation, the equation (2) reads(18) a ij,k = λ i g jk + λ j g ik + J sj λ s J ik + J si λ s J jk . Note that contracting the equation (18) with g ij we obtain that the one-form λ is the differentialof the function trace g ( A ) . In view of this, (2) could be viewed as a linear system of PDE on theentries of A only. Note also that the formula (16) is invertible: the tensor A given by (16) is nondegenerate, andthe metric ˜ g is given in the terms of g and A by ˜ g = √ det( A ) gA − g. (19) Remark . The metrics g, ˜ g are affinely equivalent, if and only if the tensor A in (16) is parallel,i.e., if and only of λ in (2) is identically zero. Remark . Since (2) is linear and the metric g is always a solution of (2), for every solution A of(2), the tensor A + const · g is again a solution. Thus, if A is degenerate, we can choose (at leastlocally) the constant such that A + const · g is a non-degenerate solution and, hence, correspondsto a metric ˜ g that is c-projectively equivalent to g .Let us denote by A ( g, J ) the linear space of symmetric hermitian solutions A of (2). The degreeof mobility D ( g, J ) of a K¨ahler structure ( g, J ) is the dimension of A ( g, J ) . Remark . If the metrics g, ˜ g are c-projectively equivalent, the spaces A ( g, J ) and A (˜ g, J ) areisomorphic and hence, D ( g, J ) = D (˜ g, J ) . This statement is probably expected and evident, theproof can be found for example in [27, Lemma 1].The results recalled above are rather classical. The next result [14, Theorem 3] is a recent oneand plays a key role in our paper. Theorem 7 ([14]) . Let ( M, g, J ) be a connected K¨ahler manifold of degree of mobility D ( g, J ) ≥ and of real dimension n ≥ . Then, there exists a unique constant B such that for every A ∈ A ( g, J ) with the corresponding -form λ there exists the unique function µ such that ( A, λ, µ ) satisfies ( ∇ Z ) A ( X, Y ) = g ( Z, X ) λ ( Y ) + g ( Z, Y ) λ ( X ) + ω ( Z, X ) λ ( JY ) + ω ( Z, Y ) λ ( JX )( ∇ Z λ )( X ) = µg ( Z, X ) + BA ( Z, X ) ∇ Z µ = 2 Bλ ( Z ) . (20) for all vector fields X, Y, Z . Conification construction and solutions of (20) as parallel tensors on theconification
Let ( M, g, J ) be a K¨ahler manifold of arbitrary signature with K¨ahler -form ω = g ( ., J. ) .We explicitly allow dim M = 2 n = 2 here. We also suppose that the form ω is an exact form, ω = dτ . This is always true if H ( M, R ) = 0 and in particular, it is always true if the manifold isdiffeomorphic to the ball. This is sufficient for our purposes since we will apply the conificationconstruction only to subsets of such kind.We consider the manifold P = R × M, where t will denote the standard coordinate on R andthe natural projection to M is denoted by π : P → M .On P , we define the -form θ = dt − τ, where for readability we omit the symbol for the pullback of τ to P (actually, the formula aboveshould be θ = dt − π ∗ τ ). We will also omit the symbols of the π -pullback in all formulas belowso if in some formula we sum or compare a (0 , k ) -tensor defined on M with a (0 , k ) -tensor definedon P , the tensor on M should be pulled back to P by π .Clearly, dθ = − dτ = − ω . Let us define the metric h on P by h = θ + g ( where θ = θ ⊗ θ ) . Remark . The freedom in the choice of τ is not essential for us since (as it is straightforwardto check) the change τ (cid:55)−→ ˜ τ = τ + df yields a metric ˜ h that is isometric to h : the mapping φ ( t, p ) = ( t − f ( p ) , p ) satisfies φ ∗ θ = ˜ θ and hence, φ ∗ h = ˜ h . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 11
Further, let us denote by H = kern( θ ) the “horizontal” distribution of P defined by θ . For anyvector field X on M we define its horizontal lift X θ on P by the properties π ∗ ( X θ ) = X and θ ( X θ ) = 0 . (21)We consider now the cone over ( P, h ) , that is we consider the n + 1) -dimensional manifold ( ˆ M = R > × P, ˆ g = dr + r h ) . Let us denote by ˆ π : ˆ M → M the natural projection ( r, t, p ) (cid:55)→ p ∈ M . The kernel of the differential of this projection is spanned by ξ = r∂ r , which will be calledthe cone vector field , and η = ∂ t . In the literature on Sasaki manifolds, η is sometimes refered toas Reeb vector field.Next, we introduce an almost complex structure ˆ J on ˆ M (which later appears to be a complexstructure) by the formula ˆ Jξ = η, ˆ Jη = − ξ, and ˆ JX θ = ( JX ) θ . We will call the n + 1) -dimensional manifold ( ˆ M , ˆ g, ˆ J ) the conification of ( M, g, J ) .We took the name “conification” from the recent paper [1] of Alekseevsky et al where thisconstruction has been obtained in a more general situation. The relation between the two con-structions is explained in [1, Example 1].We have Theorem 8 (essentially, [1]) . Let ( M, g, J ) be a K¨ahler manifold of real dimension n ≥ andsuppose that the K¨ahler -form ω = g ( ., J. ) satisfies ω = dτ for a certain -from τ . Then theconification ( ˆ M , ˆ g, ˆ J ) is a K¨ahler manifold.Conversely, suppose ( ˆ M , ˆ g, ˆ J ) is a K¨ahler manifold which locally, in a neighborhood of everypoint of a dense open subset, is a cone, i.e. ( ˆ M , ˆ g ) is of the form ( ˆ M = R > × P, ˆ g = dr + r h ) fora certain pseudo-Riemannian manifold ( P, h ) . Then, ( ˆ M , ˆ g, ˆ J ) arises locally as the conification ofits K¨ahler quotient ( M, g, J ) . The K¨ahler quotient is taken with respect to the action of the hamiltonian Killing vector field η = ˆ Jξ , where ξ = r∂ r is the cone vector field. This will be explained in more detail in the proofof Theorem 8.We could also assume the less restrictive condition that the K¨ahler class [ ω ] ∈ H ( M, R ) isinteger, and hence, ω , up to scale, is the curvature of a certain connection one-form θ on some S -bundle P over M as in [1]. Then, the metrics h on P and ˆ g on the cone ˆ M over P can bedefined in the same way as above and the proof of Theorem 8 will be literally the same as theproof that we will give below. In this more general situation, the role of η = ∂ t is then played bythe fundamental vector field η of the S -action on P that satisfies θ ( η ) = 1 .Actually from the construction it is immediately clear that ˆ J = − Id . It is also straight-forwardto check that ˆ g is hermitian w.r.t. ˆ J by checking the condition ˆ g ( ˆ Ju, ˆ Jv ) = ˆ g ( u, v ) for all possiblecombinations of tangent vectors u, v of the form ξ, η and X θ . What remains is to show that ˆ J isparallel with respect to the Levi-Civita connection of ˆ g . This will be done in Section 3.1.Let us explain how the system (20) on M relates to the parallel (0 , -tensors on ˆ M . Theorem 9.
Let ( M, g, J ) be a K¨ahler manifold of real dimension n ≥ , suppose that the K¨ahler -form ω = g ( ., J. ) satisfies ω = dτ for a certain -from τ . Then, there exists an isomorphismbetween the space of solutions ( A, λ, µ ) of (20) with B = − and the space of parallel symmetrichermitian (0 , -tensors ˆ A on ( ˆ M , ˆ g, ˆ J ) . The isomorphism is explicit and is given by ( A, λ, µ ) ↔ ˆ A = µdr − rdr (cid:12) λ + r ( µθ + θ (cid:12) λ ( J. ) + A ) , (22) where we omit the symbol for the ˆ π -pullback of µ, λ, λ ( J. ) and A to ˆ M . In the formula (22) , X (cid:12) Y = X ⊗ Y + Y ⊗ X is the symmetric tensor product.Remark . If we use ˜ τ = τ + df instead of τ to construct the conification, the diffeomorphism φ : ˆ M → ˆ M given by φ ( r, t, p ) = ( r, t − f ( p ) , p ) (compare also Remark 5) satisfies φ ∗ θ = ˜ θ . Thus, φ sends ˆ g , the K¨ahler -form ˆ ω = ˆ g ( ., ˆ J. ) = rθ ∧ dr + r ω and ˆ A given by (22) to the correspondingobjects constructed by using ˜ τ . The proof of theorems 8 and 9 is by direct calculations and will be done in Sections 3.1 and 3.2.The proof of Theorem 8 is contained in [1] and will be given for self-containedness and becausewe will need all the formulas from the proof later on.3.1.
Proof of Theorem 8.
Recall that for a vector field X on M , the natural lift of X to thehorizontal distribution H = kern( θ ) ⊆ T P will be denoted by X θ , see equation (21). Lemma 1.
Let
X, Y denote vector fields on M . The Levi-Civita connection ∇ h of the metric h on P is given by the fomulas ∇ h η η = 0 , ∇ h η X θ = ∇ h X θ η = ( JX ) θ , ∇ h X θ Y θ = ( ∇ X Y ) θ + ω ( X, Y ) η (23) Proof.
Let u, v, w be vector fields on P . Using the Koszul formula h ( ∇ h u v, w ) = uh ( v, w ) + vh ( w, u ) − wh ( u, v ) − h ( u, [ v, w ]) + h ( v, [ w, u ]) + h ( w, [ u, v ]) we calculate h ( ∇ h η η, η ) = 0 and h ( ∇ h η η, X θ ) = 0 , thus, ∇ h η η = 0 as we claimed. Now we calculate h ( ∇ h η X θ , η ) = 0 and h ( ∇ h η X θ , Y θ ) = − h ( η, [ X θ , Y θ ]) = − θ ([ X θ , Y θ ]) = dθ ( X θ , Y θ ) = − ω ( X, Y ) . This shows that ∇ h η X θ = ( JX ) θ .To verify the last equation in (23), we calculate h ( ∇ h X θ Y θ , η ) = h ( η, [ X θ , Y θ ]) = 2 ω ( X, Y ) and h ( ∇ h X θ Y θ , Z θ ) = 2 g ( ∇ X Y, Z ) . Combining these two equations gives us the third equation in (23). (cid:3)
The formulas for the Levi-Civita connection ˆ ∇ of the Riemannian cone ( ˆ M = R > × P, ˆ g = dr + r h ) over a pseudo-Riemannian manifold ( P, h ) are given by ˆ ∇ ξ = Id , ˆ ∇ X Y = ∇ X Y − h ( X, Y ) ξ, (24)where ξ = r∂ r and X, Y are vector fields on P . This is well-known, see for example [25, Fact 3.2].For ( P = R × M, h = θ + g ) as above we can combine these formulas with (23) to obtain ˆ ∇ ξ = Id , ˆ ∇ η = ˆ J, ˆ ∇ ξ X θ = X θ , ˆ ∇ η X θ = ˆ JX θ , ˆ ∇ X θ Y θ = ( ∇ X Y ) θ + ω ( X, Y ) η − g ( X, Y ) ξ. (25)Using these equations, it is easy to check that ˆ ∇ ˆ J = 0 . A straight-forward way to do it, is to showthat the equation ˆ ∇ u ( ˆ Jv ) = ˆ J ˆ ∇ u v is fulfilled for u, v of the form ξ, η and X θ .Since ˆ g is evidently symmetric, nondegenerate and hermitian with respect to ˆ J , the conification ( ˆ M , ˆ g, ˆ J ) is a K¨ahler manifold as we claimed. This completes the proof of the first statement ofTheorem 8.The other direction of Theorem 8 immediately follows from Lemma 2.
Suppose ( ˆ M n +2 , ˆ g, ˆ J ) is a K¨ahler manifold which is locally, in a neighborhood of everypoint of a dense open subset, a cone, i.e. ( ˆ M , ˆ g ) is of the form ( ˆ M = R > × P, ˆ g = dr + r h ) fora certain (2 n + 1) -dimensional pseudo-Riemannian manifold ( P, h ) . Then, ( ˆ M , ˆ g, ˆ J ) is locally theconification of its K¨ahler quotient ( M n , g, J ) , where the quotient is taken w.r.t. the action of thehamiltonian Killing vector field r ˆ J∂ r .Proof. We work on an open subset of ˆ M such that on this subset, ( ˆ M , ˆ g ) is of the form ( ˆ M = R > × P, ˆ g = dr + r h ) . Consider the vector fields ξ = r∂r and η = ˆ Jξ . Since η is orthogonal to ξ ,the derivative of r in the direction of η is zero and therefore ∂ r commutes with η . Consequently, η is essentially a vector field on the manifold P , i.e. in a coordinate system ( r, x , ..., x n +1 ) , where x , ..., x n +1 denote coordinates on P , the ∂ r -component of η is zero and the ∂ x i -components donot depend on r .Inserting η into the metric ˆ g , we see that h ( η, η ) = 1 and since η is a Killing vector field for ˆ g it follows that η is also Killing for h . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 13
Let us take the quotient of P by the action of the local flow of η , to obtain a (local quotient)bundle π : P → M , where M is a manifold of real dimension n . Since we are working locallyanyway, this bundle can be viewed as P = R × M with coordinate t on the R -component suchthat η = ∂ t . Further, we introduce the -form θ = h ( η, . ) on P and denote by H = kern θ , thehorizontal distribution. Since η is Killing, this distribution is invariant with respect to the actionof the flow of η . For tangent vectors X, Y ∈ T p M , we denote by X θ , Y θ ∈ H their horizontal liftsto a certain point in π − ( p ) ⊆ P . Defining g ( X, Y ) = h ( X θ , Y θ ) , we see that the right-hand side does not depend on the choice of the base point of X θ , Y θ in π − ( p ) , hence g defines a Riemannian metric on M .Consider the endomorphism J (cid:48) = ∇ h η : T P → T P . From (24) it immediately follows that J (cid:48) η = ∇ h η η = ˆ ∇ η η − h ( η, η ) ξ = ˆ Jη + ξ = 0 . In the same way, we obtain J (cid:48) X θ = ∇ h X θ η = ˆ ∇ X θ η = ˆ JX θ . Thus, we have that J (cid:48) : H → H defines an almost complex structure and by setting ( JX ) θ = J (cid:48) X θ , we obtain an almost complex structure J : T M → T M on M which is indeed independentof the choice of base point of the lift X θ of X ∈ T p M , since the flow of η preserves ˆ J .Using the definition of ( g, J ) and the fact that η is a Killing vector field for h , we obtain g ( X, JY ) = h ( X θ , ∇ h Y θ η ) = − dθ ( X θ , Y θ ) . On the one hand, this shows that ω = g ( ., J. ) is a -form or equivalently, that g is hermitianwith respect to J , i.e., g ( J., J. ) = g . On the other hand, since dθ is horizontal in the sense thatit vanishes when η is inserted, this shows that dθ = − π ∗ ω . From this, it also follows that ω isclosed.Since θ ([ X θ , Y θ ]) = − dθ ( X θ , Y θ ) = 2 ω ( X, Y ) , we obtain that [ X, Y ] θ = [ X θ , Y θ ] − ω ( X, Y ) η .Using this, it is straight-forward to see that the Nijenhuis torsion N J ( X, Y ) = [
X, Y ] − [ JX, JY ] + J [ JX, Y ] + J [ X, JY ] of J lifts to the corresponding Nijenhuis torsion of ˆ J which is vanishing, more precisely ( N J ( X, Y )) θ = N ˆ J ( X θ , Y θ ) = 0 . Thus, J is integrable and ( M, g, J ) is a K¨ahler manifold. From our construction, it is clear that ( ˆ M , ˆ g, ˆ J ) coincides with the conification of ( M, g, J ) . This completes the proof of the lemma. (cid:3) Remark . The construction of g from ˆ g as presented in Lemma 2 is of course well-known andcoincides with the K¨ahler quotient of ( ˆ M , ˆ g, ˆ J ) w.r.t. the action of the hamiltonian Killing vectorfield η , see [17] for a short explanation of K¨ahler quotients and symplectic reduction. The factthat the conification procedure can be reversed by taking the K¨ahler quotient was also mentionedin [1].3.2. Proof of Theorem 9.
Let us first recall the following fact proved before for example in [15,Theorem 8], [24, Lemma 1] or [25, Proposition 3.1]:
Theorem 10 ([15, 24, 25]) . There is an isomorphism between the space of symmetric parallel (0 , -tensors ˆ A on the Riemannian cone ( ˆ M = R > × P, ˆ g = dr + r h ) and solutions ( L, σ, ρ ) ∈ Γ( S T ∗ M ⊕ T ∗ M ⊕ R ) of the linear PDE system ( ∇ h Z L )( X, Y ) = h ( Z, X ) σ ( Y ) + h ( Z, Y ) σ ( X )( ∇ h Z σ )( X ) = ρh ( Z, X ) − L ( Z, X ) ∇ h Z ρ = − σ ( Z ) (26) on ( P, h ) . The isomorphism is explicitly given by ( L, σ, ρ ) ↔ ˆ A = ρdr − rdr (cid:12) σ + r L, (27) where we omitted the symbol for the pullback of objects from P to ˆ M . Now let ( P = R × M, h = θ + g ) be defined as in the previous section, where ( M, g, J ) is a n -dimensional K¨ahler manifold with exact K¨ahler -form. Let us prove a technical lemma thatgives us a characterisation of solutions ( L, σ, ρ ) of the system (26) on ( P, h ) which are invariantwith respect to the action of the flow of η : Lemma 3.
The solution ( L, σ, ρ ) of (26) on ( P, h ) is η -invariant if and only if σ ( η ) = 0 , σ (( JX ) θ ) = L ( η, X θ ) , L (( JX ) θ , ( JX ) θ ) = L ( X θ , Y θ ) and ρ = L ( η, η ) . (28) for all X, Y ∈ T M .Proof.
Using (26) and (23), we have that L η L = 0 if and only if L η L )( η, η ) = 2 σ ( η ) , L η L )( η, X θ ) = σ ( X θ ) + L ( η, ( JX ) θ ) , L η L )( X θ , Y θ ) = L (( JX ) θ , Y θ ) + L ( X θ , ( JY ) θ ) . These are the first three equations in (28). From the invariance of σ it follows L η σ )( η ) = ρ − L ( η, η ) , which is the last equation in (28). The condition L η σ )( X θ ) = − L ( η, X θ ) + σ (( JX ) θ ) is equivalent to the second equation in (28). The invariance of ρ is satisfied automatically since L η ρ = − σ ( η ) = 0 . (cid:3) Next we show
Lemma 4.
There is an isomorphism between the space of solutions ( A, λ, µ ) of (20) on ( M, g, J ) for B = − and η -invariant solutions ( L, σ, ρ ) of (26) on ( P, h ) .With respect to the decomposition T P = R η ⊕ H , the correspondence is given by L = µθ + θ ⊗ λ ( J. ) + λ ( J. ) ⊗ θ + A, σ = λ, ρ = µ, (29) where we omit the symbol for the pullback of objects from M to P .Proof. First let us show, that ( L, σ, ρ ) in (29) defines a solution of (26). By direct calculationusing the formulas (23) for the Levi-Civita connection of h we obtain ( ∇ h η L )( η, η ) = ηL ( η, η ) − L ( ∇ h η η, , η ) = 0 = h ( η, η ) σ ( η ) + h ( η, η ) σ ( η ) , ( ∇ h η L )( X θ , η ) = − L (( JX ) θ , η ) = λ ( X ) = h ( η, X θ ) σ ( η ) + h ( η, η ) σ ( X θ ) , ( ∇ h η L )( X θ , Y θ ) = − A ( JX, Y ) − A ( X, JY ) = 0 = h ( η, X θ ) σ ( Y θ ) + h ( η, Y θ ) σ ( X θ ) . Further, using the equations in (20) with B = − , we calculate ( ∇ h Z θ L )( η, η ) = Z ( µ ) + 2 λ ( Z ) = 0 = h ( Z θ , η ) σ ( η ) + h ( Z θ , η ) σ ( η ) , ( ∇ h Z θ L )( X θ , η ) = ( ∇ Z λ )( JX ) − µg ( Z, JX ) + A ( Z, JX ) = 0 = h ( Z θ , X θ ) σ ( η ) + h ( Z θ , η ) σ ( X θ ) and finally ( ∇ h Z θ L )( X θ , Y θ ) = g ( Z, X ) λ ( Y ) + g ( Z, Y ) λ ( X ) = h ( Z θ , X θ ) σ ( Y θ ) + h ( Z θ , Y θ ) σ ( X θ ) . We have shown that L defined in (29) satisfies the first equation in (26). For σ we obtain ( ∇ h η σ )( η ) = 0 = ρh ( η, η ) − L ( η, η ) , ( ∇ h η σ )( X θ ) = − σ (( JX ) θ ) = − λ ( JX ) = ρh ( η, X θ ) − L ( η, X θ ) and ( ∇ h Z θ σ )( X θ ) = µg ( Z, X ) − A ( Z, X ) = ρh ( Z θ , X θ ) − L ( Z θ , X θ ) . Thus, σ satisfies the second equation in (26). Finally, for ρ we have ∇ h η ρ = 0 = − σ ( η ) and ∇ h X θ ρ = − λ ( X ) = − σ ( X θ ) . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 15
Now we show that under the correspondence (29), the η -invariant solution ( L, σ, ρ ) of (26)descends to a solution ( A, λ, µ ) of (20) with B = − . Using (28), we calculate ( ∇ Z A )( X, Y ) = ( ∇ h Z θ L )( X θ , Y θ ) + ω ( Z, X ) σ (( JY ) θ ) + ω ( Z, Y ) σ (( JX ) θ )= h ( Z θ , X θ ) σ ( Y θ ) + h ( Z θ , Y θ ) σ ( X θ ) + ω ( Z, X ) σ (( JY ) θ ) + ω ( Z, Y ) σ (( JX ) θ )= g ( Z, X ) λ ( Y ) + g ( Z, Y ) λ ( X ) + ω ( Z, X ) λ ( JY ) + ω ( Z, Y ) λ ( JX ) , which is the first equation in (20). To verify the second equation, we calculate ( ∇ Z λ )( X ) = ( ∇ h Z θ σ )( X θ ) = ρh ( Z θ , X θ ) − L ( Z θ , X θ ) = µg ( Z, X ) − A ( Z, X ) . Finally, ∇ Z µ = − σ ( Z θ ) = − λ ( Z ) . This completes the proof of Lemma 4. (cid:3) Recall that we already have an isomorphism (27) between the space of symmetric parallel (0 , -tensors ˆ A on ( ˆ M , ˆ g, ˆ J ) and solutions ( L, σ, ρ ) of (26) on ( P, h ) . To prove Theorem 9, it remainsto show that hermitian symmetric parallel (0 , -tensors ˆ A correspond to η -invariant solutions ( L, σ, ρ ) .Using the definition of ˆ J , it is easy to see that ˆ A in (27) is hermitian, i.e., ˆ A ( ˆ J., ˆ J. ) = ˆ A , if andonly if ( L, σ, ρ ) satisfies the equations (28). By Lemma 3, it follows that ˆ A is hermitian if andonly if ( L, σ, ρ ) is η -invariant. This completes the proof of Theorem 9.4. How to reduce the investigation of A ( g, J ) of dimension ≥ locally to theinvestigation of the space of parallel hermitian (0 , -tensors on theconification If B = − in (20), the reduction was done in the previous section. For our purposes it issufficiently to assume that the manifold is diffeomorphic to the n -dimensional ball, the transition“any ball” −→ “any simply-connected manifold” will be done in Section 5.2. The goal of the sectionis to show that on every neighborhood U that is diffeomorphic to the ball and such that the closureof U is compact one can achieve B = − by replacing the metric by its c-projectively equivalent.If B (cid:54) = 0 , this could be done by a scaling of g , see the proof of Corollary 4. If B = 0 , then we doneed to change the metric by a essentially c-projectively equivalent one: Lemma 5.
Let ( M, g, J ) be a connected Riemannian K¨ahler manifold of real dimension n ≥ .Assume there exists a solution ( A, λ, µ ) of (20) with B = 0 such that λ (cid:54) = 0 .Then for every open simply connected subset U ⊆ M with compact closure, there exists aRiemannian K¨ahler metric ˜ g on U that is c-projectively equivalent to g , such that for any solution ( ˜ A, ˜ λ ) of the system (2) for the metric ˜ g there exists a function ˜ µ such that ( ˜ A, ˜ λ, ˜ µ ) satisfies (20) for the metric ˜ g and such that the corresponding constant ˜ B is different from . The proof of Lemma 5 will be given in Section 4.1.As we already remarked, D ( g, J ) is the same for all c-projectively equivalent metrics. As andirect application of Lemma 5 we obtain Corollary 4.
Let ( M, g, J ) be a connected Riemannian K¨ahler manifold of real dimension n ≥ and of degree of mobility D ( g, J ) ≥ . Suppose there exists at least one metric c-projectivelyequivalent to g and not affinely equivalent to it. Then, on each open simply connected subset U ⊆ M with compact closure, the degree of mobility D ( g | U , J | U ) is equal to the dimension of thespace of solutions of (20) with B = − for a certain positively or negatively definite K¨ahler metric ˜ g on U that is c-projectively equivalent to g .Proof. By Lemma 5, on every open simply-connected subset U with the required properties, we canfind a Riemannian K¨ahler metric g (cid:48) , c-projectively equivalent to g , such that A ( g, J ) is isomorphicto the space of solutions of the system (20) for g (cid:48) with a certain constants B (cid:48) (cid:54) = 0 . For the rescaledmetric ˜ g = − B (cid:48) g (cid:48) , the system (20) holds now with a constant ˜ B = − . Depending on the sign of B (cid:48) , the new metric ˜ g is either positively or negatively definite. (cid:3) Proof of Lemma 5: the constant B in the system (20) can be made non-zero byan arbitrary small change of the metric in the c-projective class. The proof of Lemma5 is divided into several steps. First we show how the constant B changes if one chooses anothermetric in the c-projective class. Lemma 6.
Let ( M, g, J ) be a Riemannian K¨ahler manifold of real dimension n ≥ and let D ( g, J ) ≥ . Suppose ˜ g is c-projectively equivalent to g and let A = A ( g, ˜ g ) ∈ A ( g, J ) be given byformula (16) . Let λ and µ be the -form and function respectively such that ( A, λ, µ ) constitutes asolution of the system (20) for g with constant B and let Λ = g − λ . Then the constant ˜ B in thesystem (20) for ˜ g is given by ˜ B = (det A ) ( g ( A − Λ , Λ) − µ ) . (30) Proof.
Let us view the tensor A = A ( g, ˜ g ) in equation (16) equivalently as a (1 , -tensor g − A byraising the “left index” of A by contraction with the inverse metric g − . To simplify notation, wewill denote g − A again by A such that the equation (2) now reads ∇ X A = g ( ., X )Λ + g ( ., Λ) X + g ( ., JX ) J Λ + g ( ., J Λ) JX. (31)From the defining equation (16) it follows that A − = A (˜ g, g ) , thus ˜ A = A − is a solution of (31)written down in terms of ˜ g . The corresponding vector field ˜Λ can be expressed in terms of A and Λ . Indeed, a straight-forward calculation, using (12), (16), (31) and (17) yields ˜ ∇ X ˜ A = − (det A ) (˜ g ( ., X ) A − Λ + ˜ g ( ., A − Λ) X + ˜ g ( ., JX ) JA − Λ + ˜ g ( ., JA − Λ) JX ) . Comparing this with the expected form of equation (31) for ˜ g , we see that ˜Λ = − (det A ) A − Λ . (32)We will use this equation to calculate the second equation in the system (20) for ˜ g . First we notethat ∇ X (det A ) = 2(det A ) g ( A − Λ , X ) . Using this together with (32), (12), (17) and (20), a straight-forward calculation yields ˜ ∇ X ˜Λ = (det A ) ( g ( A − Λ , A − Λ) − B ) X + (det A ) ( g ( A − Λ , Λ) − µ ) ˜ AX.
Comparing this with the expected form of the second equation in (20) for ˜ g , we see that ˜ B is givenby (30) as we claimed. (cid:3) We consider the case when for the metric g the constant B in (20) is vanishing. The thirdequation in (20) shows that the function µ is necessarily a constant. Next we show that we canalways find a solution ( A, λ, µ ) of (20) such that µ (cid:54) = 0 . Lemma 7.
Let ( M, g, J ) be a connected K¨ahler manifold of real dimension n ≥ and of degreeof mobility D ( g, J ) ≥ . Suppose the system (20) holds for B = 0 and that at least one metricc-projectively equivalent to g is not affinely equivalent to it. Then on every open simply connectedsubset U ⊆ M , we can find a solution ( A, λ, µ ) of (20) such that µ (cid:54) = 0 .Proof. Recall from Remark 2 that if ˜ g is c-projectively equivalent to g but not affinely equivalentto it, the -form λ corresponding to A = A ( g, ˜ g ) ∈ A ( g, J ) is not identically zero. Let us workwith this solution A . The equations in (20) show that for the corresponding -form λ we have ∇ X λ = µg ( X, . ) for a certain constant µ .Suppose µ = 0 , i.e. ∇ X λ = 0 . Consider the -form Ag − λ (where as always both g − : T ∗ M → T M and A : T M → T ∗ M are viewed as bundle morphisms). Calculating its covariant derivativeusing (2), we obtain ∇ X ( Ag − λ ) = ( ∇ X A ) g − λ = λ ( g − λ ) g ( X, . ) + λ ( X ) λ − λ ( JX ) λ ( J. ) . (33)Recall that λ is the differential of a function, i.e. λ = ∇ f for a certain function f . On the otherhand, on the open neighborhood U also the -form λ ( J. ) is the differential of a function f (cid:48) : U → R . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 17
This follows from the fact that λ ( J. ) is parallel (and hence closed) and U is simply connected. Letus set c = λ ( g − λ ) (which is a non-zero constant) and define the -form σ = Ag − λ − f λ + f (cid:48) λ ( J. ) . It follows from (33), that ∇ X σ = cg ( X, . ) . On the other hand, it is straight-forward to check that the symmetric hermitian (0 , -tensor ˜ A = σ ⊗ σ + σ ( J. ) ⊗ σ ( J. ) satisfies (2). The corresponding -form ˜ λ is given by cσ and satisfies ∇ X ˜ λ = c g ( X, . ) . Thus ( ˜ A, ˜ λ, ˜ µ = c ) is the desired solution of (20) with B = 0 but ˜ µ (cid:54) = 0 . (cid:3) Now we are able to give the proof of Lemma 5. Let us suppose that B = 0 and let U be anopen simply connected subset with compact closure. By Lemma 7, we can find a solution ( A, λ, µ ) of (20) on U such that µ (cid:54) = 0 and after rescaling we can suppose that µ = 1 .For arbitrary real numbers t , we define the triple A ( t ) = t ( λ ⊗ λ + λ ( J. ) ⊗ λ ( J. )) + g, λ ( t ) = tλ, µ ( t ) = tµ = t. Obviously the triple ( A ( t ) , λ ( t ) , µ ( t )) is a solution of (20) for g with B = 0 . Moreover, since U hascompact closure, we find (cid:15) > such that for all t ∈ ( − (cid:15), (cid:15) ) the solution A ( t ) is non-degenerate on U and the metric ˜ g t in the c-projective class of g which corresponds to A ( t ) (that is, ˜ g t is definedby A ( g, ˜ g t ) = A ( t ) in equation (16)) is positively definite.Using Lemma 6, we see that the constant ˜ B t in the system (20) for ˜ g t is given by ˜ B ( t ) = (det A ( t )) ( g ( A ( t ) − Λ( t ) , Λ( t )) − µ ( t )) = (det A ( t )) ( t g ( A ( t ) − Λ , Λ) − t ) , where as usual Λ = g − λ . We want to show that ˜ B ( t ) is non-zero for some t ∈ ( − (cid:15), (cid:15) ) . Since (det A ( t )) is positive anyway it suffices to show that f ( t ) = t g ( A ( t ) − Λ , Λ) − t is non-zero for some t ∈ ( − (cid:15), (cid:15) ) . Taking the derivative of f when t = 0 , we obtain dfdt (0) = − , thus, there is some t ∈ ( − (cid:15), (cid:15) ) such that ˜ B ( t ) is non-zero. The metric ˜ g t has the properties asrequired in Lemma 5. This completes the proof of Lemma 5. Remark . Though we explicitly used in the proof of Lemma 7 that the metric g is Riemannian,Lemma 7 remains true for metrics of arbitrary signature (but the proof in arbitrary signature islonger and uses nontrivial results of [13]).4.2. Conification of Einstein manifolds.
Suppose ( M n ≥ , g, J ) is a K¨ahler-Einstein manifoldof arbitrary signature. We assume that the symplectic form ω = g ( ., J. ) is exact so that we canconsider the conification ( ˆ M n +2 , ˆ g, ˆ J ) introduced in Section 3. Our goal is to show that theinvestigation of solutions of (2) on ( M, g, J ) reduces to the investigation of parallel tensors on theconification ( ˆ M , ˆ g, ˆ J ) for a Ricci flat metric ˆ g . We start with the following technical statement. Lemma 8.
Let ( M, g, J ) be a K¨ahler-Einstein manifold of real dimension n ≥ . Suppose thereexists a solution ( A, λ, µ ) of (20) such that λ (cid:54) = 0 . Then, B = − Scal( g )4 n ( n + 1) , (34) where Scal( g ) is the scalar curvature of g .Proof. Take a solution A of (20) such that λ (cid:54) = 0 . As usual we denote Λ = g − λ . It is known (see[3, Proposition 3], [14, Corollary 3] or equation (13) and the sentence below in [28]) that J Λ is aKilling vector field which in particular implies that λ is non-zero in every point of an open andeverywhere dense subset. We define the function σ = trace ∇ Λ . From the second equation in (20),we see that σ and µ are related by σ = 2 nµ + B trace( A ) . Thus, taking the covariant derivative ofthis equation and inserting the third equation of (20), we obtain ∇ σ = 2 n ∇ µ + 4 Bλ = 4 B ( n + 1) λ. (35) On the other hand, since J Λ is Killing, we have the identity ∇ X ∇ Λ = − JR ( X, J Λ) . Usingthe symmetries of the curvature tensor R of the K¨ahler metric g , we have trace( JR ( X, JY )) =2Ric( g )( X, Y ) . Combining the last two equations yields ∇ σ = − g )( ., Λ) and inserting thisinto (35), we have − Ric( g )( ., Λ) = 2 B ( n + 1) λ. Since g is K¨ahler-Einstein, that is Ric( g ) = Scal( g )2 n g , we evidently have (34). (cid:3) Lemma 9.
Let ( M, g, J ) be a K¨ahler-Einstein manifold of real dimension n ≥ and let thesymplectic form ω = g ( ., J. ) be exact. Assume Scal( g ) = 4 n ( n + 1) . Then, the conification of ( M, g, J ) is Ricci flat. Moreover, if a conification of a certain K¨ahler-Einstein manifold ( M ≥ , g, J ) is Ricci-flat, then g is Einstein with scalar curvature Scal( g ) = 4 n ( n + 1) .Proof. By direct calculation, using the formulas (25), we obtain that the curvature tensor ˆ R of ( ˆ M , ˆ g, ˆ J ) is given by the formulas ˆ R ( ., . ) ξ = ˆ R ( ., . ) Jξ = 0 , (36)where ξ = r∂ r is the cone vector field on ( ˆ M , ˆ g, ˆ J ) , and ˆ R ( X θ , Y θ ) Z θ = ( R ( X, Y ) Z − H ( X, Y ) Z ) θ , (37)where R denotes the curvature tensor of ( M, g, J ) , H ( X, Y ) Z = 14 ( g ( Z, Y ) X − g ( Z, X ) Y + ω ( Z, Y ) JX − ω ( Z, X ) JY + 2 ω ( X, Y ) JZ ) is the algebraic curvature tensor of constant holomorphic curvature equal to one and X θ denotesthe horizontal lift of tangent vectors X ∈ T M to the distribution H = span { ξ, Jξ } ⊥ ⊆ T ˆ M (see also Section 3 for the notation). Having these formulas, Lemma 9 follows from simple linearalgebra:From (36) it is clear that Ric(ˆ g )( ξ, . ) = Ric(ˆ g )( Jξ, . ) = 0 , where ξ denotes the cone vector fieldon ˆ M . A straight-forward calculation using (37) yields Ric(ˆ g )( X θ , Y θ ) = r (Ric( g )( X, Y ) − n + 1) g ( X, Y )) . implying that if Scal( g ) = 4 n ( n + 1) then Ric(ˆ g )( X θ , Y θ ) = r (cid:18) Ric( g )( X, Y ) − Scal( g )2 n g ( X, Y ) (cid:19) = 0 so ˆ g is Ricci flat, and if ˆ g is Ricci flat then g is Einstein with Scal( g ) = 4 n ( n + 1) . (cid:3) Lemma 10.
Let ( M, g, J ) be a K¨ahler-Einstein manifold of real dimension n ≥ . Suppose thereexists a solution ( A, λ, µ ) of (20) such that λ (cid:54) = 0 . Then, every metric ˜ g , c-projectively equivalentto g , is also K¨ahler-Einstein.Proof. Let A = A ( g, ˜ g ) be the solution of (2) given by (16). In the second equation ∇ λ = µg + BA of (20), we express λ in terms of the -form Φ by using the relation (17). A straight-forwardcalculation yields ( µ − g ( A − Λ , Λ)) gA − + Bg = −∇ Φ + Φ ⊗ Φ − Φ( J. ) ⊗ Φ( J. ) , where A is equivalently viewed as a (1 , -tensor and Λ = g − λ as usual. Using ˜ g = (det A ) − gA − and the transformation rule (30) for B , we can rewrite this into the form Bg − ˜ B ˜ g = −∇ Φ + Φ ⊗ Φ − Φ( J. ) ⊗ Φ( J. ) , (38)On the other hand, using the transformation rule (12) for the Levi-Civita connections of thetwo metrics, it is straight-forward to show and well-known (see [37, equation (1.11)]) that theRicci-tensors corresponding to g and ˜ g are related by Ric(˜ g ) = Ric( g ) − n + 1)( ∇ Φ − Φ ⊗ Φ + Φ( J. ) ⊗ Φ( J. )) . (39)Combining this equation with (38), we obtain Ric(˜ g ) + 2( n + 1) ˜ B ˜ g = Ric( g ) + 2( n + 1) Bg.
ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 19
Since the assumptions of Lemma 8 are satisfied and g is Einstein, we see from formula (34) thatthe right-hand side of the last equation is vanishing identically. Then, ˜ g is Einstein. (cid:3) Combining Lemma 5 with Lemmas 8 and 10, we obtain
Corollary 5.
Let ( M, g, J ) be a Riemannian K¨ahler-Einstein manifold of real dimension n ≥ and of degree of mobility D ( g, J ) ≥ . Suppose there exists a solution ( A, λ, µ ) of (20) such that λ (cid:54) = 0 .Then, on each open simply connected subset U ⊆ M with compact closure, the degree of mobility D ( g | U , J | U ) is equal to the dimension of the space of solutions of (20) with B = − Scal(˜ g )4 n ( n +1) = − for a certain positively or negatively definite K¨ahler-Einstein metric ˜ g on U that is c-projectivelyequivalent to g . It follows from Corollary 5, Theorem 9 and Lemma 9 that (at least in the local setting) wereduced the study of the degrees of mobility of n -dimensional K¨ahler-Einstein Riemannian metricsto the study of the possible dimensions of the space of parallel hermitian symmetric (0 , -tensorsof Ricci-flat cone K¨ahler manifolds ( ˆ M , ˆ g, ˆ J ) of dimension n + 1) where ˆ g is positively definiteor has signature (2 , n ) .In the proof of Theorem 3 we will need one more observation. Lemma 11.
Let ( ˆ M = R > × P, ˆ g = dr + r h, ˆ J ) be a K¨ahler manifold which is the cone overan (2 n + 1) -dimensional pseudo-Riemannian manifold ( P, h ) . (1) If dim ˆ M < and ˆ g is Ricci flat, then ˆ g is flat. (2) Let ˆ g have signature (2 , n ) . If dim ˆ M < , ˆ g is Ricci flat and X is a non-zero parallelnull-vector field on ˆ M , then ˆ g is flat.Proof. (1) Using (24) it is straight-forward to calculate that the curvature tensor ˆ R of ˆ g is givenby the formulas ˆ R ( ., . ) ∂ r = 0 and ˆ R ( X, Y ) Z = R ( h )( X, Y ) Z − ( h ( Z, Y ) X − h ( Z, X ) Y ) , (40)where X, Y, Z ∈ T P and R ( h ) is the curvature tensor of h . Then, calculating the Ricci tensor of ˆ g , it is straight-forward to see that if ˆ g is Ricci flat, h is Einstein with scalar curvature Scal( h ) =(dim P )(dim P − . If in addition dim ˆ M < , we have that dim P < and therefore h hasconstant curvature equal to . Inserting this back into (40), we obtain that ˆ R = 0 as we claimed.(2) Let ξ = r∂ r denote the cone vector field on ˆ M and let X be the parallel non-zero nullvector field. Since ˆ ∇ ξ = Id , we obtain that ˆ g ( X, ξ ) (cid:54) = 0 on a dense and open subset of ˆ M . Indeed,if ˆ g ( X, ξ ) = 0 on an open subset U we can take the covariant derivative of this equation in thedirection of Y ∈ T ˆ M to obtain that ˆ g ( X, Y ) = 0 in every point p ∈ U for all Y ∈ T p ˆ M . Thisimplies X = 0 on ˆ M which contradicts our assumption. By similar arguments one also obtainsthat ˆ g ( JX, ξ ) (cid:54) = 0 on a dense and open subset of ˆ M . Let us work in a point p ∈ ˆ M such that ˆ g ( X, ξ ) (cid:54) = 0 and ˆ g ( JX, ξ ) (cid:54) = 0 at p . We suppose that dim ˆ M = 6 and show that ˆ g is flat.It is an easy exercise to show that there exist a basis X, Y, Z, JX, JY, JZ of T p ˆ M in which ˆ g takes the form ˆ g = and such that span { X, Y, ˆ JX, ˆ JY } = span { X, ξ, ˆ JX, ˆ Jξ } . Hence any endomorphism of T p ˆ M thatcommutes with ˆ J and vanishes on ξ and X has to vanish on Y as well. This holds in particular forthe curvature endomorphisms ˆ R = ˆ R ( u, v ) : T p ˆ M → T p ˆ M for every pair of vectors u, v ∈ T p ˆ M . Since ˆ R commutes with ˆ J and is skew-symmetric with respect to ˆ g it takes the form ˆ R = a b − A − B − C − a c − D − A − E − c − b − E − C − FA B C a bD A E − a cE C F − c − b . in the basis X, Y, Z, JX, JY, JZ from above. Since ˆ R vanishes on X, Y, JX, JY it implies that a = b = c = A = B = C = D = E = 0 . Furthermore, using the condition that ˆ g is Ricci flat, i.e., trace( ˆ J ˆ R ) = 0 yields F = 0 . Thus the curvature tensor ˆ R is vanishing in every point of a denseand open subset, implying that ˆ g is flat as we claimed. (cid:3) Proof of Theorems 1 and 3
Proof of the first statement of the Theorems 1 and 3 in the local situation.
Let ( M, g, J ) be a K¨ahler manifold of real dimension n ≥ . Our goal is to show that for every opensimply connected subset U ⊆ M with compact closure and the property that the K¨ahler form ω is exact on U , the degree of mobility D ( g | U , J | U ) is given by one of the values either in the list ofTheorem 1 for a general metric or in the list of Theorem 3 under the additional assumption thatthe metric is Einstein. We will prove this simultaneously.By Corollary 4 and Theorem 9, the number D ( g | U , J | U ) is precisely the dimension of the spaceof parallel hermitian symmetric (0 , -tensors on the conification ( ˆ U , ˆ˜ g, ˆ J ) of ( U, ˜ g | U , J | U ) , where ˜ g is a certain metric on U that is c-projectively equivalent to g . Moreover, since ˜ g is either positivelyor negatively definite, the metric ˆ˜ g will be either positively definite or has signature (2 , n ) . Wealso know in view of Lemma 10 that if the metric g is Einstein, then the metric ˜ g is also Einsteinso the metric ˆ˜ g is Ricci-flat.To avoid cumbersome notations, we will drop the “hat” and the “tilde” in the notation for theconification. The local version of the Theorems 1 and 3 that we are going to prove in this sectionreads Theorem 11.
Let ( M, g, J ) be a simply connected K¨ahler manifold of real dimension n + 2 ≥ which is a cone over a (2 n + 1) dimensional manifold. Further, let g be either positively definiteor have signature (2 , n ) . Then, the dimension D of the space of parallel hermitian symmetric (0 , -tensors is given by one of the values in the list of Theorem 1. Moreover, if the metric g is Ricci-flat, then the dimension D of the space of parallel hermitian symmetric (0 , -tensors isgiven by one of the values in the list of Theorem 3. The proof of Theorem 11 in the case when the metric g is positively definite is more simplethan in the case when the signature is (2 , n ) . Moreover, the arguments for the proof when g ispositively definite are implicitly contained in the proof when the signature is (2 , n ) . We thereforerestrict to the latter case and assume that g has signature (2 , n ) in what follows; the Riemanniansignature is explained in Section 1.4 and we leave it as an easy exercise. Proof.
Let p be an arbitrary point in M . Consider a maximal orthogonal holonomy decompositionof T p M . T p M = T ⊗ T ⊗ ... ⊗ T (cid:96) . (41)Here T is a nondegenerate subspace of T p M such that the holonomy group acts trivially and suchthat it is J -invariant, and T i for i ≥ are J -invariant nondegenerate subspace invariant w.r.t. theaction of the holonomy group. We assume that the decomposition is maximal in the sense thatno T i , i ≥ has a holonomy-invariant nontrivial nondegenerate subspace and therefore can not bedecomposed further. The existence of such a decomposition is standard and follows for examplefrom [38]. ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 21
If in addition the initial manifold is Ricci-flat, then the restriction of the curvature tensor toeach T i is also Ricci-flat.It is well known that symmetric hermitian parallel (0 , -tensor fields on M are in one-onecorrespondence with symmetric hermitian (0 , -tensors on T p M that are invariant w.r.t. theaction of the holonomy group. As it was shown in [15, Theorem 5], every symmetric holonomy-invariant (0 , -tensor on T p M has the form(42) k (cid:88) α,β =1 c αβ τ α ⊗ τ β + C g + ... + C (cid:96) g (cid:96) . Here { τ i } i =1 ,..., k is a basis in the subspace of T ∗ M consisting of those elements that are invariantw.r.t. the holonomy group, and g i , i = 1 , ..., (cid:96) is the restriction of g to T i viewed as (0 , -tensors on T p M . Note that in the case of indefinite signature the number k must not coincidewith k := dim T , since it might exist a light-like holonomy-invariant vector such that it isorthogonal to all vectors from T . In the signature (2 , n ) we have k = k or k = k + 1 . Thecoefficients c αβ satisfy c αβ = c βα so ( c αβ ) is a symmetric matrix. Our assumption that the paralleltensor is hermitian implies that the matrix c αβ is hermitian.Clearly, the dimension of the space of the tensors of the form (42) is the number of free pa-rameters c αβ , C i . It is well known that the space of symmetric hermitian k × k matrices hasdimension k so the first term of (42) gives us k dimensions and we obtain k + (cid:96) in total whichis as we claimed. Our goal is to show that k and (cid:96) satisfy the restrictions in the Theorems 1 and3. Suppose (cid:96) = 0 (that is g is flat and hence, the initial n -dimensional metric has constantholomorphic curvature). Then, k = n + 1 and we obtain that the dimension of the space of theparallel hermitian tensors is ( n + 1) as we want.Suppose (cid:96) ≥ and take i ≥ . By [15, Lemma 2], the dimension of T i is ≥ and since it iseven, we have dim( T i ) ≥ . Moreover, under the additional assumption that T i is Ricci-flat, wehave dim( T i ) ≥ by Lemma 11.Suppose now T i with i ≥ contains a nonzero holonomy-invariant vector. Let us show thatthen the dimension of this T i is ≥ . Let us denote this vector by v . Note that the vector Jv isalso holonomy-invariant and any linear combination of v and Jv is light-like since otherwise therewould exist a nontrivial holonomy-invariant nondegenerate (two-dimensional) subspace.We extend T i and also v, Jv ∈ T i to the whole manifold by parallel translating these objectsalong all possible ways starting at p . The extension of T i is well-defined and gives us an integrabledistribution on M . The extensions of v and Jv are also well-defined and are parallel vector fields.It is sufficient to show that under the assumption dim( T i ) = 4 the restriction of the curvatureto this distribution vanishes, since this will imply in view of the theorem of Ambrose-Singerthat the holonomy group acts trivially on T i which contradicts the assumption that T i has nonontrivial holonomy-invariant nondegenerate subspaces. We choose a generic point q . Since thepoint is generic, then by [15, Lemma 5 and Lemma 2] there exists a vector u ∈ T i ( q ) such that g ( u, u ) (cid:54) = 0 and such that R ( u, . ) . = 0 . We consider the basis { u, Ju, v, Jv } of T i ( q ) . This isindeed a basis since the vectors u and Ju (resp. v and Jv ) are nonproportional and thereforelinearly independent and no nontrivial linear combination of u and Ju can be equal to a nontriviallinear combination of v abd Jv since any linear combination of v and Jv is light-like and anylinear combination of u and Ju is not light-like. Now, the vectors u, v, Ju, Jv satisfy the condition R ( u, . ) . = R ( Ju, . ) . = R ( v, . ) . = R ( Jv, . ) = 0 . Indeed, R ( u, . ) . = 0 is essentially the choice ofour vector, R ( Ju, . ) . = 0 is because the Riemanian curvature of a K¨ahler metric is J -invariant, R ( v, . ) . = R ( Jv, . ) = 0 is fulfilled because v and Jv are parallel. Thus, dim( T i ) ≥ .Let us now suppose that T i is Ricci flat and contains a nonzero (and therefore light-like)holonomy-invariant vector. Then, it follows from Lemma 11 that dim( T i ) ≥ .We obtain that the number k is at most n − and the number (cid:96) is at most (cid:2) n − k − (cid:3) . Indeed,suppose there exists a nonzero holonomy-invariant vector contained in one T i with i ≥ . Withoutloss of generality we may assume i = 1 . As we explained above, the dimension of T is ≥ and the dimension of all other T j for j ≥ is at least . The dimension of T is k − . Then,(43) dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) k − + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + ... + dim( T (cid:96) ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ = 2( n + 1) implying k ≤ n − and (cid:96) ≤ (cid:2) n − k − (cid:3) as we want.Suppose now there exists no nonzero holonomy-invariant vector contained in one T i with i ≥ .Then, dim( T ) = 2 k and the dimension of all T j for j ≥ is at least . Here we obtain (cid:96) ≤ (cid:2) n − k − (cid:3) by the same argument. Indeed, in this case(44) dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) k + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + ... + dim( T (cid:96) ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ = 2( n + 1) implying (cid:96) ≤ (cid:2) n − k − (cid:3) as we want.Assume now the metric ˆ g is Ricci-flat. Then, each T i is Ricci flat, so its dimension is ≥ .As we have shown above, if T i (with i ≥ ) contains a nonzero holonomy-invariant vector, then dim( T i ) ≥ , and the analog of (43) looks dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) k − + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + ... + dim( T (cid:96) ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ = 2( n + 1) . implying k ≤ n − and (cid:96) ≤ (cid:2) n − k − (cid:3) as we want. If there exists no nonzero holonomy-invariantvector contained in one T i with i ≥ . Then, dim( T ) = 2 k , the dimension of all T j for j ≥ is atleast and we obtain k ≤ n − and (cid:96) ≤ (cid:2) n − k − (cid:3) by the same argument. Indeed, in this case dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) k + dim( T ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ + ... + dim( T (cid:96) ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ = 2( n + 1) implying k ≤ n − and (cid:96) ≤ (cid:2) n − k − (cid:3) as we want. Theorem 11 is proved. (cid:3) Proof of the first parts of Theorems 1 and 3 in the global situation.
In this section,we complete the proof of the first parts of the Theorems 1 and 3. Let ( M, g, J ) be a simplyconnected K¨ahler manifold of real dimension n ≥ .We call a subset U ⊆ M a ball if U is open, homeomorphic to an open n -ball in R n and hascompact closure ¯ U .Since a ball U satisfies all the assumptions in Corollary 4, Theorem 9 and Theorem 11, weobtain that the degree of mobility D ( g | U , J | U ) of the restriction of the K¨ahler structure is givenby one of the values in the list of Theorem 1.If in addition g is Einstein, it follows from Corollary 5, Theorem 9 and Theorem 11 that D ( g | U , J | U ) is given by one of the values in the list of Theorem 3.Recall from [3, Proposition 4] or [29, equation (1.3)] that the space A ( g, J ) of hermitian sym-metric solutions of equation (2) is isomorphic to the subspace Par( E, ∇ E ) of the space of sectionsof a certain vector bundle π : E → M whose elements are parallel with respect to a certainconnection ∇ E on E . In particular, we have D ( g, J ) = dim Par( E, ∇ E ) .The next statement will complete the proof of the first parts of the Theorems 1 and 3. Lemma 12.
Let M be a simply connected manifold and let π : E → M be a vector bundle over M with a connection ∇ E . Let I ⊆ N be a set of nonnegative integers and suppose that for everyball U ⊆ M , we have dim Par( E | U , ∇ E ) ∈ I . Then, dim Par( E, ∇ E ) ∈ I .Proof. First let us introduce some notions. Let p ∈ U , where U is any simply connected open subsetof M , and let H ( U, p ) be the holonomy group of the restriction ∇ E : Γ( E | U ) → Γ( T ∗ U ⊗ E | U ) inthe point p . The space Par( E | U , ∇ E ) is isomorphic to the holonomy-invariant elements P ( U, p ) = { u ∈ E p : hu = u ∀ h ∈ H ( U, p ) } in the fiber E p , the isomorphism is given by parallel extensionof u ∈ P ( U, p ) to a parallel section on U . Since U is simply connected, the group H ( U, p ) isconnected. Then, P ( U, p ) coincides with(45) { u ∈ E p : hu = 0 ∀ h ∈ h ( U, p ) } , ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 23 Up c c c c c N − c N − c N Figure 3.
We can choose a tubular neighborhood U of the union (cid:83) Ni =1 c i ([0 , of the curves c , ..., c N : [0 , → M such that U is a ball.where h ( U, p ) is the Lie algebra of H ( U, p ) . Let R E ∈ Γ(Λ T ∗ M ⊗ End( E )) be the curvature of ∇ E . By the theorem of Ambrose-Singer (see e.g. [23]), h ( U, p ) as a vector space is generated byelements of the form ( τ c ) − R E ( X, Y ) τ c : E p → E p , (46)where X, Y ∈ T q M , q ∈ M , and τ c : E c (0) → E c (1) is the parallel displacement along a certainpiece-wise smooth curve c : [0 , → U with c (0) = p and c (1) = q .Since h ( M, p ) is a finite-dimensional vector space there exist finitely many curves c , ..., c N :[0 , → M starting at p such that h ( M, p ) as a vector space is generated by finitely many elementsof the form ( τ c i ) − R E ( X, Y ) τ c i : E p → E p . (47)If we sligtly perturbe these curves, the corresponding elements (47) will still generate h ( M, p ) sowe may assume that the curves have no intersections and self-intersections. Then, a sufficientlythin tubular neighborhood U of the union of the curves c i is a ball, see figure 3.The degree of mobility of the restriction of the K¨ahler structure to the ball U clearly coincideswith the degree of mobility of the K¨ahler structure on the whole manifold since the holonomygroups have the same algebras and therefore coincide. (cid:3) Proof of the second “realization” part of Theorems 1 and 3.
Let n ≥ , k ∈{ , ..., n − } and (cid:96) ∈ { , ..., [ n +1 − l ] } . We need to construct a n -dimensional simply-connectedRiemannian K¨ahler manifold ( M, g, J ) such that D ( g, J ) = k + (cid:96) and such that in the case k + (cid:96) ≥ there exists a metric ˜ g that is c-projectively but not affinely equivalent to g .The construction is as follows: we consider the direct product ( ˆ M , ˆ g, ˆ J ) = ( M , g , J ) × ( M , g , J ) × ... × ( M (cid:96) , g (cid:96) , J (cid:96) ) (48)of Riemannian K¨ahler manifolds. The manifold ( M , g , J ) is the standard R k with the standardflat metric and the standard complex structure. The Riemannian K¨ahler manifolds ( M i , g i , J i ) for i ≥ satisfy the following conditions: they have dimension ≥ , admit no nontrivial parallelhermitian symmetric (0 , -tensor field, are cone manifolds, and the sum of their dimensions is n +1 − k ) . The existence of such ( M i , g i , J i ) is trivial: because of the condition (cid:96) ∈ { , ..., [ n +1 − l ] } there exists a decomposition of n + 1 − k ) in the sum of the integer numbers k + ... + 2 k (cid:96) such that every k i ≥ . Now, as the manifold ( M i , g i , J i ) we take the conification of the standard ( R k i − , g flat , J standard ) . They are cone manifolds and they admit no nontrivial parallel hermitiansymmetric (0 , -tensor fields since for example by Theorem 9 the existence of such a tensor field will imply that the constant B of the standard ( R k i − , g flat , J standard ) is B = − though it isequal to zero.Let us also note, in view of the proof of Theorem 8, that ( M , g , J ) = ( R k \ { } , g flat , J ) coincides (at least locally) with the conification of ( C P ( k − , g F S , J standard ) via the Hopf fibration S k − → C P ( n ) .The direct product (48) is clearly a Riemannian K¨ahler manifold. By [15, Lemma 5], it is a conemanifold, so by Theorem 8 it is (at least in a neighborhood of almost every point) the conificiationof a certain n -dimensional K¨ahler manifold. This manifold has degree of mobility k + (cid:96) sincethe dimension of parallel symmetric hermitian (0 , tensors on its conification (which is (48)) is k + (cid:96) as we want. This completes the proof of Theorem 1.Now, in order to construct a K¨ahler-Einstein metric with degree of mobility D ( g, J ) = k + (cid:96) where k ∈ { , ..., n − } and (cid:96) ∈ { , ..., [ n +1 − l ] } we proceed along the same lines of ideas used abovebut assume in addition that the manifolds ( M i , g i , J i ) are Ricci-flat (and as such manifolds we cantake conifications of K¨ahler-Einstein manifolds with scalar curvature chosen in correspondencewith section 4.2) and irreducible. The restrictions k ∈ { , ..., n − } and (cid:96) ∈ { , ..., [ n +1 − l ] } implythat this is possible. This completes the proof of Theorem 3.6. Proof of Theorems 2 and 4
We need to show that dim( c ( g, J ) / i ( g, J )) is given by one of the values in the list of Theorem 2for a generic metric or by one of the values in the list of Theorem 4 if the metric is Einstein. Weassume that the manifold is simply connected and that the K¨ahler metric is Riemannian.Denote by h ( g, J ) the Lie algebra of homothetic vector fields of ( M, g, J ) , i.e., vector fields v satisfying L v g = const · g , where L v denotes the Lie derivative with respect to v . Consider thefollowing sequence → h ( g, J ) (cid:44) → c ( g, J ) f −→ A ( g, J ) / R g → , (49)where the mapping f is given by f ( v ) = − (cid:18) L v g − trace( g − L v g )2( n + 1) g (cid:19) mod R g. It is straight-forward to check that the tensor contained in the brackets on the right-hand side isindeed a solution of (2), for a proof see [27, Lemma 2]. From the formula for f , it is straight-forward to see that f ( v ) = 0 mod R g if and only if v is a homothetic vector field. Thus, the kernelof f coincides with the image of the inclusion map from h ( g, J ) to c ( g, J ) and the sequence (49)is exact at the first two stages. In particular, it follows that dim( c ( g, J ) / h ( g, J )) ≤ D ( g, J ) − . (50)From this inequality we see that in the case D ( g, J ) = 2 the codimension of i ( g, J ) in c ( g, J ) is atmost equal to one so dim( c ( g, J ) / i ( g, J )) is or . These two values are equal to dim( c ( g, J ) / i ( g, J )) for certain n ≥ -dimensional Riemannian K¨ahler-Einstein manifolds ( M, g, J ) admitting a c-projectively equivalent metric which is not affinely equivalent to it. Indeed, most of the closedRiemannian K¨ahler-Einstein manifolds ( M, g, J ) constructed in [4] that admit c-projectively equiv-alent metrics which are not affinely equivalent to g , are of non-constant holomorphic curvatureand therefore admit no non-killing c-projective vector field by the Yano-Obata conjecture [27].These examples have therefore dim( c ( g, J ) / i ( g, J )) = 0 . In order to consider the case D ( g, J ) ≥ ,and also to construct examples with dim( c ( g, J ) / i ( g, J )) = 1 , we need the following Lemma 13.
Let ( M, g, J ) be a connected K¨ahler manifold of real dimension n ≥ such that theequation (20) admits a solution ( A, λ, µ ) with λ (cid:54) = 0 . Assume B (cid:54) = 0 . Then, dim( c ( g, J ) / i ( g, J )) = D ( g, J ) − . (51) Proof.
Without loss of generality we can assume that B = − . Let us first show that g admitsno (local) homothety which is not an isometry. Suppose F : M → M is a homothety for g , i.e. F ∗ g = cg for a certain constant c . For the new metric ˜ g = cg , the system (20) holds for theconstant ˜ B = B ( cg ) = c B . But the constant B is unique: since cg and g are isometric via F , ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 25 the constants B and B ( cg ) (i.e., the constant B corresponding to the metric cg ) must coincide. Itfollows that c = 1 and consequently every homothety is an isometry.In view of this, the sequence (49) reads → i ( g, J ) (cid:44) → c ( g, J ) f −→ A ( g, J ) / R g → , (52)Let us now show that the sequence (49) is exact which of course immediately implies the equality(51).In order to do it, we show the existence of a splitting h : A ( g, J ) / R g → c ( g, J ) of the sequence (49). The mapping h is explicit and sends a solution A to the corresponding vectorfield Λ = g − λ . Using the system (20), it is straight-forward to check that Λ is a c-projective vectorfield (for an explicit proof see [36, Proposition 10.3]). Moreover, the vector field Λ is the samefor A and A + const · g , hence, h is well-defined and linear. For the composition of f and h , wecalculate f ( h ( A mod R g )) = f (Λ) = − (cid:18) ∇ λ − trace( ∇ Λ) n + 1 g (cid:19) mod R g (20) = − (cid:18) µg − n + 1) A + trace( A ) gn + 1 (cid:19) mod R g. The third equation in (20) implies dµ = − λ = − d trace A . Thus, the functions µ and − trace A coincide up to adding a constant. This shows that f ( h ( A mod R g )) = A mod R g, implying that (49) is a splitting exact sequence and (51) holds. (cid:3) Remark . The image of the map h is precisely the “canonical” space of essential c-projectivevector fields whose existence we announced in the introduction.Combining Lemma 13 with Theorem 1, we obtain the list from Theorem 2. Combining Lemma13 with Theorem 3, we obtain the list from Theorem 4 (under the additional assumption that B (cid:54) = 0 ).Note that by using Lemma 13, also the values from the lists of Theorem 2 or Theorem 3 can beobtained as the number dim( c ( g, J ) / i ( g, J )) since in Section 5.3 we also constructed metrics in allconsidered dimensions admitting solutions ( A, λ, µ ) of (20) with λ (cid:54) = 0 such that their degree ofmobility is two. The only case that cannot be constructed in this way is a -dimensional K¨ahler-Einstein structure ( g, J ) with dim( c ( g, J ) / i ( g, J )) = 1 since we can only produce examples ofconstant holomorphic curvature by using the procedure from Section 5.3. We therefore constructan explicit example: consider the local -dimensional Riemannian K¨ahler structure ( g, ω, J = − g − ω ) given in coordinates x, y, s, t by g = ( x − y )( dx + dy ) + x − y (cid:104) ( ds + xdt ) + ( ds + ydt ) (cid:105) , ˆ ω = dx ∧ ( ds + ydt ) + dy ∧ ( ds + xdt ) . This K¨ahler structure is a special case of those obtained in [3, 9]. It is straight-forward to checkthat g is Ricci-flat but non-flat and that the (1 , -tensor A = x ∂ x ⊗ dx + y ∂ y ⊗ dy + ( x + y ) ∂ s ⊗ ds + xy ∂ s ⊗ dt − ∂ t ⊗ ds is contained in A ( g, J ) (when viewed as (0 , -tensor) and is non-parallel (and thus, correspondsto a K¨ahler metric ˜ g , that is c-projectively equivalent to g and not affinely equivalent). Moreover,the vector field v = x ∂ x + y ∂ y + 2 s ∂ s + t ∂ t is a c-projective vector field for g and it is not Killing(thought, it is in fact an infinitesimal homothety, i.e. we have L v g = 3 g ). For this metric we have dim( c ( g, J ) / i ( g, J )) = 1 .Let us now consider the case B = 0 . In this situation, as in the proof of Theorem 1, we changethe metric in the c-projective class to make B non-zero. This can be done on every open connectedsubset with compact closure, see Corollary 4 and Corollary 5. The next lemma shows that thenumber dim( c ( g, J ) / i ( g, J )) remains the same. Lemma 14 (follows from [13]) . Let ( M, g, J ) be a simply connected K¨ahler manifold of real di-mension n ≥ . Then, dim i ( g, J ) = dim i (˜ g, J ) for any metric ˜ g that is c-projectively equivalentto g .Proof. We give a shorter version of the proof from [13]. Let K be a Killing vector field for ( g, J ) .It follows that K is also symplectic for the K¨ahler -form ω = g ( ., J. ) . Since we are working ona simply connected space, every symplectic vector field arises from a hamiltonian function f , i.e. K = X f , where X f is defined by g ( X f , J. ) = df . The condition that K is Killing is equivalent tothe condition that ∇∇ f is hermitian.Let φ be the function given by (14) and consider the function e φ f . Let us show that thisfunction is the hamiltonian function for a Killing vector field for ˜ g . The geometry behind thisstatement is explained in [13].We need to show that the symmetric (0 , -tensor field ˜ ∇ ˜ ∇ ( e φ f ) is hermitian.First of all, it is well-known, see for example [3, Proposition 3] or [20, Lemma 3.2], that thefunction e − φ is the hamiltonian for a Killing vector field for g and, swapping the metrics g and ˜ g , that e φ is the hamiltonian function for a Killing vector field for ˜ g . Consequently, its hessian ˜ ∇ ˜ ∇ ( e φ ) is hermitian.Using the transformation law (12), we calculate ˜ ∇ ˜ ∇ ( e φ f ) = f ( ˜ ∇ ˜ ∇ e φ ) + e φ ( ˜ ∇ df + 2Φ (cid:12) df )= f ( ˜ ∇ ˜ ∇ e φ ) + e φ ( ∇ df + Φ (cid:12) df + Φ( J. ) (cid:12) df ( J. )) . We see that the right-hand side of the above equation is hermitian, thus, the Hamiltonian vectorfield of e φ f is a Killing vector field for ˜ g . If we choose another hamiltonian function f + const for K , the mapping K (cid:55)−→ ˜ K = ˜ X e φ f , where ˜ X f is defined by ˜ g ( ˜ X f , J. ) = df , is only defined up to adding constant multiples of X e φ .Thus, dim i ( g, J ) coincides with dim i (˜ g, J ) as we claimed. (cid:3) Since obviously c ( g, J ) = c (˜ g, J ) , it follows from the lemma that on each open simply connectedneighborhood U , the number dim( c ( g, J ) / i ( g, J )) does not depend on the choice of the metric inthe c-projective class. Suppose in addition that U has compact closure. Then, by Corollary 4 thereexists a Riemannian metric ˜ g on U which is c-projectively equivalent to g and such that the system(20) for ˜ g holds with a constant ˜ B = − . Thus, by the already proven part, when restricted toa simply connected open subset U with compact closure, the number dim( c ( g, J ) / i ( g, J )) is givenby one of the values from the list of Theorem 2 and, in the Einstein situation, it is given by oneof the values from the list of Theorem 4.In order to prove Theorem 2 and 4 on the whole manifold, we again use Lemma 12. It isknown that Killing vector fields could be viewed as parallel sections of a certain vector bundle.The same is true for c-projective vector fields, for example because c-projective geometry is aparabolic geometry, see for example [11, Section 3.3], [10, Section 4.6] or [18], and infinitesimalsymmetries of parabolic geometries are sections of a certain vector bundle. Actually, in [13] thevector bundle and also the connection on it are explicitly constructed. By Lemma 12, the number dim( c ( g, J ) / i ( g, J )) on the whole manifold is the same as this number for the restriction of theK¨ahler structure to a certain ball and above we have shown that this value is contained in the listof Theorem 2 or, in the Einstein situation, in the list of Theorem 4.7. Proof of Theorem 5
Let us first recall the following statement from [22] (and give a full proof since the publicationis not easy to find)
Lemma 15.
Let g, ˜ g be c-projectively equivalent K¨ahler-Einstein metrics on the connected complexmanifold ( M, J ) of real dimension n ≥ . Then, for every A ∈ A ( g, J ) with corresponding -form λ , there exists a function µ such that ( A, λ, µ ) satisfies (20) with B = − Scal( g )4 n ( n +1) . ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 27
Proof.
Denote by A = A ( g, ˜ g ) the solution of (2) given by (16). We can insert the relation (17)between the -forms Φ in (12) and λ in (2) into (39) to obtain the change of the Ricci-tensors interms of A and λ . Denoting by Λ = g − λ the vector field corresponding to λ , a straight-forwardcalculation shows Ric(˜ g ) = Ric( g ) + 2( n + 1)( g ( A − ∇ Λ ., . ) − g ( A − Λ , Λ) g ( A − ., . )) . Now suppose both metrics are Einstein, that is
Ric(˜ g ) = ˜ c ˜ g and Ric( g ) = cg for constants c = Scal( g )2 n , ˜ c = Scal(˜ g )2 n . Inserting this into the last equation and multiplying with g − from the leftyields ˜ cg − ˜ g = c Id + 2( n + 1)( A − ∇ Λ − g ( A − Λ , Λ) A − ) . By (16), ˜ g can be written as ˜ g = (det A ) − gA − . Inserting this into the last equation andmultiplying with A from the left, we obtain ˜ c (det A ) − Id = cA + 2( n + 1)( ∇ Λ − g ( A − Λ , Λ)Id) . Rearranging terms yields ∇ Λ = µ Id +
BA, (53)where we defined µ = ¯ c (det A ) − n + 1) + g ( A − Λ , Λ) and B = − c n + 1) . (54)Equation (53) is exactly the second equation in (20). It remains to show that the third equation onthe function µ is satisfied as well. In [14, Remark 5] it was noted that if the second equation in thesystem (20) holds for B equal to a constant, the third equation in (20) is satisfied automatically.This is sufficient for our purposes, however, we show that the third equation can be obtaineddirectly by taking the covariant derivative of (53). We obtain ∇ X ∇ Λ = ( ∇ X µ )Id + B ∇ X A (2) = ( ∇ X µ )Id + B ( g ( ., X )Λ + g ( ., Λ) X + g ( ., JX ) J Λ + g ( ., J Λ) JX ) . Taking the trace of this equation yields trace( ∇ X ∇ Λ) = 2 n ∇ X µ + 4 Bg ( X, Λ) . (55)As in the proof of Lemma 8, we can use that J Λ is Killing to obtain the identity ∇ X ∇ Λ = − JR ( X, J Λ) . Together with the usual identities for the Ricci-tensor of a K¨ahler metric, thisyields trace( ∇ X ∇ Λ) = − trace( JR ( X, J
Λ)) = − X, Λ) = − cg ( X, Λ) , where we used the Einstein condition in the last step. Inserting the above formula and B from(54) into (55), we obtain the third equation in (20).We have shown that when g, ˜ g are c-projectively equivalent K¨ahler-Einstein metrics, there existsa function µ and a constant B such that the triple ( A = A ( g, ˜ g ) , λ, µ ) satisfies (20) for the metric g (of course, by interchanging the roles of g, ˜ g this can be obtained also in terms of ˜ g ). Since in thecase D ( g, J ) = 2 every A (cid:48) ∈ A ( g, J ) is a linear combination of Id and the solution A from above,we obtain the proof of Lemma 15 for D ( g, J ) = 2 . On the other hand, in the case D ( g, J ) ≥ ,Lemma 15 follows as a direct application of Theorem 7 above. (cid:3) Remark . Combining Lemmas 10 and 15, we obtain Theorem 6.Let us now prove Theorem 5, that is, let us show that two c-projectively equivalent K¨ahler-Einstein metrics on a closed connected complex manifold have constant holomorphic curvatureunless they are affinely equivalent.We have shown that the triple ( A, λ, µ ) , where A is the tensor from (16) constructed by the twometrics and λ is the corresponding -form from (2), satisfies the system (20) for a certain constant B . If this constant is zero, we see from (20) that the function µ is constant and ∇ λ is parallel.Since λ is the differential of a function and the manifold is closed, there are points where ∇ λ is positively and negatively definite respectively (corresponding to the minimum and maximumvalue respectively of the function). Since ∇ λ is parallel, it actually has to vanish identically, thus, λ is parallel. Since it vanishes at points where the corresponding function is maximal, λ has to beidentically zero. Using Remark 2, this implies that the two Einstein metrics are affinely equivalent.Now suppose that the metrics are not affinely equivalent. In particular, B is not zero and thefunction µ is not constant. Using the equations from the system (20), we can succesively replacethe covariant derivatives of A and λ to obtain that µ satisfies the third order system ( ∇∇∇ µ )( X, Y, Z ) = B [2( ∇ X µ ) g ( Y, Z ) + ( ∇ Z µ ) g ( X, Y ) + ( ∇ Y µ ) g ( X, Z ) − ( ∇ JZ µ ) g ( JX, Y ) − ( ∇ JY µ ) g ( JX, Z )] . (56)of partial differential equations. This equation was studied in [16, 36]. There it was shown that theexistence of non-constant solutions of this equation on a closed connected K¨ahler manifold impliesthat B < and the metric g has constant holomorphic curvature equal to − B . By interchangingthe roles of g and ˜ g , this statement holds for ˜ g as well. This completes the proof of Theorem 5. Acknowledgements.
We are grateful to D. V. Alekseevsky, D. Calderbank, M. Eastwood, A.Ghigi, V. Kiosak and C. T ø nnesen-Friedman for discussions and useful comments to this paper.Also, we thank Deutsche Forschungsgemeinschaft (Research training group 1523 — Quantum andGravitational Fields) and FSU Jena for partial financial support. References [1] D. V. Alekseevsky, V. Cortes, T. Mohaupt,
Conification of K¨ahler and hyper-K¨ahler manifolds , accepted toComm. Math. Phys., arXiv:1205.2964, 2012[2] V. Apostolov, D. Calderbank, P. Gauduchon,
The geometry of weakly self-dual K¨ahler surfaces , CompositioMath., , no. 3, 279–322, 2003, MR1956815[3] V. Apostolov, D. Calderbank, P. Gauduchon,
Hamiltonian 2-forms in K¨ahler geometry. I. General theory , J.Differential Geom. , no. 3, 359–412, 2006[4] V. Apostolov, D. Calderbank, P. Gauduchon, C. T ø nnesen-Friedman, Hamiltonian 2-forms in K¨ahler geome-try. II. Global classification , J. Differential Geom. , no. 2, 277–345, 2004[5] V. Apostolov, D. Calderbank, P. Gauduchon, C. T ø nnesen-Friedman Hamiltonian 2-forms in K¨ahler geometry.III. Extremal metrics and stability , Invent. Math. , no. 3, 547–601, 2008[6] V. Apostolov, D. Calderbank, P. Gauduchon, C. T ø nnesen-Friedman, Hamiltonian 2-forms in K¨ahler geome-try. IV. Weakly Bochner-flat K¨ahler manifolds , Comm. Anal. Geom. , no. 1, 91–126, 2008[7] S. Bando, T. Mabuchi, Uniqueness of Einstein K¨ahler metrics modulo connected group actions , Algebraicgeometry, Sendai, 1985, 11–40, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, MR0946233[8] A. Besse,
Einstein manifolds , Springer, 1987[9] A. V. Bolsinov, V. S. Matveev, T. Mettler, S. Rosemann,
Four-dimensional K¨ahler metrics admitting essentialc-projective vector fields, in preparation.[10] A. ˇCap,
Correspondence spaces and twistor spaces for parabolic geometries , J. Reine Angew. Math. ,143–172, 2005, MR2139714[11] A. ˇCap, A. R. Gover, M. Hammerl,
Holonomy reductions of Cartan geometries and curved orbit decompositions ,arXiv:1103.4497 [math.DG], 2011[12] G. De Rham,
Sur la reductibilit´e d’un espace de Riemann,
Comment. Math. Helv. , 328–344, 1952.[13] M. Eastwood, V. Matveev, K. Neusser, C-projective geometry: background and open problems , in preparation.[14] A. Fedorova, V. Kiosak, V. Matveev, S. Rosemann,
The only K¨ahler manifold with degree of mobility at least3 is ( CP ( n ) , g Fubini − Study ) , Proc. London Math. Soc., , no. 1, 153–188, 2012, doi: 10.1112/plms/pdr053,2012[15] A. Fedorova, V. Matveev, Degree of mobility for metrics of lorentzian signature and parallel (0,2)-tensor fieldson cone manifolds , arXiv:1212.5807 [math.DG], 2012[16] H. Hiramatu,
Integral inequalities in K¨ahlerian manifolds and their applications , Period. Math. Hungar. ,no. 1, 37–47, 1981, MR0607627, Zbl 0427.53032.[17] N. J. Hitchin, A. Karlhede, U. Lindstr¨om, M. Rocek, Hyper-K¨ahler metrics and supersymmetry , Com. Math.Phys. 108 (4) 535–589, 1987, MR877637[18] J. Hrdina,
Almost complex projective structures and their morphisms , Arch. Math. (Brno) , no. 4, 255–264,2009, MR2591680[19] K. Kiyohara, Two classes of Riemannian manifolds whose geodesic flows are integrable , Mem. Amer. Math.Soc. , no. 619, viii+143 pp., 1997[20] K. Kiyohara, P. J. Topalov,
On Liouville integrability of h-projectively equivalent K¨ahler metrics , Proc. Amer.Math. Soc. , 231–242, 2011.[21] V. Kiosak, M. Haddad,
On A-harmonic K¨ahler spaces,
Geometry of generalized spaces, 41–45, Penz. Gos.Ped. Inst., Penza, 1992.[22] V. Kiosak, M. Haddad,
On holomorphic-projective transformations of A-harmonic K¨ahler spaces, preprintedin Ukr. NIINTI 20.08.1991 no. 1217-UK91.
ONIFICATION CONSTRUCTION AND ITS APPLICATIONS 29 [23] S. Kobayashi, K. Nomizu,
Foundations of Differential Geometry II , John Wiley and Sons, Inc., 1996.[24] V. S. Matveev,
Gallot-Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable conesover closed complete pseudo-Riemannian manifolds do not exist , J. Diff. Geom. Appl. , no. 2, 236–240, 2010[25] V. S. Matveev, P. Mounoud, Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds andapplications , Ann. Glob. Anal. Geom. , 259–271, 2010[26] V. S. Matveev, Geodesically equivalent metrics in general relativity,
J. Geom. Phys. , no. 3, 675–691, 2012.[27] V. S. Matveev, S. Rosemann, Proof of the Yano-Obata conjecture for h-projective transformations , J. Differ-ential Geom. , no. 1, 221–261, 2012[28] J. Mikes, V. V. Domashev, On The Theory Of Holomorphically Projective Mappings Of Kaehlerian Spaces ,Math. Zametki , no. 2, 297–303, 1978[29] J. Mikes, Holomorphically projective mappings and their generalizations. , J. Math. Sci. (New York) , no. 3,1334–1353, 1998[30] A. Moroianu, Lecture Notes on K¨ahler geometry ∼ moroianu/tex/kg.pdf[31] A. Moroianu, U. Semmelmann, Twistor forms on K¨ahler manifolds , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , no. 4, 823–845, 2003, MR2040645[32] T. Otsuki, Y. Tashiro, On curves in Kaehlerian spaces , Math. Journal of Okayama University , 57–78, 1954[33] U. Semmelmann, Conformal Killing forms on Riemannian manifolds , Math. Z. , no. 3, 503–527, 2003,MR2021568[34] N. S. Sinjukov,
Geodesic mappings of Riemannian spaces. (in Russian) “Nauka”, Moscow, 1979, MR0552022,Zbl 0637.53020.[35] N. S. Sinyukov, E. N. Sinyukova,
Holomorphically projective mappings of special K¨ahlerian spaces , (Russian)Mat. Zametki , no. 3, 417–423, 1984, MR0767221[36] S. Tanno, Some Differential Equations On Riemannian Manifolds , J. Math. Soc. Japan , no. 3, 509–531,1978[37] Y. Tashiro, On A Holomorphically Projective Correspondence In An Almost Complex Space , Math. Journalof Okayama University , 147–152, 1956[38] H. Wu, On the de Rham decomposition theorem,
Illinois J. Math. , 291–311, 1964.[39] K. Yano, Differential geometry on complex and almost complex spaces.
International Series of Monographs inPure and Applied Mathematics, Vol. , A Pergamon Press Book. The Macmillan Co., New York 1965 xii+326pp. Institute of Mathematics, Friedrich-Schiller-Universit¨at Jena, Jena, Germany.
E-mail address : [email protected] Institute of Mathematics, Friedrich-Schiller-Universit¨at Jena, Jena, Germany.
E-mail address ::