December 7, 2001

A class of Osserman spaces

Dmitri V. Alekseevsky^{1}, Novica Blažić^{2}, Vicente Cortés^{3},

and Srdjan Vukmirović^{4}

1 ����� ������ ���� ���� �������� � � ��� ������ ������

Department of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK D.V.Alekseevsky@maths.hull.ac.uk

3 Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany vicente@math.uni-bonn.de

2,4 Departement of Mathematics, University of Belgrade Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia blazicn@matf.bg.ac.yu, vsrdjan@matf.bg.ac.yu

Abstract

We prove that a symmetric space is Osserman if its complexification is a complex hyper-Kähler symmetric space. This includes all pseudo-hyper-Kähler as well as para-hyper-Kähler symmetric spaces. We extend the classification of pseudo-hyper-Kähler symmetric spaces obtained by the first and the third author to the class of para-hyper-Kähler symmetric spaces.

Keywords: symmetric spaces, hyper-Kähler manifolds, para-hyper-Kähler manifolds, Osserman spaces MSC 2000: 53C35, 53C26, 53C25, 53C50

This work was supported by the priority programme “String Theory”of the Deutsche Forschungsgemeinschaft.

1 Introduction

A (pseudo-) Riemannian manifold is called an Osserman space if the characteristic polynomial of the Jacobi operator R_{X} = R(·, X)X is the same for all unit vectors X. In this paper we present a new class of Ossermann spaces with many interesting properties, see Theorem 4. Our examples are pseudo-Riemannian symmetric spaces M = G/H, where G is a 3-step nilpotent group and H = Hol(M) is Abelian. Moreover, they carry an invariant hyper-Kähler or para-hyper-Kähler structure. Simply connected symmetric hyper-Kähler manifolds were classified in [AC]. Here we give also the classification of simply connected symmetric para-hyper-Kähler manifolds, see Theorem 3. We believe that this classification can be useful for physical applications, since there is evidence that para-hyper-Kähler manifolds are precisely the allowed targets for the scalars of rigid N=2 supersymmetric field theories with hypermultiplets on four-dimensional space-times with Euclidean signature.

2 Basic facts about symmetric spaces

In this section we recall some general facts about symmetric spaces.

2.1 Real and complex symmetric spaces

A pseudo-Riemannian symmetric space is a pseudo-Riemannian manifold (M, g) such that any point is an isolated fixed point of an isometric involution. Such a pseudo-Riemannian manifold admits a transitive Lie group of isometries L and can be identified with L/L_{o}, where L_{o} is the stabilizer of a point o. More precisely, any simply connected pseudo-Riemannian symmetric space M = G/K is associated with a symmetric decomposition

𝔤 = 𝔨 + 𝔪 , [𝔨, 𝔨] ⊂ 𝔨 , [𝔨, 𝔪] ⊂ 𝔪 , [𝔪, 𝔪] ⊂ 𝔨  (2.1)

of the Lie algebra 𝔤 = Lie G together with an Ad_{K}-invariant pseudo-Euclidean scalar product on 𝔪. We will assume that G acts almost effectively on M, i.e. 𝔨 does not contain any nontrivial ideal of 𝔤, that M and G are simply connected and that K is connected. Then, under the natural identification of the tangent space T_{o}M at the canonical base point o = eK with 𝔪, the holonomy group Hol ⊂ Ad_{K}|𝔪. We will denote by 𝔥 the holonomy Lie algebra. Since the covariant derivative of the curvature tensor R of a symmetric space vanishes, the holonomy algebra 𝔥 is spanned by the curvature operators R(x, y), x, y ∈ 𝔪, according to the Ambrose-Singer theorem. Recall that the curvature tensor R of a symmetric space M = G/K at o is ad𝔨-invariant and is given by

R(x, y) = −ad_{[x,y]}|𝔪.

Since the isotropy representation is faithful, we can identify the holonomy algebra 𝔥 = span{R(x, y)|x, y ∈ 𝔪} with the subalgebra 𝔥 = [𝔪, 𝔪] of 𝔤. It is easy to see that the group generated by the ideal 𝔥 + 𝔪 ⊂ 𝔤 = 𝔨 + 𝔪 acts transitively on M. So replacing the symmetric decomposition 𝔨+𝔪 by 𝔥+𝔪, if necessary, we can assume that 𝔨 = 𝔥 = [𝔪, 𝔪].

Note that the symmetric decomposition can be reconstructed from the curvature tensor R. More precisely, let 𝔨 ⊂ gl(𝔪) be a linear Lie algebra which preserves a pseudo-Euclidean scalar product on 𝔪 and let R be a 𝔨-invariant 𝔨-valued 2-form on 𝔪 which satisfies the Bianchi identity

R(x, y)z + R(y, z)x + R(z, x)y = 0 for all x, y, z ∈ 𝔪.

Then the formulas

[A, x] = Ax for A ∈ 𝔨, x ∈ 𝔪 and [x, y] = −R(x, y) for x, y ∈ 𝔪

define the structure of a Lie algebra with a symmetric decomposition 𝔤 = 𝔨 + 𝔪. The corresponding simply connected pseudo-Riemannian symmetric space M = G/K has the holonomy algebra 𝔥 = [𝔪, 𝔪].

A complex Riemannian manifold is a complex manifold M equipped with a complex metric g, i.e. a holomorphic section g ∈ Γ(S^{2}T^{∗}M) which defines a nondegenerate complex quadratic form. As in the real case any such manifold has a unique holomorphic torsionfree and metric connection (Levi-Civita connection).

A complex Riemannian symmetric space is a complex Riemannian manifold (M, g) such that any point is an isolated fixed point of an isometric holomorphic involution. Like in the real case one can prove that it admits a transitive complex Lie group of holomorphic isometries and that any simply connected complex Riemannian symmetric M is associated to a complex symmetric decomposition

𝔤 = 𝔨 + 𝔪 , [𝔨, 𝔨] ⊂ 𝔨 , [𝔨, 𝔪] ⊂ 𝔪 , [𝔪, 𝔪] = 𝔨  (2.2)

of a complex Lie algebra 𝔤 together with an ad𝔨-invariant complex scalar product on 𝔪. More precisely M = G/K, where G is the simply connected complex Lie group with the Lie algebra 𝔤 and K is the (closed) connected subgroup associated with 𝔨. The holonomy group of such manifold is H = Ad_{K}|𝔪.

Any pseudo-Riemannian symmetric space M = G/K associated with a symmetric decomposition 𝔤 = 𝔨 + 𝔪 has a canonical complexification M^{ℂ} = G^{ℂ}/K^{ℂ} defined by the complexification 𝔤^{ℂ} = 𝔨^{ℂ} + 𝔪^{ℂ} of the symmetric decomposition.

2.2 Hyper-Kähler and para-hyper-Kähler symmetric spaces

A (possibly indefinite) (ε)-hyper-Kähler manifold is a pseudo-Riemannian manifold (M^{4n}, g) together with a parallel (ε)-hypercomplex structure, i.e. three anticommuting parallel endomorphism fields (J_{1}, J_{2}, J_{3} = J_{1}J_{2}), which are skew symmetric with respect to g and which satisfy J_{1}^{2} = J_{2}^{2} = ε Id, ε ∈ {+, −}. Notice that the product J_{3} is a parallel skew symmetric complex structure. The notion of (ε)-hyper-Kähler manifold unifies the notion of hyper-Kähler manifold (ε = −) and of para-hyper-Kähler manifold (ε = +). Notice that the triple (J_{1}, J_{2}, J_{3}) defines the structure of a vector space over the quaternions (ε = −) or over the para-quaternions (ε = +) on the tangent space.

The above conditions mean that the holonomy group Hol ⊂ Sp(k, l), n = k + l, in the case ε = − and Hol ⊂ Sp(n, ℝ) := Sp(ℝ^{2n}) in the case ε = +. Two (ε)-hyper-K”ahler manifolds (M, g, J_{α}) (α = 1, 2, 3) and (M′, g′, J′_{α}) are called isomorphic if there is a diffeomorphism φ : M → M′ such that φ^{∗}J′_{α} = J_{α} and φ^{∗}g′ = g.

An (ε)-hyper-Kähler symmetric space is a pseudo-Riemannian symmetric space (M = G/K, g) together with an invariant (ε)-hypercomplex structure. Consider now a simply connected (ε)-hyper-Kähler symmetric space (M = G/K, g, J_{α}). Without restriction of generality we will assume that G acts almost effectively. M being (ε)-hyper-Kähler is equivalent to

Ad_{K}|𝔪 ⊂ \{ \begin{matrix}Sp(k, l) for & ε = − \\ Sp(n, ℝ) for & ε = + \end{matrix}

or, since K is connected, to

ad𝔨|𝔪 ⊂ \{ \begin{matrix}sp(k, l) for & ε = − \\ sp(n, ℝ) for & ε = + . \end{matrix}

This condition means that 𝔨 commutes with the Lie algebra

Q := span\{J_{1}, J_{2}, J_{3}\} = \{ \begin{matrix}sp(1) ⊂ so(𝔪) = so(4k, 4l) for & ε = − \\ sl(2, ℝ) ⊂ so(𝔪) = so(2n, 2n) for & ε = + \end{matrix}

spanned by the three anticommuting structures J_{1}, J_{2}, J_{3}.

A complex hyper-Kähler manifold is a complex Riemannian manifold (M^{4n}, g) of complex dimension 4n together with a compatible hypercomplex structure, i.e. three gorthogonal parallel complex linear endomorphisms (J_{1}, J_{2}, J_{3} = J_{1}J_{2}) with J_{α}^{2} = −1. This means that the holonomy group Hol ⊂ Sp(n, ℂ) = Z_{O(4n,ℂ)}(Sp(1, ℂ)). The linear group Sp(n, ℂ) is diagonally embedded into Sp(n, ℂ) × Sp(n, ℂ) ⊂ GL(4n, ℂ). Two complex hyper-Kähler manifolds (M, g, J_{α}) (α = 1, 2, 3) and (M′, g′, J′_{α}) are called isomorphic if there exists a holomorphic diffeomorphism φ : M → M′ such that φ^{∗}J′_{α} = J_{α} and φ^{∗}g′ = g. We notice that the complexification of an (ε)-hyper-Kähler symmetric space is a complex hyper-Kähler symmetric space.

3 Classification of real and complex hyper-Kähler

symmetric spaces

Now we recall the classification of real and complex hyper-Kähler symmetric spaces [AC]. Let (E, ω) be a complex symplectic vector space of dimension 2n and E = E^{+} ⊕ E^{−} a Lagrangian decomposition. Then any element S ∈ S^{4}E^{+} defines a simply connected complex symmetric space M_{S}^{c} of dimension 4n which is associated with the symmetric decomposition

𝔤 = 𝔥 + 𝔪 ,

where 𝔪 = H ⊗ E, H = ℂ^{2} = ℂ1 ⊕ ℂj = ℍ = span{1, i, j, k} with its standard complex symplectic structure ω^{H}, 𝔥 = span{S_{e,e′}| e, e′ ∈ E} ⊂ sp(E) with the natural action on 𝔪 ⊂ H ⊗ E and the Lie bracket 𝔪 ∧ 𝔪 → 𝔥 is given by

[h ⊗ e, h′ ⊗ e′] = ω^{H}(h, h′)S_{e,e′}.

Here S_{e,e′} ∈ S^{2}E = sp(E) denotes the contraction of S ∈ S^{4}E^{+} ⊂ S^{4}E with ee′ ∈ S^{2}E by means of ω.

Theorem 1 [AC] Let M_{S}^{c} be the complex symmetric space associated to S ∈ S^{4}E^{+}. Then it is a complex hyper-Kähler symmetric space with complex Riemannian metric g defined by ω^{H} ⊗ ω. and compatible hypercomplex structure (J_{1}, J_{2}, J_{3}) defined by (R_{i} ⊗ Id, R_{j} ⊗ Id, −R_{k} ⊗ Id), where R_{q} denotes the right multiplication by the quaternion q. Moreover, M_{S}^{c} has no flat factor if and only if S_{E,E}E := span{S_{e,e′}e′′|e, e′, e′′ ∈ E} = E^{+}. Conversely, any simply connected complex hyper-Kähler symmetric space is of the form M_{S}^{c}.

To obtain real hyper-Kähler symmetric spaces, let us asume that on (E, ω) we have a compatible quaternionic structure j : E → E, i.e. an antilinear map such that j^{2} = −Id and j^{∗}ω = ω̄, and a j-invariant Lagrangian decomposition E = E^{+} ⊕ E^{−}. Such a decomposition exists if and only if the Hermitian form γ = ω(·, j·) has real signature (4m, 4m), where m is related to the complex dimension 4n of M_{S}^{c} by n = 2m. On H = ℍ we fix the standard quaternionic structure j^{H} = L_{j}, the left-multiplication with j. It satisfies (j^{H})^{∗}ω^{H} = ω̄^{H}. We denote by τ the real structure on S^{2r}E induced by the quaternionic structure j on E: τ(e_{1}e_{2} … e_{2r}) := j(e_{1})j(e_{2}) … j(e_{2r}), e_{i} ∈ E. On 𝔪 = H ⊗ E we have the real structure ρ := j^{H} ⊗ j. We assume that S ∈ S^{4}E^{+} is real, i.e. τS = S. Then S defines a (real) symmetric space M_{S}, which is associated with the symmetric decomposition

𝔤_{0} = 𝔥_{0} + 𝔪_{0} ,

where

𝔥_{0} := 𝔥^{τ} := 𝔥 ∩ (S^{2}E)^{τ} , 𝔪_{0} := 𝔪^{ρ} = (H ⊗ E)^{ρ} .

Theorem 2 [AC] Let M_{S} be the symmetric space associated to S ∈ (S^{4}E^{+})^{τ}. Then it is a (real) hyper-Kähler symmetric space with pseudo-Riemannian metric defined by (ω^{H} ⊗ ω)|_{𝔪_{0}} and compatible hypercomplex structure (J_{1}, J_{2}, J_{3}) defined by J_{1} := (R_{i} ⊗ Id)|_{𝔪_{0}}, J_{2} := (R_{j} ⊗ Id)|_{𝔪_{0}} and J_{3} := J_{1}J_{2} = −(R_{k} ⊗ Id)|_{𝔪_{0}}. The metric has signature (4m, 4m), 4m = dim_{ℂ} E. The complexification of M_{S} coincides with the complex hyper-Kähler symmetric space M_{S}^{c} associated to S ∈ S^{4}E^{+}. Any simply connected hyper-Kähler symmetric space is the Riemannian product of a flat hyper-Kähler symmetric space (a hyper-Hermitian vector space) of arbitrary signature (4p, 4q) and a hyper-Kähler symmetric space of the form M_{S}.

4 Classification of para-hyper-Kähler symmetric spa-

ces

In this section we give the classification of simply connected para-hyper-Kähler symmetric spaces. The basic data for the construction of such spaces are the following: (E_{0}, ω_{0}) a real symplectic vector space of real dimension 2n, E_{0} = E_{0}^{+} ⊕E_{0}^{−} a Lagrangian decomposition and H_{0} = ℝ^{2} with its standard symplectic structure ω_{0}^{H} and para-hypercomplex structure

j_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , j_{2} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , and j_{3} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

Then any element S ∈ S^{4}E_{0}^{+} defines a simply connected symmetric space M_{S} of dimension 4n which is associated with the symmetric decomposition

𝔤_{0} = 𝔥_{0} + 𝔪_{0} ,

where 𝔪 = H_{0} ⊗ E_{0}, 𝔥 = span{S_{e,e′}| e, e′ ∈ E_{0}} ⊂ sp(E_{0}) with the natural action on 𝔪_{0} = H_{0} ⊗ E_{0} and the Lie bracket 𝔪_{0} ∧ 𝔪_{0} → 𝔥_{0} is given by

[h ⊗ e, h′ ⊗ e′] = ω_{0}^{H}(h, h′)S_{e,e′}.

Here S_{e,e′} ∈ S^{2}E_{0} = sp(E_{0}) denotes the contraction of S ∈ S^{4}E_{0}^{+} ⊂ S^{4}E_{0} with ee′ ∈ S^{2}E_{0} by means of ω_{0}.

Theorem 3 Let M_{S} be the symmetric space associated to S ∈ S^{4}E_{0}^{+}. Then it is a parahyper-Kähler symmetric space with pseudo-Riemannian metric defined by ω_{0}^{H} ⊗ ω_{0} and compatible para-hypercomplex structure (J_{1}, J_{2}, J_{3}) defined by J_{1} := j_{1} ⊗ Id, J_{2} := j_{2} ⊗ Id and J_{3} := J_{1}J_{2} = j_{3} ⊗ Id. The metric has signature (2n, 2n), 2n = dim E_{0}. The complexification of M_{S} coincides with the complex hyper-Kähler symmetric space M_{S^{c}}^{c} associated to the complex extension S^{c} ∈ S^{4}E^{+}, E^{+} := E_{0}^{+} ⊗ ℂ, of S ∈ S^{4}E_{0}^{+}. Any simply connected para-hyper-Kähler symmetric space is of the form M_{S}.

Proof. It is easy to check that M_{S} is a para-hyper-Kähler symmetric space of signature (2n, 2n) whose complexification coincides with the complex hyper-Kähler symmetric space M_{S^{c}}^{c}. We now prove that any simply connected para-hyper-Kähler symmetric space is of the form M_{S}. Let M be such a symmetric space associated with a symmetric decomposition

𝔤_{0} = 𝔥_{0} + 𝔪_{0} .

Since M is para-hyper-Kähler, we can identify 𝔪_{0} = H_{0} ⊗ E_{0} with the tensor product of two symplectic vector spaces H_{0} = ℝ^{2} and E_{0}. The pseudo-Riemannian metric on M corresponds to the product of the two symplectic structures and the holonomy Lie algebra 𝔥_{0} ⊂ Id⊗sp(E_{0}) = sp(E_{0}) acts only on the second factor. The complexification M^{c} of M is the complex hyper-Kähler symmetric space associated to the symmetric decomposition

𝔤 = 𝔥 + 𝔪 , 𝔥 := 𝔥_{0} ⊗ ℂ , 𝔪 := 𝔪_{0} ⊗ ℂ = H ⊗ E ,

where (E, ω) and (H, ω^{H}) are the complexifications of the real symplectic vector spaces (E_{0}, ω_{0}) and (H_{0}, ω_{0}^{H}) respectively. By Theorem 1, M^{c} = M_{S^{c}} is associated to a quartic polynomial S^{c} ∈ S^{4}E^{+} ⊂ S^{4}E = S^{4}E^{∗}, where E^{+} ⊂ E is a Lagrangian subspace. Now we prove that the multilinear form S^{c} ∈ S^{4}E^{∗} has real values on S^{4}E_{0} and hence defines an element of S^{4}E_{0}. First we remark that by the definition of the Lie bracket 𝔪 × 𝔪 → 𝔥

S^{2}E_{0} = sp(E_{0}) ⊃ 𝔥_{0} = [𝔪_{0}, 𝔪_{0}] = span{S_{e,e′}^{c}ω^{H}(h, h′)|e, e′ ∈ E_{0} , h, h′ ∈ H_{0}}
= span{S_{e,e′}^{c}|e, e′ ∈ E_{0}} =: S_{E_{0},E_{0}}^{c} .

This shows that S_{E_{0},E_{0}}^{c} ⊂ S^{2}E_{0} and hence that S^{c} is real valued. We denote by S the corresponding element of S^{4}E_{0}. Then we have [h ⊗ e, h′ ⊗ e′] = S_{e,e′}ω_{0}^{H}(h, h′) for all e, e′ ∈ E_{0} and h, h′ ∈ H_{0} and 𝔥_{0} = [𝔪_{0}, 𝔪_{0}] = S_{E_{0},E_{0}}. The (real) subspace E_{0} ∩ E^{+} ⊂ E_{0} is isotropic. Therefore there exists a Lagrangian subspace E_{0}^{+} ⊂ E_{0}, which contains E_{0} ∩ E^{+}. Thus

S_{E_{0},E_{0}}E_{0} = S_{E_{0},E_{0}}^{c}E_{0} ⊂ E_{0} ∩ E^{+} ⊂ E_{0}^{+} .

Now the next lemma implies that S ∈ S^{4}E_{0}^{+} and we can conclude that M = M_{S}.

Lemma 1 Let E be a (real or complex) symplectic vector space, F ⊂ E a subspace, S ∈ S^{4}E and S_{E,E}E ⊂ F. Then S ∈ S^{4}F.

Proof. Let C ⊂ E be a complement of F. We denote by C^{∧} and F^{∧} the annihilators of C and F in E^{∗} respectively. We can consider them again as subspaces of E via the identification E^{∗} ≅ E given by the symplectic form. The decomposition

S^{4}E = ⊕_{p+q=4}S^{p}C · S^{q}F .

gives rise to a decomposition S = ∑_{p+q=4} S^{p,q}, where S^{p,q} ∈ S^{p}C · S^{q}F. Now from

F ⊃ S_{E,E}E ⊃ S_{F^{∧},F^{∧}}F^{∧} = S_{F^{∧},F^{∧}}^{4,0}F^{∧} ⊕ S_{F^{∧},F^{∧}}^{3,1}F^{∧} ,
S_{F^{∧},F^{∧}}^{4,0}F^{∧} ⊂ C and S_{F^{∧},F^{∧}}^{3,1}F^{∧} ⊂ F

we conclude that S^{4,0} = 0. Then, similarly, by considering successively S_{F^{∧},F^{∧}}C^{∧} ⊂ F, S_{F^{∧},C^{∧}}C^{∧} ⊂ F and S_{C^{∧},C^{∧}}C^{∧} ⊂ F we can conclude that S^{3,1} = 0, S^{2,2} = 0 and S^{1,3} = 0 respectively. This shows that S = S^{0,4} ∈ S^{4}F.

5 Hyper-Kähler symmetric spaces are Osserman

Let (M, g) be a pseudo-Riemannian (or complex Riemannian) manifold, R its curvature tensor and X ∈ T_{p}M a tangent vector. The symmetric operator R_{X} : T_{p}M ∋ Y ↦ R(Y, X)X ∈ T_{p}M is called the Jacobi operator of X. We denote by S(T M) := {X ∈ T M|g(X, X) = ±1} the bundle of unit vectors.

Definition 1 (M, g) is called an Osserman space if the characteristic polynomial P_{X}(t) = det(t Id − R_{X}) of R_{X} is independent of X ∈ S(T M).

Theorem 4 Let (M, g) be an (ε)-hyper-Kähler symmetric space (or a complex hyper-Kähler symmetric space) of dimension 4n. Then (M, g) is an Osserman space. More precisely, the product of any two Jacobi operators is zero. In particular, all Jacobi operators are nilpotent and P_{X}(t) = t^{4n}.

Proof. Since the complexification of any (ε)-hyper-Kähler symmetric space is a complex hyper-Kähler symmetric space it is sufficient to prove the theorem for complex hyper-Kähler symmetric spaces. By Theorem 1 any complex hyper-Kähler symmetric space M is of the form M = M_{S} for some S ∈ S^{4}E^{+}. It is sufficient to check that R_{X}R_{Y} = 0 for all X, Y ∈ T_{o}M, where o is the canonical base point of the symmetric space M_{S}. Any tangent vector X ∈ T_{o}M = 𝔪 = H ⊗ E can be decomposed as X = ∑_{i=1}^{2} h_{i} ⊗ e_{i}, where h_{i} ∈ H and e_{i} ∈ E. So R_{X} is given by:

R_{X} = ∑_{i,j} R(·, h_{i} ⊗ e_{i})h_{j} ⊗ e_{j} .

This shows that it is suffient to check that the product of any two operators of the form R(·, h ⊗ e)h′ ⊗ e′ is zero. Let h_{1}, …, h_{4} ∈ H and e_{1}, …, e_{4} ∈ E. Applying the operator R(·, h_{1} ⊗ e_{1})h_{2} ⊗ e_{2} to h ⊗ e we have:

R(h ⊗ e, h_{1} ⊗ e_{1})h_{2} ⊗ e_{2} = ω^{H}(h, h_{1})h_{2} ⊗ S_{e,e_{1}}e_{2} = ω^{H}(h, h_{1})h_{2} ⊗ S_{e_{1},e_{2}}e .

Next we apply R(·, h_{3} ⊗ e_{3})h_{4} ⊗ e_{4} to the result, which yields:

ω^{H}(h, h_{1})R(h_{2} ⊗ S_{e_{1},e_{2}}e, h_{3} ⊗ e_{3})h_{4} ⊗ e_{4} = ω^{H}(h, h_{1})ω^{H}(h_{2}, h_{3})h_{4} ⊗ S_{(S_{e_{1},e_{2}}e),e_{3}}e_{4}
= ω^{H}(h, h_{1})ω^{H}(h_{2}, h_{3})h_{4} ⊗ S_{e_{3},e_{4}}S_{e_{1},e_{2}}e = 0 .

Here we have used the complete symmetry of S, the fact that S_{e_{1},e_{2}}e ∈ E^{+} and that S_{E,E} ⊂ S^{2}E^{+} vanishes on E^{+}. This shows that the composition of any two operators of the form R(·, h ⊗ e)h′ ⊗ e′ is zero and, hence, that R_{X}R_{Y} = 0 for all X, Y.

Remark 1: If S ≠ 0 then the Jordan normal form of the Jacobi operators R_{X} depends on the direction X. In fact R_{X} = 0 if X ∈ H ⊗ E^{+} and, if S ≠ 0, then there exists X ∈ H ⊗ E^{−} such that R_{X} ≠ 0. Remark 2: In dimension 4, pseudo-Riemannian manifolds satisfying the Osserman condition pointwise are characterised as self-dual Einstein 4-manifolds [ABBR]. This implies that symmetric Ricci-flat Osserman spaces of dimension 4 are the same as symmetric (ε)-hyper-Kähler manifolds. Similarly, complex symmetric Ricci-flat Osserman spaces of dimension 4 are the same as complex symmetric hyper-Kähler manifolds. It follows from Theorem 1 that any complex hyper-Kähler symmetric space of dimension 4 is defined by a quartic polynomial S = e^{4}, e ∈ E. Up to isomorphism, there are only two such manifolds: the flat one corresponding to e = 0 and the non-flat one corresponding to e ≠ 0. In the real case, it follows from Theorem 2 that any 4-dimensional hyper-Kähler symmetric space is flat. However, by Theorem 3 there exists two non-flat para-hyper-Kähler symmetric spaces which correspond to the polynomials ±e^{4}, e ∈ E_{0}. These manifolds ocurr in [BBR] as examples of Osserman spaces of signature (2, 2).

References

[ABBR] D.V. Alekseevsky, N. Blažić, N. Bokan, Z. Rakić, Self-duality and pointwise Osserman manifolds, Arch. Math. (Brno) 35 (1999), no. 3, 193-201

[AC] D.V. Alekseevsky and V. Cortés, Classification of indefinite hyper-Kähler symmetric spaces, to appear in Asian J. Math., preprint 2000-79, Max-Planck-Institut für Mathematik, Bonn, math.DG/0007189.

[BBR] N. Blažić, N. Bokan, Z. Rakić, Osserman pseudo-Riemannian manifolds of signature (2, 2), to appear in Bull. Australian Math. Soc.