STRONG DUALITY PRINCIPLE FOR FOUR-DIMENSIONAL OSSERMAN MANIFOLDS. Vladica Andrejić

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1 17 Kragujevac J. Math. 33 (010) STRONG DUALITY PRINCIPLE FOR FOUR-DIMENSIONAL OSSERMAN MANIFOLDS Vladica Andrejić University of Belgrade, Faculty of Mathematics, Belgrade, Serbia ( (Received November 18, 009) Abstract. In this paper we deal with pseudo-riemannian Osserman curvature tensors. We define a concept of strong duality principle, and we prove that strong duality principle holds for four-dimensional Osserman curvature tensor. 010 Mathematics Subject Classification: 53B30, 53C50. Key words: Duality principle, Osserman algebraic curvature tensor, Jacobi operator. 1. INTRODUCTION Let us introduce the basic notation and terminology which are used throughout this work. Let R be an algebraic curvature tensor on a vector space V equipped with indefinite metric g of signature (ν, n ν). The norm of X V is ε X = g(x, X) and it determines various types of vectors. We say that X V is timelike (if ε X < 0), spacelike (ε X > 0), null (ε X = 0), nonnull (ε X 0), or unit (ε X { 1, 1}). The link between R and curvature operator R is given by R(X, Y, Z, W ) = g(r(x, Y )Z, W ). If (E 1, E,..., E n ) is an orthonormal basis of V, we shall use short notation ε i = ε Ei

2 18 and R ijkl = R(E i, E j, E k, E l ) for 1 i, j, k, l n. For the initial definitions and deeper explanations of this topic, the reader can consult Gilkey s book [7]. The Polarized Jacobi operator J (X, Y ) : V V is given by J (X, Y )(Z) = 1 {R(Z, X)Y + R(Z, Y )X}. The Jacobi operator J X : V V is defined by J X = J (X, X), which means that J X (Z) = R(Z, X)(X), for every Z V. For nonnull X V, J X preserves nondegenerate hyperspace {X} = {Y V : X Y }, and we have reduced Jacobi operator J X : {X} {X}, given by J X = J X {X}. We say that R is Osserman curvature tensor if the characteristic polynomial of J X is constant on both unit pseudo-spheres, in particular on positive (ε X = 1) and negative (ε X = 1) one. R is Jordan Osserman if the Jordan normal form of J X is constant on both unit pseudo-spheres. R is diagonalizable if the related Jacobi operator is diagonalizable. We say that R is k-stein if there exist constants C j for 1 j k, such that equations Tr((J X ) j ) = (ε X ) j C j hold for any X V. Special cases are Einstein (k = 1) and zwei-stein (k = ), and it is well known that every Osserman R is k-stein for every k. In our previous work [3] we defined the duality principle on the following way. Definition 1. Let R be an algebraic curvature tensor on vector space V. We say that duality principle holds for R if for every λ R, and for all mutually orthogonal units X, Y V holds J X (Y ) = ε X λy J Y (X) = ε Y λx. (1) Our natural aim is to prove duality principle for Osserman curvature tensor in general or construction of eventual counterexamples. Unfortunately, this problem is too hard, and we are able to give just results under some specific conditions. The first restriction can be a low index ν of the metric g. In the case of ν = 0 (Riemannian settings) Rakić proved that duality principle holds [10]. In the case of ν = 1 (Lorentzian settings) R has constant sectional curvature [5] and duality

3 19 principle holds. The second restriction can be a small number of eigenvalues of reduced Jacobi operator. In the case of diagonalizable R with two eigenvalues [], the present author recently gave some results. The last restriction can be a low dimension n. The case of four-dimensional R is solved in our previous work [1, 3], but this proof was complicated and based on the discussion of the possibly Jordan normal forms. In this article we give the much easier proof. Moreover, in the four-dimensional case we prove that strong duality principle holds.. PRELIMINARIES The following theorem contains the well known equations for Einstein and zweistein curvature tensor. Theorem 1. For a zwei-stein curvature tensor R in any orthonormal basis hold ε i ε p R piip = C 1, () 1 p n ε p R pijp = 0, (3) 1 p n 1 p,q n 1 p,q n ε p ε q R piiq R pjjq ε i ε j C + 1 p,q n 1 p,q n ε p ε q (R piiq ) = C, (4) ε p ε q R piiq R pijq = 0, (5) ε p ε q (R pijq + R pjiq ) = 0, (6) for every 1 i j n. We left the theorem without a proof, but reader can consult [1] and [3] for the details. Let us remark that equations () and (3) represent Einstein condition. Let us finish the section with lemma from [], which we need later. Lemma 1. Every null N 0 from a nondegenerate space V can be decomposed as N = P + Q, where P, Q V and ε P = ε Q = 1.

4 0 Proof. A nondegenerate vector space V can be decomposed as a direct sum V = V + V of complementary orthogonal subspaces, where V + is a maximal spacelike subspace and V is a maximal timelike subspace. This is why we can decompose N = S + T, with S V +, T V. S T gives ε S = ε T > 0, and if we set ( εs + 1 P = S + ε ) ( S 1 εs 1 T, Q = S + ε ) S + 1 T, ε S ε S ε S ε S it is easy to check that P + Q = S + T = N, and ε P = ε Q = STRONG DUALITY PRINCIPLE In Definition 1. we can demand nonnull X and Y, because such X and Y can be painlessly scaled to the units X/ ε X and Y/ ε Y with the same properties. Which are the optimal conditions for vectors X and Y from the formula (1)? For null X we feel some problems, because assumption become J X (Y ) = 0, which is independent of λ, but λ is relevant in the case of nonnull Y. Moreover, case ε X = 0 fall already in the dimension 4. For example, pseudo-riemannian manifold (R 4, g) with the metric g = x x 3 dx 1 dx 1 x 1 x 4 dx dx + dx 1 dx + dx dx 1 + dx 1 dx 3 + dx 3 dx 1 + dx dx 4 + dx 4 dx is globally Jordan-Osserman [6]. For this manifold we have J ( x x 3 1 ) = 0 and J ( x x 1 3 ) = 1 x 4, and for X = x 3 and Y = x 1 formula (1) does not work at any point. However, there is no visible reason for other restrictions, which is good ground for the introduction of strong duality principle. Definition. Let R be an algebraic curvature tensor on vector space V. We say that strong duality principle holds for R if for every λ R, and for all X, Y V with ε X 0 holds J X (Y ) = ε X λy J Y (X) = ε Y λx. The rest of this section contains the results which follow the arguments from our previous work [1, 3]. Lemma. If formula (1) holds for all mutually orthogonal X and Y with ε X 0, then it holds with only restriction ε X 0.

5 1 Proof. Let us suppose J X (Y ) = ε X λy for ε X 0 with g(x, Y ) 0. The decomposition Y = αx + Z with g(z, X) = 0 gives α 0. Image of J X is orthogonal to X, so 0 = g(j X (αx + Z), X) = g(ε X λ(αx + Z), X) = g(ε X λαx, X) = ε Xαλ, and because of ε X 0 and α 0, we have λ = 0. From J X (X) = 0 follows J X (Z) = J X (αx + Z) = ε X λ(αx + Z) = 0, and because of g(x, Z) = 0 and ε X 0, the lemma assumption gives J Z (X) = 0. Thus arose J Y (X) = J αx+z (X) = αr(x, Z)X + R(X, Z)Z = αj X (Z) + J Z (X) = 0, which proves J Y (X) = 0 = ε Y λx. Theorem. If R is a diagonalizable algebraic curvature tensor, then duality principle and strong duality principle are equivalent. Proof. Let us suppose J X (Y ) = ε X λy, g(x, Y ) = 0, ε X 0, and ε Y = 0. Diagonalizable R implies nondegenerate eigensubspace ker( J X ε X λid) Y. The decomposition Y = S + T from Lemma 1 with S, T ker( J X ε X λid) and ε S = ε T = 1 gives g(s, T ) = g(s, X) = g(t, X) = 0 with J X (S) = ε X λs, J X (T ) = ε X λt, Y = S + T. Since ε S+θT = ε S + θ ε T = (1 θ ), for θ 1, vectors S + θt, S and T are nonnull, orthogonal on X, eigenvectors of operator J X duality principle holds we have for eigenvalue ε X λ, and therefore if J S+θT (X) = ε S+θT λx, J S (X) = ε S λx, J T (X) = ε T λx. After the substitution in the standard calculation J S+θT (X) = J S (X) + θ J T (X) + θj (S, T )X, (7) we get ε S+θT λx = ε S λx + θ ε T λx + θj (S, T )X. For θ { 1, 0, 1} it gives J (S, T )X = 0 and (7) becomes J S+θT (X) = J S (X) + θ J T (X). Especially for θ = 1

6 holds J Y (X) = J S+T (X) = J S (X) + J T (X) = ε S λx + ε T λx = 0 = ε Y λx, which proves (1) for X Y and ε X 0. Lemma eliminates the condition X Y, which completes the proof. Theorem is very useful for the extension of duality principle. For example, in Riemannian settings duality principle holds [10], and such curvature tensor is diagonalizable, so by Theorem strong duality principle holds. It is worth noting that the condition of diagonalizability is pretty natural, because every Jordan Osserman curvature tensor of non-balanced signature is necessarily diagonalizable [8]. 4. FOUR-DIMENSIONAL OSSERMAN Let R be four-dimensional zwei-stein curvature tensor. Theorem 1 gives useful information. From () for 1 i 4 we have four equations ε 1 ε R 11 + ε 1 ε 3 R ε 1 ε 4 R 4114 = C 1, ε ε 1 R 11 + ε ε 3 R 33 + ε ε 4 R 44 = C 1, ε 3 ε 1 R ε 3 ε R 33 + ε 3 ε 4 R 4334 = C 1, ε 4 ε 1 R ε 4 ε R 44 + ε 4 ε 3 R 3443 = C 1, and after solve the system of equations ε ε 3 R 33 = ε 1 ε 4 R 4114, ε ε 4 R 44 = ε 1 ε 3 R 3113, (8) ε 3 ε 4 R 4334 = ε 1 ε R 11. From (3) for 1 i j 4 we get six equations R 443 = ε 1 ε 4 R 113, R 144 = ε 3 ε 4 R 133, R 334 = ε 1 ε 3 R 114, R 1443 = ε ε 4 R 13, (9) R 34 = ε 1 ε R 3114, R 1334 = ε ε 3 R 14.

7 3 From (5) for 1 i 4 we have four equations R11 + R R ε ε 3 R113 + ε ε 4 R114 + ε 3 ε 4 R3114 = C, R11 + R33 + R44 + ε 1 ε 3 R13 + ε 1 ε 4 R14 + ε 3 ε 4 R34 = C, R R33 + R ε 1 ε R133 + ε 1 ε 4 R ε ε 4 R334 = C, R R44 + R ε 1 ε R144 + ε 1 ε 3 R ε ε 3 R443 = C, and after use (8) and (9) hold ε ε 3 R113 + ε ε 4 R114 + ε 3 ε 4 R3114 = ε 1 ε 3 R13 + ε 1 ε 4 R14 + ε 3 ε 4 R3114 = ε 1 ε R133 + ε 1 ε 4 R14 + ε ε 4 R114 = ε 1 ε R133 + ε 1 ε 3 R13 + ε ε 3 R113. The solution of the previous system of equations is ε ε 3 R113 = ε 1 ε 4 R14, ε ε 4 R114 = ε 1 ε 3 R13, (10) ε 3 ε 4 R3114 = ε 1 ε R133. The derived equations are enough to prove duality principle. Theorem 3. For four-dimensional zwei-stein curvature tensor duality principle holds. Proof. Let J X (Y ) = ε X λy holds for mutually orthogonal units X and Y from V. Let us set E 1 = X, E = Y and extend them to an orthonormal basis (E 1, E, E 3, E 4 ) of V. Initial J E1 (E ) = ε 1 λe, gives R 11 = ε 1 ε λ, R 113 = 0, and R 114 = 0. By (10) it gives R 14 = 0 and R 13 = 0. Finally J E (E 1 ) = ε 1 R 11 E 1 +ε 3 R 13 E 3 +ε 4 R 14 E 4 = ε λe 1, and duality principle holds. Moreover, under the same initial conditions strong duality principle holds. In the Riemannian settings (signature (0,4) or (4,0)) we know that four-dimensional zwei-stein is necessarily Osserman [4, 9]. For Riemannian Osserman manifold duality principle holds [10], and (because Riemannian R is diagonalizable) by Theorem

8 4 strong duality principle holds. In the Lorentzian settings (signature (1,3) or (3,1)) R is necessarily of constant sectional curvature [5], and it is easy to check that strong duality principle holds. This is why we should check only the signature (,). Without loss of generality we can set ε 1 = ε = ε 3 = ε 4. If we apply (5) for i =, j = 3 and for i = 3, j =, we get R 11 R 131 R 14 R 134 R 31 R R 34 R 334 R 41 R R 44 R 434 = 0, R 1331 R 131 R 1334 R R 331 R 31 R 334 R 34 R 4331 R R 4334 R 434 = 0, and after some arrangement and use of (8) and (9) R 11 R 113 R 14 R R 13 R R 3114 R 114 R 14 R R 3113 R 113 = 0, R 3113 R 113 R 14 R 134 R 133 R 13 R 114 R 3114 R 14 R R 11 R 113 = 0. The sum and the difference of the previous equations are R 11 R R 3113 R 113 R 14 R 134 R 14 R 134 = 0, (11) R 13 R R 114 R 3114 = 0. (1) From (6) for i =, j = 3 we get (R 11 R 1331 R 14 R R 44 R 4334 ) + (R11 + R R4114 R113 R114 + R3114) + (R R 131 ) + R13 R133 (R R 134 ) + R31 R33 R34 R331 R33 + R334 (R R 431 ) R43 + R433 + (R R 434 ) = 0,

9 5 which after use of (8), (9), and (10) becomes R 11 R R113 + R 3113 R 11 + R11 + R R4114 4R113 4R R R113 + R114 R3114 (R R 134 ) + R114 R4114 R3114 R3114 R R114 (R R 134 ) R R R113 = 0. The easy calculation gives 4R 11 R R11 + R3113 (R R 134 ) = 0, and finally Theorem 4. principle holds. (R 11 + R 3113 ) = (R R 134 ). (13) For four-dimensional zwei-stein curvature tensor strong duality Proof. As we know that duality principle holds by Theorem 3, it is enough to prove (1) for X Y with ε X = 1 and ε Y = 0 in the view of Lemma. Let us set E 1 = X, and decompose Y = E +E 3 by Lemma 1, with mutually orthogonal units E and E 3, which are orthogonal to X, such that ε E1 = ε E = ε E3. Let E 4 be a vector which extend them to an orthonormal basis (E 1, E, E 3, E 4 ). Because of the signature (, ), we have ε 1 = ε = ε 3 = ε 4, which enable use of the previous derived equations. Our aim is to prove the formula J E1 (E + E 3 ) = ε 1 λ(e + E 3 ) J E +E 3 E 1 = 0. The assumption is ε 1 λ(e + E 3 ) = J E1 (E + E 3 ) = ε (R 11 + R 311 )E + ε 3 (R R 3113 )E 3 + ε 4 (R R 3114 )E 4, hence ε (R 11 + R 311 ) = ε 1 λ = ε 3 (R R 3113 ), ε 4 (R R 3114 ) = 0,

10 6 and finally R 11 + R R 113 = 0, (14) R R 3114 = 0. (15) The substitution (15) into (1) gives R 13 R 133 = R 114 R 3114 = R 114 = R 13, and therefore R 13 = 0 or R 133 = R 13. However from (10) and (15) R 13 = 0 R 114 = 0 R 3114 = 0 R 133 = 0 and holds anyway. By substitution (14) into (11) we get R 133 = R 13, (16) (R R 134 )R 14 = (R 11 + R 3113 )R 113 = R 113 = R 14. Thus arise R 14 = 0 or R R 134 = R 14. Similarly we use (10), (14), and (13) for R 14 = 0 R 113 = 0 R 11 + R 3113 = 0 R R 134 = 0, and undoubtedly R R 134 = R 14 = (R 14 + R 1334 ). (17) Everything is ready for the final computation of J E +E 3 E 1. J E +E 3 E 1 = p ε p (R 1p + R 13p + R 13p + R 133p )E p = ε 1 (R 11 + R R 113 )E 1 + ε (R 133 R 13 )E + ε 3 (R 13 R 133 )E 3 + ε 4 (R 14 + R R R 134 )E 4 The equations (14), (16), and (17) finally gives J E +E 3 E 1 = 0, which completes the proof.

11 7 Of course, Osserman curvature tensor is zwei-stein and strong duality principle holds in four-dimensional case. Acknowledgements: The author is partially supported by the Serbian Ministry of Science and Technological Development, project No D References [1] V. Andrejić, Prilog teoriji pseudo-rimanovih Osermanovih mnogostrukosti, magistarska teza, Beograd, 006. [] V. Andrejić, Quasi-special Osserman manifolds, Preprint (008). [3] V. Andrejić, Z. Rakić, On the duality principle in pseudo-riemannian Osserman manifolds, Journal of Geometry and Physics 57 (007), [4] A. Besse, Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin, [5] N. Blažić, N. Bokan, P. Gilkey A Note on Osserman Lorentzian manifolds, Bull. London Math. Soc. 9 (1997), [6] E. Garcia-Río, D. Kupeli, R. Vázquez-Lorenzo, Osserman Manifolds in Semi- Riemannian Geometry, Springer-Verlag, 00. [7] P. Gilkey, The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds, Imperial College Press, 007. [8] P. Gilkey, R. Ivanova, Spacelike Jordan Osserman algebraic curvature tensors in the higher signature setting, Differential Geometry, Valencia 001, World Scientific (00),

12 8 [9] P. Gilkey, A. Swann, L. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacoby operator, Quart. J. Math. Oxford 46 (1995), [10] Z. Rakić, On duality principle in Osserman manifolds, Linear Algebra and its Applications 96 (1999),