Complex Osserman Kaehler Manifolds
aa r X i v : . [ m a t h . DG ] M a r COMPLEX OSSERMAN K ¨AHLER MANIFOLDS IN DIMENSIONFOUR
M. BROZOS-V ´AZQUEZ AND P. GILKEY
Abstract.
Let H be a 4-dimensional almost-Hermitian manifold which sat-isfies the K¨ahler identity. We show that H is complex Osserman if and onlyif H has constant holomorphic sectional curvature. We also classify in arbi-trary dimensions all the complex Osserman K¨ahler models which do not have3 eigenvalues.MSC 2000:53B35. Introduction
The geometric context.
Let R ( x, y ) := ∇ x ∇ y − ∇ y ∇ x − ∇ [ x,y ] be the cur-vature operator of a Riemannian manifold M := ( M, g ) of dimension n and let R ( x, y, z, w ) := g ( R ( x, y ) z, w ) be the associated curvature tensor. The eigenvaluestructure of various operators naturally associated to R has been studied in the lastdecade to obtain geometric information about M . We recall some notation: Definition 1.1.
Let M = ( M, g ) be a Riemannian manifold. (1) If R ( x, y, y, x ) = c for any orthonormal set { x, y } , then R is said to have constant sectional curvature . (2) The
Jacobi operator is defined by J ( x ) : y → R ( y, x ) x . (3) Let { e , ..., e k } be an orthonormal basis for a subspace σ . Following [21] ,the higher order Jacobi operator J ( σ ) is defined by setting: J ( σ ) := J ( e ) + ... + J ( e k ) . This is independent of the particular orthonormal basis chosen for σ . (4) M is said to be Osserman if the eigenvalues of J ( x ) are constant on thebundle of unit tangent vectors S ( M ) . Any locally 2-point homogeneous space is clearly Osserman. Osserman [20] won-dered if the converse held. This converse implication has been established exceptfor the (possibly) exceptional case n = 16; the Jacobi operator of M has con-stant eigenvalues on S ( M ) if and only if M is locally a 2-point homogeneous space[6, 9, 15, 17]. There are related problems defined by other natural operators. Theconformal Jacobi operator has been investigated [1, 2, 18], the skew-symmetriccurvature operator has been investigated [12, 13, 16], and the higher order Jacobioperator has been investigated [10]; we refer to [8, 11] for further details.We now pass to the complex setting. Recall that H := ( M, g, J ) is said to bean almost-Hermitian manifold if J is an endomorphism of the tangent bundle T M which satisfies J = − Id and J ∗ g = g . Again, we recall some notation: Definition 1.2.
Let H be an almost-Hermitian manifold. (1) A -dimensional subspace π of T M is said to be a complex line if Jπ = π .Let CP ( H ) be the bundle of complex lines. The map x → π x := Span { x, Jx } defines the Hopf fibration S ( H ) → CP ( H ) . If π ∈ CP ( H ) , the complex cur-vature operator and the complex Jacobi operator are defined, respectively,by setting R ( π ) := R ( x, Jx ) and J ( π ) := J ( x ) + J ( Jx ) ; these operatorsare independent of the particular unit vector x ∈ S ( π ) which is chosen. (2) Let s ( x ) := R ( x, Jx, Jx, x ) be the holomorphic sectional curvature; H issaid to have constant holomorphic sectional curvature if s ( · ) = c on S ( H ) . (3) If R ( Jx, Jy ) = R ( x, y ) for all x, y , then H is said to satisfy the K¨ahleridentity . (4) An endomorphism Ξ of T M is said to be complex if J Ξ = Ξ J . (5) H is said to be complex Osserman if J ( π ) is complex for every π ∈ CP ( H ) and if the eigenvalues of J ( π ) are constant on CP ( H ) . In this setting, theeigenvalues and eigenvalue multiplicities of J ( π ) for any π ∈ CP ( H ) aresaid to be the eigenvalues and eigenvalue multiplicities of H . We shall be assuming for the most part that H satisfies the K¨ahler identity;Lemma 3.1 below shows that necessarily J ( π ) is complex in this setting.1.2. The algebraic context.
Let V be an n -dimensional real vector space. Anelement A ∈ ⊗ V ∗ is said to be an algebraic curvature tensor if A has the symmetriesof the Riemann curvature tensor, i.e. if for all x, y, z, w ∈ V , we have: A ( x, y, z, w ) = − A ( y, x, z, w ) = A ( z, w, x, y ) , (1) A ( x, y, z, w ) + A ( y, z, x, w ) + A ( z, x, y, w ) = 0 . (2)One says that H := ( V, h· , ·i , J, A ) is an almost-Hermitian curvature model if A isan algebraic curvature tensor on V , if h· , ·i is a positive definite inner product on V , and if J is an endomorphism of V satisfying J = − Id and J ∗ h· , ·i = h· , ·i . Let A ( x, y ) be the corresponding curvature operator. The notions of Definition 1.1 andDefinition 1.2 then extend immediately to this setting.An almost-Hermitian manifold H is said to be K¨ahler if ∇ J = 0; such a mani-fold satisfies the K¨ahler identity R ( Jx, Jy ) = R ( x, y ) discussed above. An almost-Hermitian curvature model H is said to be a K¨ahler model if H satisfies the K¨ahleridentity. Every K¨ahler model can be geometrically realized by a K¨ahler manifold[5].The eigenvalue structure of a complex Osserman K¨ahler model is very restrictive.We shall establish the following result in Section 3: Theorem 1.3.
Let H be a complex Osserman K¨ahler model of dimension n ≥ which is not flat. Then one of the following holds: (1) There are 2 eigenvalues of multiplicities ( n − , . (2) There are 3 eigenvalues of multiplicities ( n − , , with n = 4 k ≥ . Algebraic classification.
Let V be a vector space of dimension n = 2 m ,let h· , ·i be a positive definite symmetric inner product on V . If φ ∈ S ( V ∗ ) is asymmetric 2-tensor, we define A φ ( x, y, z, w ) = φ ( x, w ) φ ( y, z ) − φ ( x, z ) φ ( y, w ) . Similarly if ψ ∈ Λ ( V ∗ ) is an anti-symmetric 2-tensor, we define: A ψ ( x, y, z, w ) = ψ ( x, w ) ψ ( y, z ) − ψ ( x, z ) ψ ( y, w ) − ψ ( x, y ) ψ ( z, w ) . It is an easy calculation to show that A φ and A ψ are algebraic curvature tensors– see, for example, Lemma 1.8.1 of [11]. Let J be an endomorphism of V with J = − Id and with J ∗ h· , ·i = h· , ·i . The following examples will play a crucial rolein our development. Example 1.4.
Let A nµ = ( V, h· , ·i , µA ) be an n -dimensional real model where A = A φ is defined by taking φ = h· , ·i , i.e.: A ( x, y, z, w ) := h x, w ih y, z i − h x, z ih y, w i . Example 1.5.
Let B nµ = ( V, h· , ·i , J, µ ( A + A J )) where A J is defined by taking ψ := h J · , ·i to be negative of the K¨ahler form, i.e.: A J ( x, y, z, w ) := h Jx, w ih Jy, z i − h
Jx, z ih Jy, w i − h Jx, y ih Jz, w i . OMPLEX OSSERMAN K¨AHLER MANIFOLDS 3
The non-zero curvatures are determined for i < j , up to the usual Z symmetries,by the relations: B nµ ( x i , Jx i , Jx i , x i ) = µ, B nµ ( x i , x j , x j , x i ) = µ,B nµ ( Jx i , x j , x j , Jx i ) = µ, B nµ ( Jx i , Jx j , Jx j , Jx i ) = µ,B nµ ( x i , x j , Jx j , Jx i ) = µ, B nµ ( x i , Jx j , Jx i , x j ) = µ,B nµ ( x i , Jx i , Jx j , x j ) = µ . Example 1.6.
Fix an orthonormal basis { x , Jx , ..., x m , Jx m } on the vector space V of dimension n . Let C nµ = ( V, h· , ·i , J, C nµ ) where the non-zero components of C nµ are determined by the following identities modulo the usual Z symmetries for i = j : C nµ ( x i , Jx i , Jx i , x i ) = µ, C nµ ( x i , x j , x j , x i ) = − µ ,C nµ ( Jx i , x j , x j , Jx i ) = µ , C nµ ( Jx i , Jx j , Jx j , Jx i ) = − µ ,C nµ ( x i , x j , Jx j , Jx i ) = − µ , C nµ ( x i , Jx j , Jx i , x j ) = µ ,C nµ ( x i , Jx i , Jx j , x j ) = 0 . We will verify in Section 2 that B nµ and C nµ are complex Osserman K¨ahler models;as they are flat if µ = 0, we shall usually restrict to the case µ = 0. The realmodel A nµ has constant sectional curvature µ . The complex model B nµ has constantholomorphic sectional curvature µ and satisfies the K¨ahler identity. One has thefollowing converse – see, for example, [14, 19]: Lemma 1.7. (1) If R = ( V, h· , ·i , A ) is a real model of dimension n with constant sectionalcurvature µ , then R is isomorphic to A nµ . (2) If H is a K¨ahler model of dimension n with constant holomorphic sectionalcurvature µ , then H is isomorphic to B nµ . We will establish the following classification result in Section 4:
Theorem 1.8.
Let n ≥ . If H is a complex Osserman K¨ahler model of dimension n with 2 eigenvalues which does not have constant holomorphic sectional curvature,then H is isomorphic to C nµ for some µ . Geometric classification in dimension . We shall show in Section 5 that C µ is not geometrically realizable for µ = 0; the proof uses the self-dual and the anti-self-dual Weyl tensors and does not generalize to the higher dimensional context.Combining this result with Theorem 1.8 then yields the main result of this paperwhich motivated our investigations in the first instance: Theorem 1.9.
Let H be a -dimensional almost-Hermitian manifold which satisfiesthe K¨ahler identity. Then H is complex Osserman if and only if H has constantholomorphic sectional curvature. Remark 1.10.
In fact, we shall prove a bit more. In Theorem 1.9, it is onlynecessary to assume that H is pointwise complex Osserman, i.e. the eigenvaluestructure is a priori permitted to vary with the point in question. The scalarcurvatures are given by τ B µ = 6 µ and τ C µ = 4 µ . Since by Lemma 3.2 H is Einstein,the scalar curvature (and hence µ ) is constant so H is in fact globally complexOsserman in this setting. We also note that H need not be K¨ahler to satisfy theK¨ahler identity; there are flat 4-dimensional almost-Hermitian manifolds which arenot integrable. 2. The models A nµ , B nµ , and C nµ In this section, we establish the basic properties of these models.
M. BROZOS-V´AZQUEZ AND P. GILKEY
Definition 2.1.
Let { y , y , y , y } belong to the set { x , ..., x m , Jx , ..., Jx m } . Wesay that an index i with 1 ≤ i ≤ m is an impacted index if x i = y a or Jx i = y a forsome a . It is then immediate that B nµ ( y , y , y , y ) = 0 and C nµ ( y , y , y , y ) = 0 ifthere are more than 2 impacted indices or if any impacted index appears with oddmultiplicity. Finally, B nµ and C nµ vanish if J appears an odd number of times.2.1. The model B nµ .Lemma 2.2. The model B nµ has constant holomorphic sectional curvature µ andis complex Osserman K¨ahler with 2 eigenvalues ( µ, µ ) of multiplicities ( n − , ,respectively.Proof. As noted above A and A J are algebraic curvature tensors. It is an easycomputation that B nµ has constant holomorphic sectional curvature µ . We showthat B nµ is K¨ahler by verifying that µ { A ( x, y, Jz, Jw ) + A J ( x, y, Jz, Jw ) } = µ {h x, Jw ih y, Jz i − h x, Jz ih y, Jw i + h Jx, Jw ih Jy, Jz i − h
Jx, Jz ih Jy, Jw i − h Jx, y ih JJz, Jw i} = µ {h Jx, w ih Jy, z i − h
Jx, z ih Jy, w i + h x, w ih y, z i − h x, z ih y, w i − h Jx, y ih Jz, w i} = µ { A ( x, y, z, w ) + A J ( x, y, z, w ) } .The Jacobi operators are given by: J A ( x ) y = (cid:26) y = xy if y ⊥ x (cid:27) , J A J ( x ) y = (cid:26) y if y = Jx y ⊥ Jx (cid:27) , (3) J B nµ ( π x ) y = µ (cid:26) y if y ∈ π x y if y ⊥ π x (cid:27) . This shows that J B nµ ( π x ) is complex Osserman with eigenvalues ( µ, µ ) of multi-plicities ( n − , (cid:3) The model C µ . There is an auxiliary complex structure L , which commuteswith J , that will be important in our investigations which is defined by: Lx = x , Lx = − x , LJx = Jx , LJx = − Jx . Let ρ be the Ricci operator and let τ be the scalar curvature. Lemma 2.3.
Adopt the notation established above. (1) A C µ = µ A + µ A J − µ A L . (2) C µ is a complex Osserman K¨ahler model with 2 eigenvalues ( µ, of multi-plicities (2 , , respectively. (3) ρ A = ρ A J = ρ A L = 3 Id , τ A = τ A J = τ A L = 12 , τ B µ = 6 µ , τ C µ = 4 µ .Proof. We have A ( x, y, z, w ) := h x, w ih y, z i − h x, z ih y, w i A J ( x, y, z, w ) := h Jx, w ih Jy, z i − h
Jx, z ih Jy, w i − h Jx, y ih Jz, w i ,A L ( x, y, z, w ) := h Lx, w ih Ly, z i − h
Lx, z ih Ly, w i − h Lx, y ih Lz, w i . OMPLEX OSSERMAN K¨AHLER MANIFOLDS 5
Let i = j . We may verify Assertion (1) by computing:Monomial A A J A L C µ ( x i , Jx i , Jx i , x i ) 1 3 0 µ ( x i , x j , x j , x i ) 1 0 3 − µ ( Jx i , x j , x j , Jx i ) 1 0 0 µ ( Jx i , Jx j , Jx j , Jx i ) 1 0 3 − µ ( x i , x j , Jx j , Jx i ) 0 1 2 − µ ( x i , Jx j , Jx i , x j ) 0 1 − µ ( x i , Jx i , Jx j , x j ) 0 2 1 0Since A , A J , and A L are algebraic curvature tensors, A C µ is an algebraic curva-ture tensor. The K¨ahler identity is immediate from the defining relations. We useEquation (3) to compute: µ J A ( π x ) + µ J A J ( π x ) = Id µ . Since L is a Hermitian almost complex structure commuting with J , Lπ x is acomplex 2-plane as well. We complete the proof of Assertion (2) by verifying: J L ( x ) y = (cid:26) y if y = Lx y ⊥ Lx (cid:27) , J L ( Jx ) y = (cid:26) y if y = LJx y ⊥ LJx (cid:27) , J L ( π x ) y = (cid:26) y if y ∈ Lπ x y ⊥ Lπ x (cid:27) , J C µ ( π x ) y = (cid:26) y ∈ Lπ x µy if y ⊥ Lπ x (cid:27) . Assertion (3) is immediate from the definitions. (cid:3)
The model C nµ . We begin by studying the group of symmetries of the model.
Lemma 2.4.
Let O ( m ) act diagonally on R m . If Θ ∈ O ( m ) , then Θ ∗ C nµ = C nµ .Proof. If Θ ∈ O ( m ), then Θ J = J Θ and Θ ∗ h· , ·i = h· , ·i . Let C = C nµ ; this tensor isinvariant under permutations of the coordinate indices. Since O ( m ) is generated bycoordinate permutations and by rotations in the first 2 indices, it suffices to provethe lemma for the elementsΘ θ ( x ) = cos θx + sin θx , Θ θ ( Jx ) = cos θJx + sin θJx , Θ θ ( x ) = − sin θx + cos θx , Θ θ ( Jx ) = − sin θJx + cos θJx , Θ θ ( x i ) = x i for i ≥ , Θ θ ( Jx i ) = Jx i for i ≥ . We compute representative terms:Θ ∗ θ A ( x , Jx , Jx , x ) = cos θA ( x , Jx , Jx , x ) + sin θA ( x , Jx , Jx , x )+ cos θ sin θ { A ( x , Jx , Jx , x ) + A ( x , Jx , Jx , x ) + A ( x , Jx , Jx , x )+ A ( x , Jx , Jx , x ) + A ( x , Jx , Jx , x ) + A ( x , Jx , Jx , x ) } = cos θµ + sin θµ + cos θ sin θ { µ + µ + µ + µ + 0 } = µ ,Θ ∗ θ A ( Jx , x , x , Jx ) = cos θA ( Jx , x , x , Jx ) + sin θA ( Jx , x , x , Jx )+ sin θ cos θ {− A ( Jx , x , x , Jx ) − A ( Jx , x , x , Jx )+ A ( Jx , x , x , Jx ) + A ( Jx , x , x , Jx ) − A ( Jx , x , x , Jx ) − A ( Jx , x , x , Jx ) } = (cos θ + sin θ ) µ + cos θ sin θ ( − µ − µ + µ − − µ } = µ ,Θ ∗ θ A ( Jx , x , x , Jx ) = cos θA ( Jx , x , x , Jx ) + sin θA ( Jx , x , x , Jx )= µ (cos θ + sin θ ) = µ ,Θ ∗ θ A ( x , x , x , x ) = cos θA ( x , x , x , x ) + sin θA ( x , x , x , x ) − cos θ sin θA ( x , x , x , x ) − cos θ sin θA ( x , x , x , x )= − µ { cos θ + 2 cos θ sin θ + sin θ } = − µ ,Θ ∗ θ A ( x , x , x , x ) = cos θA ( x , x , x , x ) + sin θA ( x , x , x , x )= − µ { cos θ + sin θ } = − µ , M. BROZOS-V´AZQUEZ AND P. GILKEY
Since Θ J = J Θ and since A is K¨ahler, necessarilyΘ ∗ θ A ( x i , x j , Jx j , Jx i ) = A (Θ x i , Θ x j , Θ Jx j , Θ Jx i ) = A (Θ x i , Θ x j , J Θ x j , J Θ x i )= A (Θ x i , Θ x j , Θ x j , Θ x i ) = A ( x i , x j , x j , x i ) = A ( x i , x j , Jx j , Jx i ),Θ ∗ θ A ( Jx i , Jx j , Jx j , Jx i ) = A (Θ Jx i , Θ Jx j , Θ Jx j , Θ Jx i )= A ( J Θ x i , J Θ x j , J Θ x j , J Θ x i ) = A (Θ x i , Θ x j , Θ x j , Θ x i )= A ( x i , x j , x j , x i ) = A ( Jx i , Jx j , Jx j , Jx i ). (cid:3) We remark that cos θ Id + sin θJ also is an isometry of C n µ . We now show: Lemma 2.5. C nµ is a complex Osserman K¨ahler model with 2 eigenvalues ( µ, ofmultiplicities (2 , n − , respectively. Let L := Span { x , ..., x m } and let ξ ∈ S ( V ) . (1) J ( π ξ ) ξ = µξ if and only if ξ ∈ π x for some x ∈ S ( L ) . (2) J ( π ξ ) ξ = 0 if and only if ξ = ( x + Jy ) / √ for x, y ∈ S ( L ) with x ⊥ y .Proof. It is immediate that C nµ satisfies the Z symmetries of Equation (1) andthat C nµ satisfies the Bianchi identity of Equation (2). Furthermore, one sees byinspection that C nµ is K¨ahler. Thus C nµ is a K¨ahler model.We wish to study the eigenvalue structure of J ( π ξ ) for ξ ∈ S ( V ). We expand ξ = P i a i x i + P j b j Jx i . By replacing ξ by Jξ if necessary, we may assume that P a i = 0. We use Lemma 2.4. By applying an appropriate element of O ( m ), wemay assume that a = 0 and that a i = 0 for i ≥
2. We then apply an appropriateelement of O ( m −
1) to assume b j = 0 for j ≥
3. Thus without loss of generality,we may assume that ξ = a x + b Jx + b Jx ∈ R = Span { x , x , Jx , Jx } where a = 0 and a + b + b = 1. Let η ∈ { x , ..., x m , Jx , ..., Jx m } . Since C nµ vanishes if there are more than 2 impacted indices, we have J ( π ξ ) η = {J ( a x + b Jx ) + J ( − b x + a Jx ) } η + {J ( a x + b Jx ) + J ( − b x + a Jx ) } η = ( a + b ) J ( π x ) η + ( a + b ) J ( π x ) η . One computes directly that J ( π x i ) x j = 0 for i = j . This shows J ( π ξ ) η = 0 for η ∈ ( R ) ⊥ . On the other hand, R is invariant under J ( π ξ ) and we have alreadydetermined the eigenvalue structure to be ( µ,
0) of multiplicities (2 , C nµ is a complex Osserman model with eigenvalues ( µ,
0) ofmultiplicities (2 , n − O ( m ) and by replacing ξ by Jξ . Thus, as above, we may assume ξ = a x + b Jx + b Jx for a = 0. Weuse the analysis used to prove Lemma 2.3. We have Jξ = − b x − b x + a Jx and Lξ = a x − b Jx + b Jx . We have the following two chains of equivalences: J ( π ξ ) ξ = µξ ⇔ ξ ⊥ Lπ ξ ⇔ Jξ ⊥ Lξ ⇔ a b = 0 ⇔ b = 0 ⇔ ξ ∈ π x ; J ( π ξ ) ξ = 0 ⇔ ξ ∈ Lπ ξ ⇔ Jξ = ± Lξ ⇔ b = 0 and a = ± b ⇔ ξ = √ ( x ± Jx ) . Assertions (1) and (2) now follow. (cid:3) Complex Osserman K¨ahler models
In this section, we present some general results we shall need subsequently.
OMPLEX OSSERMAN K¨AHLER MANIFOLDS 7
Basic results.
We refer to [4] for the proof of the following Lemma:
Lemma 3.1.
Let H be a K¨ahler model. Then: (1) J ∗ A = A . (2) J ( π ) is complex for all π ∈ CP ( H ) . (3) J ( π ) y = A ( x, Jx ) Jy . (4) If J ( π ) = 0 for all π ∈ CP ( H ) , then A = 0 . If H is complex Osserman K¨ahler and if λ is an eigenvalue of H , let E λ ( π ) be thecorresponding eigenspace of J ( π ) for π ∈ CP ( H ). Lemma 3.2.
Let H be a complex Osserman K¨ahler model which has 2 eigenvalues λ < λ . Then: (1) H is Einstein. (2) If x, y ∈ S ( H ) , then y ∈ E λ i ( π x ) implies x ∈ E λ i ( π y ) .Proof. Let x ∈ S ( H ). By Lemma 3.1, J ∗ A = A . Thus A ( y, Jx, Jx, z ) = A ( Jy, x, x, Jz ) so J ( Jx ) = − J J ( x ) J .
Consequently the Ricci tensor satisfies ρ ( x, x ) = Tr {J ( x ) } = Tr {J ( Jx ) } = Tr {J ( π x ) } = P i λ i dim { E λ i ( π x ) } . As ρ ( x, x ) is constant on S ( H ), ρ ( · , · ) = c h· , ·i so H is Einstein. We suppose i = 2so λ is the maximal eigenvalue; the case i = 1 is similar. We have λ = max z ∈ S ( H ) hJ ( π x ) z, z i , and if z is a unit vector which realizes the maximum, then z is an eigenvector.Hence Assertion (2) follows from the following sequence of equalities: λ = hJ ( π x ) y, y i = A ( y, x, x, y ) + A ( y, Jx, Jx, y )= A ( x, y, y, x ) + A ( x, Jy, Jy, x ) = hJ ( π y ) x, x i . (cid:3) Eigenvalue multiplicities for complex Osserman models.
Methods ofalgebraic topology can be used to control the eigenvalue structure of a complexOsserman model. In particular, no more than 3 distinct eigenvalues may occur. Werefer to [3] for the proof of the following result:
Theorem 3.3. If H is complex Osserman, then one of the following holds: (1) There is just 1 eigenvalue. (2)
There are eigenvalues with multiplicities ( n − , with n ≥ . (3) There are eigenvalues with multiplicities ( n − , with n ≥ . (4) There are eigenvalues with multiplicities ( n − , , with n ≥ . We now impose the K¨ahler identity and apply the relations of Lemma 3.1. Webegin with a simple observation.
Lemma 3.4. If H is a complex Osserman K¨ahler model with only one eigenvalue,then H is flat.Proof. Suppose that the complex Jacobi operator has only 1 eigenvalue µ . Then hJ ( π x ) x, x i = A ( x, Jx, Jx, x ) = s ( x )is constant on S ( H ) so H has constant holomorphic sectional curvature. Thus byLemma 1.7, H is isomorphic to B nµ for some µ . By Lemma 2.2, the eigenvalues of B nµ are { µ, µ } . Thus µ = µ so µ = 0 and H is flat. (cid:3) Remark 3.5.
It follows from Equation (3) that µ A + µ A J is a complex Ossermantensor with constant sectional curvature which has only one eigenvalue; this doesnot contradict Lemma 3.4 since this tensor is not K¨ahler. M. BROZOS-V´AZQUEZ AND P. GILKEY
Critical points of the sectional curvature.
We assume that H has 2 eigen-values henceforth; this is, of course, the case if n = 4 and H is not flat. Lemma 3.6.
Let H be a complex Osserman K¨ahler model which has 2 eigenvalues λ < λ . Let x ∈ S ( H ) . Then x is a critical point of the holomorphic sectionalcurvature function s if and only if x ∈ E λ i ( x ) for some i .Proof. Suppose y ∈ S ( E λ i ( x )). Let α := h y, x i . Then R ( x, Jx, Jy, x ) = R ( x, Jx, Jx, y ) = hJ ( π x ) y, x i = λ i α . Consider the variation v y ( ε ) := (1 + 2 εα + ε ) − / ( x + εy ) ∈ S ( H ). Expand: s ( v y ( ε )) = (1 + 2 εα ) − (cid:8) s ( x ) + εR ( x, Jx, Jx, y ) + εR ( x, Jx, Jy, x )+ εR ( x, Jy, Jx, x ) + εR ( y, Jx, Jx, x ) (cid:9) + O ( ε )= (1 − εα ) { s ( x ) + 4 εR ( x, Jx, Jx, y ) } + O ( ε )= s ( x ) + ε {− αs ( x ) + 4 αλ i } + O ( ε ) ,∂ ε s ( v y ( ε )) | ε =0 = 4 α ( λ i − s ( x )) . Suppose that x is a critical point of the sectional curvature function. Expand x = α y + α y for y i ∈ S ( E λ i ( x )). Then ∂ ε s ( v y i ( ε )) | ε =0 = 0. Since x is non-zero,at least one of the α i must be non-zero. Thus s ( x ) = λ i . Furthermore, if λ j is theother eigenvalue, then λ i = λ j so λ j − s ( x ) = 0 so α j = 0 and thus x ∈ E λ i ( x ).Conversely, suppose that x ∈ E λ i ( x ). If y ∈ E λ i ( x ), then λ i − s ( x ) = 0 and ∂ ε ( s ( v y ( ε )) | ε =0 = 0. If y ∈ E j ( x ) for i = j , then α j = 0 and so ∂ ε ( s ( v y ( ε )) | ε =0 = 0as well. Thus we conclude ∂ ε ( s ( v y ( ε )) | ε =0 = 0 for all such variations. Since thederivatives of all such variations form a spanning set for the tangent space T x S ( H ),we conclude x is a critical point of the sectional curvature function. (cid:3) Lemma 3.7.
Let H be a complex Osserman K¨ahler model which has 2 eigenvalues λ < λ and which does not have constant holomorphic sectional curvature. (1) For i = 1 , , there exist x i ∈ S ( H ) so that x i ∈ E λ i ( x i ) and s ( x i ) = λ i . (2) If dim E λ i ≥ , then λ i = 0 . (3) We do not have both dim E λ ≥ and dim E λ ≥ .Proof. Let s attain its minimum at x ∈ S ( H ) and its maximum at x ∈ S ( H ).Since H does not have constant holomorphic sectional curvature, s ( x ) < s ( x ). As x and x are critical points of s , Assertion (1) follows from Lemma 3.6; note that x ∈ E λ i ( x ) implies s ( x ) = λ i .Suppose that dim E λ i ≥
4. Choose x i so x i ∈ E λ i ( x i ). Since dim E λ i ( x i ) ≥ z i ∈ E λ i ( x i ) with z i ⊥ π x i . Let x i ( θ ) := cos θx i + sin θz i ∈ E λ i ( x i ). ByLemma 3.2 (2), x i ∈ E λ ( x i ( θ )). We use Lemma 3.1 (3) to see: λ i x i = A (cos θx i + sin θz i , cos θJx i + sin θJz i ) Jx i = cos θA ( x i , Jx i ) Jx i + 2 cos θ sin θA ( x i , Jz i ) Jx i + sin θA ( z i , Jz i ) Jx i = λ i x i + 2 cos θ sin θA ( x i , Jz i ) Jx i . Thus A ( x i , Jz i ) Jx i = A ( z i , Jx i ) Jx i = 0 and, similarly, A ( z i , x i ) x i = 0. Hence λ i z i = A ( x i , Jx i ) z i = A ( z i , x i ) x i + A ( z i , Jx i ) Jx i = 0 . This shows λ i = 0 and establishes Assertion (2); Assertion (3) follows from Assertion(2) since 0 = λ < λ = 0 is not possible. (cid:3) The proof of Theorem 1.3.
Let H be a complex Osserman K¨ahler modelwhich is not flat. By Lemma 3.7, the eigenvalue multiplicity ( n − ,
4) with n ≥ n ) is not possible. Theorem 3.3then shows the multiplicities to be ( n − ,
2) or ( n − , ,
2) with n = 4 k ≥ (cid:3) OMPLEX OSSERMAN K¨AHLER MANIFOLDS 9 The proof of Theorem 1.8
Through out this section, let H be a complex Osserman K¨ahler model with2 eigenvalues ( µ, λ ) of multiplicities (2 , n − n ≥
6, then λ = 0 by Lemma 3.7and hence µ = 0. On the other hand, if n = 4, we may assume without loss ofgenerality that the notation is chosen so that µ = 0 since both µ and λ can notvanish simultaneously. Thus we shall always assume that µ = 0 henceforth. Definition 4.1.
Complex lines { π , ..., π k } in CP ( H ) will be said to form a µ -configuration if π i ⊂ E µ ( π i ) and if π i ⊥ π j for i = j ; this then implies π j ⊂ E λ ( π i )for i = j . Lemma 4.2.
Given H as above, there exist complex lines { π , ..., π n } which forma µ -configuration.Proof. Suppose first n = 4. Since H does not have constant holomorphic sectionalcurvature, we may apply Lemma 3.7 to choose x ∈ S ( H ) so that x ∈ E µ ( π x ). Let π := π x and let π := π ⊥ . We have π = E λ ( π ) and hence, dually, π = E λ ( π )by Lemma 3.2. Thus π = E µ ( π ) and { π , π } form a µ -configuration.Suppose next that n >
4. We proceed by induction on n . By Lemma 3.7, λ = 0.Use Lemma 3.7 to choose π so π = E µ ( π ). Let H := ( π ⊥ , h· , ·i| π ⊥ , J | π ⊥ , A π ⊥ )be the restriction of the model H to the subspace π ⊥ . As the restriction of aK¨ahler model to a complex subspace is K¨ahler, we have H is a K¨ahler model. If y ∈ S ( π ⊥ ), then y ∈ E ( π ) and hence dually π ⊂ E ( π y ). This shows J ( π y )preserves π and hence as J ( π y ) is self-adjoint, J ( π y ) preserves π ⊥ . Furthermore, J ( π y ) has eigenvalues ( µ,
0) of multiplicities (2 , n −
4) on π ⊥ . Thus H is a complexOsserman K¨ahler model of dimension n − µ,
0) of multiplicities(2 , n − λ = 0 and µ = 0, H does not have constant holomorphic sectionalcurvature. Consequently we may proceed inductively to construct a µ -configuration { π , ..., π n } for H ; { π , ..., π n } is then a µ -configuration for H . (cid:3) If n = 4, then Theorem 1.8 follows from the following result: Lemma 4.3.
Let n = 4 and let { π , π } be a µ -configuration for H . Fix x ∈ S ( π ) .There exists x ∈ S ( π ) and constants α, β with α + β = λ so that: (1) J ( x ) x = αx , J ( Jx ) x = βx , J ( x ) Jx = βx , J ( Jx ) Jx = αx , J ( x ) x = αx , J ( Jx ) x = βx , J ( x ) Jx = βx , J ( Jx ) Jx = αx . (2) The non-zero curvatures of A are given up to the usual Z symmetries by: A ( x , Jx , Jx , x ) = A ( x , Jx , Jx , x ) = µ , A ( x , Jx , Jx , x ) = λ , A ( x , x , x , x ) = A ( Jx , Jx , Jx , Jx ) = A ( Jx , Jx , x , x ) = α , A ( x , Jx , Jx , x ) = A ( Jx , x , x , Jx ) = A ( x , Jx , Jx , x ) = β . (3) We can choose x ∈ S ( π ) so that β = µ and so that α = − µ ; the model H is then equal to C µ .Proof. Since J ( x ) x = 0 and since J ( x ) Jx = J ( π ) Jx = µJx , J ( x ) pre-serves π and hence as J ( x ) is self-adjoint, J ( x ) preserves π as well. Thus wemay choose an orthonormal basis { x , Jx } for π so that J ( x ) x = αx and J ( x ) Jx = βJx . With this choice of basis, we have that A ( x , x , x , x ) = hJ ( x ) x , x i = α , A ( Jx , x , x , Jx ) = hJ ( x ) Jx , Jx i = β , A ( Jx , x , x , x ) = hJ ( x ) Jx , x i = 0. If n = 4, it is a priori possible to have both a µ -configuration and a λ -configuration. Similarly J ( Jx ) preserves π , J ( x ) preserves π , and J ( Jx ) preserves π . Since { x i , Jx i } is an orthonormal basis for π i , the remaining identities of Assertion (1)follow from the calculations: hJ ( Jx ) x , x i = A ( x , Jx , Jx , x ) = A ( Jx , x , x , Jx ) = β , hJ ( Jx ) x , Jx i = A ( x , Jx , Jx , Jx ) = − A ( Jx , x , x , x ) = 0, hJ ( Jx ) Jx , Jx i = A ( Jx , Jx , Jx , Jx ) = A ( x , x , x , x ) = α , hJ ( x ) x , x i = A ( x , x , x , x ) = α , hJ ( x ) x , Jx i = A ( x , x , x , Jx ) = − A ( x , x , x , Jx ) = 0, hJ ( x ) Jx , Jx i = A ( x , Jx , Jx , x ) = A ( Jx , x , x , Jx ) = β , hJ ( Jx ) x , x i = A ( Jx , x , x , Jx ) = β , hJ ( Jx ) x , Jx i = A ( x , Jx , Jx , Jx ) = A ( x , x , x , Jx ) = 0, hJ ( Jx ) Jx , Jx i = A ( Jx , Jx , Jx , Jx ) = A ( x , x , x , x ) = α .The curvatures listed in Assertion (2) follow from the facts that H is K¨ahler,that π i ∈ E µ ( π i ), that π i ∈ E λ ( π j ) for i = j , and from Assertion (1). We considerpossible missing terms. If there is only one ‘ x i ’ term, the curvature must have aform like R ( x , Jx , x , x ). Such a term vanishes since J ( x ) x ⊥ Jx . Thus wemay assume there are two x terms and two x terms. If two J terms touch in eitherthe first or the last two arguments, we can use the K¨ahler identity to remove a J . Ifthere is one J term it must look like A ( x , x , x , Jx ) modulo the K¨ahler identity;this vanishes by Assertion (1). The terms with no J in them are A ( x , x , x , x )which is known. Thus there are two J terms A ( J ∗ , ∗ , J ∗ , ∗ ) which, modulo theK¨ahler identity, are of the form A ( Jx , x , Jx , x ) or A ( Jx , x , Jx , x ) whichhave already been discussed. This proves Assertion (2).Consider z = ( x + x ) / √
2. We then have J ( π z ) = A ( z, Jz ) J = A ( x + x , Jx + Jx ) J = A ( x , Jx ) J + A ( x , Jx ) J + A ( x , Jx ) J = {J ( π ) + J ( π ) } + A ( x , Jx ) J .
It is clear that J ( π ) + J ( π ) = ( λ + µ ) Id. We compute: A ( x , Jx ) Jx = A ( x , Jx ) Jx = J ( Jx ) x = βx ,A ( x , Jx ) Jx = J ( Jx ) x = βx . It now follows that J ( π z ) x = ( λ + µ ) x + βx , J ( π ) x = ( λ + µ ) x + βx . Since J ( π z ) J = J J ( π z ), we obtain as well J ( π z ) Jx = ( λ + µ ) Jx + βJx , J ( π ) Jx = ( λ + µ ) Jx + βJx . Thus the eigenvalues of J ( π z ) are ( λ + µ ) ± β . We interchange the roles of α and β by considering instead ˜ z := ( x + Jx ) / √
2. A similar argument shows theeigenvalues of J ( π ˜ z ) are ( λ + µ ) ± α . It now follows that α = ± β . Suppose first that α = β . Then λ = α + β = 2 α so α = β = λ . Thus theeigenvalues of J ( π z ) are ( λ + µ ) − λ = µ and ( λ + µ ) + λ = λ + µ . Since µ = 0, λ + µ = λ and thus we must have λ + µ = µ so λ = µ = 0. We substitutethe values α = β = λ = µ into Assertion (2) to determine the curvature tensor andsee it agrees with the expression given in Example 1.5. This is not possible as weassumed H does not have constant holomorphic sectional curvature.Next suppose that α = − β . We then have λ = 0. By interchanging the rolesof x and Jx if need be, we may assume that µ + β = µ and µ − β = 0. We OMPLEX OSSERMAN K¨AHLER MANIFOLDS 11 substitute the values − α = β = µ into Assertion (2) to determine the curvaturetensor and see it agrees with the expression given in Example 1.6. (cid:3) We suppose n ≥ λ = 0. Lemma 4.4.
Let ≤ k < n . Let { π , ..., π k } be a µ -configuration for H . Let D := π + ... + π k and let D = ( D, h· , ·i| D , J | D , A D ) . (1) D is a complex Osserman K¨ahler model with 2 eigenvalues ( µ, of multi-plicities (2 , k − , respectively, which does not have constant holomorphicsectional curvature. (2) If u, v, w ∈ D , then A ( u, v ) w ∈ D and J ( u ) v ∈ D .Proof. We apply the argument used in the proof of Lemma 4.2 to establish Assertion(1). Let y ∈ D ⊥ . Then y ∈ E ( π i ) so dually π i ⊂ E ( π y ). Thus D ⊂ E ( π y ).Let z ∈ S ( D ). Then z ∈ E ( π y ) and hence y ∈ E ( π z ). Thus D ⊥ ⊂ E ( π z ).Consequently J ( π z ) D ⊥ ⊂ D ⊥ and as J ( π z ) is self adjoint, J ( π z ) D ⊂ D . Itnow follows that D is a complex Osserman K¨ahler model with eigenvalues ( µ, , k − D does not have constant holomorphicsectional curvature since µ = 0 and λ = 0. This proves Assertion (1).Let { x i , Jx i } be an orthonormal basis for π i ; { x , Jx , ..., x k , Jx k } is an orthonor-mal basis for D . Let i = j . Let z := cos( θ ) x i + sin( θ ) x j ∈ S ( D ). We noted abovethat J ( π z ) D ⊂ D . We expand: J ( π z ) = A (cos( θ ) x i + sin( θ ) x j , cos( θ ) Jx i + sin( θ ) Jx j ) J = cos ( θ ) J ( π i ) + sin θ J ( π j ) + 2 cos θ sin θA ( x i , Jx j ) J .
This shows that A ( x i , Jx j ) preserves D ; similarly A ( x i , x j ) and A ( Jx i , Jx j ) pre-serve D . Since A ( x i , Jx i ) also preserves D , we conclude A ( u, v ) w ∈ D for any u, v, w ∈ D . Taking v = w shows J ( u ) preserves D . (cid:3) The proof of Theorem 1.8
Let n ≥
6. Let { π , ..., π n } be a µ -configuration.Choose x ∈ S ( π ). We apply Lemma 4.3 to choose x i ∈ S ( π i ) for i ≥ x and x j for 2 ≤ j ≤ n aregiven by Example 1.6. Lemma 4.4 shows that A ( · , · , · , · ) vanishes if there are 3impacted indices. Thus to show H is isomorphic to C nµ , we need only examine thecurvature tensor on π j + π k for 2 ≤ j < k ≤ n . To simplify the notation, we set j = 2 and k = 3. Let D be the K¨ahler model determined by the µ -configuration { π , π , π } ; let D ij be the K¨ahler model determined by the µ -configuration { π i , π j } where i < j ; D and D ij are complex Osserman K¨ahler models with eigenvalues { µ, } with multiplicities (2 ,
4) and (2 , π := π √ ( x + x + x ) , π ij := π √ ( x i + x j ) for i < j . We may expand J ( π i ) = A ( x i , Jx i ) J, J ( π ij ) = A ( x i , Jx i ) J + A ( x j , Jx j ) J + 2 A ( x i , Jx j ) J, (4) 3 J ( π ) = P ≤ i ≤ A ( x i , Jx i ) J + 2 P ≤ i Since J ( π ) + J ( π ) + J ( π ) = µ Id on D , we may express(5) 3 J ( π ) = 2 J ( π ) + 2 J ( π ) + 2 J ( π ) − µ Id . By replacing A by − A if necessary, we may assume without loss of generality that µ > 0. As µ is the largest eigenvalue, if σ is any complex line, hJ ( σ ) w, w i ≤ µ | w | and equality holds if and only if J ( σ ) w = µw . Choose w ∈ S ( D ) so J ( π ) w = µw . By replacing w by cos θw + sin θJw if necessary, we may suppose h w, Jx i = 0 so w = a x + a x + a x + b Jx + b Jx . Set w := a x + a x + b Jx , w := a x + a x + b Jx ,w := a x + a x + b Jx + b Jx . We then have | w | + | w | + | w | = 2. Since the curvature vanishes if there are3 impacted indices, J ( π ij ) w = J ( π ij ) w ij . We use Equation (5) to estimate:4 µ = h (3 J ( π ) + µ Id) w, w i = 2 P i 3. Since b = 0, this implies a = a = a and b = b = 0.Consequently w = ± ( x + x + x ) / √ J ( π )( x + x ) = µ ( x + x )or, equivalently by Equation (4), that µ ( x + x ) = 2 A ( x , Jx ) J ( x + x ) . We take the inner product with x to conclude A ( x , Jx , Jx , x ) = µ , i.e. J ( x ) Jx = µ Jx . By Lemma 4.3, the eigenvalues of J ( x ) on π are ± µ . Con-sequently J ( x ) x = − µ x . Lemma 4.3 now shows that the curvatures of D aregiven by Example 1.6. This completes the proof. (cid:3) The proof of Theorem 1.9 Definition 5.1. We say that ( M, g ) is a regular curvature homogeneous manifold ifgiven any point P ∈ M , there exists an open neighborhood O of P and an orthonor-mal frame field { e , ..., e n } so that the curvature tensor R ( e i , e j , e k , e l )( x ) = C ijkl is independent of the point x ∈ O . Lemma 5.2. Let H = ( M, g, J ) be a complex Osserman manifold whose curva-ture is modeled on C µ at every point of M . Then ( M, g ) is a regular curvaturehomogeneous Einstein manifold.Proof. Fix P ∈ M and let O be a contractible open neighborhood of P . Let R L := 3 { µ A g + µ A J − R } define a smooth algebraic curvature tensor on T ( O ).The analysis of Lemma 2.3 shows that R L is Osserman with eigenvalues { , µ } ofmultiplicities (3 , σ ( s ) be orthogonal projection on E µ ( J R L ( s ))for s ∈ S ( T M ). The Cauchy integral formula σ ( s ) = 12 πi Z | z − µ | = µ {J R L ( s ) − z Id } − dz shows that σ varies smoothly. Consequently, Σ := Range( σ ) is a smooth line bundleover S ( T O ) = O × S . Consequently Σ is trivial and admits a smooth unit section L . Since L x = ± Lx , this analysis shows that the auxiliary almost complex structure L of Lemma 2.3 can be chosen to vary smoothly with the point of the manifold.We have ( JL ) = 1 and ( JL ) ∗ = LJ = JL is self-adjoint. Let E ± ( JL ) be theassociated ± JL . These comprise orthogonal J -invariant 2-planes.Let s be a smooth unit tangent vector such that Js = ± Ls . Form: s ± := | s ± JLs | − ( s ± JLs ) ∈ S ( E ± ( JL )) and x = ( s + + s − ) / √ . We then have g ( Jx , Lx ) = − g ( JLx , x ) = − g ( s + − s − , s + + s − ) = 0 . OMPLEX OSSERMAN K¨AHLER MANIFOLDS 13 Let π := Span { x , Jx } . Then π = E µ ( π ). We use the Cauchy integral formulato choose a smooth unit section to the − µ eigenspace of J ( x ) on π ⊥ . The proofthat the curvature tensor has the requisite form relative to this local frame nowfollows using the argument to establish Lemma 4.3. (cid:3) Let ( M, g ) be an Einstein regular curvature homogeneous manifold of dimension n = 4. Let R Λ be the induced action of the curvature tensor on Λ . Choose a localorientation for M and decompose Λ = Λ ⊕ Λ − . Let W be the Weyl conformalcurvature tensor. Since ( M, g ) is Einstein, R = W + cA and hence R Λ = W Λ + c Idfor some suitably chosen constant c . Since W Λ : Λ ± → Λ ± , R = R + ⊕ R − where R ± ∈ Λ ± ⊗ Λ ± . Let { λ ± , λ ± , λ ± } be the eigenvalues of R Λ ± ; these are constant as( M, g ) is curvature homogeneous. We let ω ± i be an orthonormal frame for { Λ } sothat | ω ± i | = 2 and R Λ ± ω ± i = λ ± i ω ± i where ω ± i = P ab ω ± i,ab e a ∧ e b . Since ( M, g ) is curvature homogeneous, the coefficients ω ij,ab are constant. Wespecialize an argument of Derdzi´nski [7] to show: Lemma 5.3. Let ( M, g ) be an Einstein regular curvature homogeneous manifoldof dimension n = 4 . If λ +1 = λ +2 = λ +3 and if the matrix ω +1 ,ab is invertible, then ∇ ω +1 = 0 .Proof. As ( M, g ) is Einstein, the Ricci tensor ρ is parallel. The second Bianchiidentity yields:( δ R )( y, z, w ) := R ( e i , y, z, w ; e i ) = − R ( e i , y, w, e i ; z ) − R ( e i , y, e i , z ; w )= − ρ ( y, w ; z ) + ρ ( y, z ; w ) = 0 . As the decomposition Λ = Λ ⊕ Λ − is parallel, we have δ R ± = 0 individually.Furthermore, there exist smooth 1-forms φ + ij = − φ + ji so that ∇ ω + i = P j φ + ij ⊗ ω + j where φ + ij = P a φ ij,a e a . We use the eigenvalue decomposition to express: R + = { λ +1 ω +1 ⊗ ω +1 + λ +2 ω +2 ⊗ ω +2 + λ +3 ω ⊗ ω +3 } . Since the eigenvalues λ + i are constant and since φ + ij = − φ + ji , we have: ∇ R + = P ij λ + i φ + ij ⊗ { ω + j ⊗ ω + i + ω + i ⊗ ω + j } = P ij { λ + i − λ + j } φ + ij ⊗ ω + j ⊗ ω + i ,δ R + = P ijab { λ + i − λ + j } φ + ij,a ω + j,ab e b ⊗ ω + i . We set i = 2 and i = 3 in this identity. Since λ +2 = λ +3 , we set j = 1 to show:0 = { λ +2 − λ +1 } P ab φ +21 ,a ω +1 ,ab e b , { λ +3 − λ +1 } P ab φ +31 ,a ω +1 ,ab e b . As ω +1 ,ab is invertible, φ +21 = 0 and φ +31 = 0 so ∇ ω +1 = 0. (cid:3) We may combine Lemma 5.3 and Lemma 5.2 to show: Lemma 5.4. Let H = ( M, g, J ) be a connected -dimensional complex Ossermanmanifold whose curvature is modeled on C µ at every point. Then ∇ R = 0 .Proof. Fix P ∈ M . We may use Lemma 5.2 to choose a local orthonormal frame { x , Jx , ... } realizing the curvature relations of Example 1.6. Let L be defined asabove. Define: φ ± := x ∧ Jx ± x ∧ Jx , φ ± := x ∧ x ∓ Jx ∧ Jx ,φ ± := x ∧ Jx ± Jx ∧ x . This forms an orthogonal frame for Λ ± where || φ ± i || = 2. We use the structureequations of Example 1.6 to compute: Rφ ± = { R ( x , Jx , x , Jx ) ± R ( x , Jx , x , Jx ) + R ( x , Jx , x , Jx ) } φ ± = {− µ ± − µ } φ ± , Rφ ± = { R ( x , x , x , x ) ∓ R ( x , x , Jx , Jx ) + R ( Jx , Jx , Jx , Jx } φ ± = { µ ∓ µ + µ } φ ± , Rφ ± = { R ( x , Jx , x , Jx ) ± R ( x , Jx , Jx , x ) + R ( Jx , x , Jx , x ) } φ ± = {− µ ± µ − µ } φ ± .This yields Rφ +1 = − µφ +1 , Rφ +2 = 0 φ +2 , Rφ +3 = 0 φ +3 ,Rφ − = − µφ − , Rφ − = µφ − , Rφ − = − µφ − . Thus these are eigenvectors of R . Since φ +1 is the K¨ahler form of J , φ +1 ,ab is aninvertible matrix. As λ +1 = λ +2 = λ +3 , Lemma 5.3 yields ∇ φ +1 = 0 and, equivalently, ∇ J = 0. Since λ − = λ − = λ − , and since the K¨ahler form of L is φ +2 , a similarargument shows ∇ L = 0. We have ∇ A = 0. Since ∇ J = 0 and ∇ L = 0, we have ∇ A J = 0 and ∇ A L = 0 as well. It now follows that ∇ R = 0 at P . Since P wasarbitrary, the Lemma follows. (cid:3) The proof of Theorem 1.9 Suppose H = ( M, g, J ) is a connected complexOsserman manifold which does not have constant holomorphic sectional curvatureat at least one point Q of the manifold; we argue for a contradiction. We useTheorem 1.8 to see that at any point P of M , the curvature is either modeled on B µ or on C µ . If the curvature at P is modeled on B µ , then µ = 2 λ = 0 while ifthe curvature is modeled on C µ , then λ = 0. If λ = 0 at some point, then λ = 0on an open set O and hence µ = 2 λ on O . Since ( M, g ) is Einstein, the scalarcurvature is constant. Since the scalar curvature in this setting is 6 µ , µ and hence λ are constant on O so λ = 0 on the closure of O . This implies O = M andcontradicts the assumption that H does not have constant holomorphic sectionalcurvature at some point. Thus at every point, the curvature is modeled on C µ ; since τ = 4 µ we have µ is constant. We may therefore use Lemma 5.4 to see ∇ R = 0.Let R ( u , u , u , u ; u , u ) be the components of ∇ R . The curvature tensor of alocally symmetric space is very restrictive. In particular, we have:0 = R ( u , u , u , u ; u , u ) − R ( u , u , u , u ; u , u )= R ( R ( u , u ) u , u , u , u ) + R ( u , R ( u , u ) u , u , u )+ R ( u , u , R ( u , u ) u , u ) + R ( u , u , u , R ( u , u ) u ) . We apply this identity with u = x , u = x , u = x , u = Jx , u = x , u = Jx to compute:0 = R ( R ( Jx , x ) x , x , x , Jx ) + R ( x , R ( Jx , x ) x , x , Jx )+ R ( x , x , R ( Jx , x ) x , Jx ) + R ( x , x , x , R ( Jx , x ) Jx ) . We substitute the relations of Example 1.6 to conclude:0 = µR ( Jx , x , x , Jx ) + 0 + 0 − µR ( x , x , x , x )= µ + µ . This shows that µ = 0. Hence ( M, g ) is flat, contrary to our assumption. (cid:3) OMPLEX OSSERMAN K¨AHLER MANIFOLDS 15 Acknowledgements The authors would like to thank Prof. E. Garc´ıa-R´ıo for many useful conver-sations. Research of M. Brozos-V´azquez supported by projects MTM2009-07756and INCITE09 207 151 PR (Spain). Research of P. Gilkey partially supported byproject MTM2009-07756 (Spain). References [1] N. Blaˇzi´c, and P. Gilkey; Conformally Osserman manifolds and conformally complex spaceforms, Int. J. Geom. Methods Mod. Phys. (2004), 97–106.[2] N. Blaˇzi´c, and P. Gilkey; Conformally Osserman manifolds and self-duality in Riemanniangeometry, Differential Geom. Appl. , 15–18, Matfyzpress, Prague, 2005.[3] M. Brozos-V´azquez, and P. Gilkey; Complex Osserman algebraic curvature tensors andClifford families, Houston J. Math. (2008), 677–702.[4] M. 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