PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 43–45 DOI: 10.2298/PIM1308043N

A NOTE ON RAKIĆ DUALITY PRINCIPLE FOR OSSERMAN MANIFOLDS

Yuri Nikolayevsky and Zoran Rakić

Abstract. We prove that for a Riemannian manifold, the pointwise Osserman condition is equivalent to the Rakić duality principle.

1. Introduction

Let R be an algebraic curvature tensor on a Euclidean space ℝ^{n} and let for X ∈ ℝ^{n}, R_{X} : Y ↦ R(Y, X)X be the corresponding Jacobi operator. An algebraic curvature tensor R is called Osserman, if the spectrum of the Jacobi operator R_{X} does not depend on the choice of a unit vector X ∈ ℝ^{n}.

Let M^{n} be a Riemannian manifold, R be its curvature tensor, and R_{X} be the corresponding Jacobi operator. It is well known that the properties of R_{X} are intimately related with the underlying geometry of the manifold. The manifold M^{n} is called pointwise Osserman, if R is Osserman at every point p ∈ M^{n}, and is called globally Osserman, if the spectrum of R_{X} is the same for all X in the unit tangent bundle of M^{n}. Locally two-point homogeneous spaces are globally Osserman, since the isometry group of each of these spaces acts transitively on the unit tangent bundle. Osserman [8] conjectured that the converse is also true. This gives a very nice characterization of local two-point homogeneous spaces in terms of the geometry of the Jacobi operator.

At present, the Osserman Conjecture is almost completely solved by the results of Chi [3], who proved the Conjecture in dimensions n ≠ 4k, k > 1 and n = 4, and the first author [5, 6, 7], who proved it in all the remaining cases, except for some cases in dimension n = 16.

One of the crucial steps in the existing proofs of the Osserman Conjecture is the fact that any Osserman algebraic curvature tensor satisfies the Rakić duality principle [9]. We say that an algebraic curvature tensor R on a Euclidean space ℝ^{n} satisfies the Rakić duality principle, if for any unit vectors X, Y ∈ ℝ^{n}, the vector Y is an eigenvector of R_{X} if and only if the vector X is an eigenvector of R_{Y} (with the same eigenvalue).

2010 Mathematics Subject Classification: Primary 53B20, 53C25.

Key words and phrases: Jacobi operator, Osserman manifold, duality principle.

Z. R. is partially supported by the Serbian Ministry of Education and Science, project No. 174012. Y. N. is partially supported by the La Trobe University DGS grant.

The duality principle is extended to the pseudo-Riemannian settings in [1].

2. Equivalence of duality principle and Osserman pointwise condition

Recently, for an algebraic curvature tensor in Riemannian signature, Brozos-Vázquez and Merino [2] proved the equivalence of the Osserman condition and the duality principle for spaces of dimension less than 5. We show that this holds in an arbitrary dimension.

Theorem 2.1. The following two conditions for an algebraic curvature tensor R in the Riemannian signature are equivalent:

(a) R satisfies the duality principle;

(b) R is Osserman.

Proof. The implication (b) ⟹ (a) is proved in [9].

To establish the converse, consider the characteristic polynomial χ_{X} (t) of the Jacobi operator R_{X}, where X is a unit vector. As the coefficients of χ_{X} are analytic functions on the unit sphere S ⊂ ℝ^{n}, there is an open and dense subset S′ ⊂ S such that for all X ∈ S′, the number and the multiplicity of the eigenvalues of R_{X} are constant, the eigenvalues are analytic functions of X, and the eigendistributions of R_{X} are analytic (viewed as submanifolds of the appropriate Grassmannians) [4, 10].

Let X ∈ S′ and let Y ∈ S be orthogonal to X. Suppose λ_{0} is an eigenvalue of R_{X} with a unit eigenvector e_{0}. For small ϕ, the vector cosϕX + sinϕY belongs to S′, so there exist a differentiable (in fact, analytic) eigenvalue function λ(ϕ) of the operator R_{cos ϕX+sin ϕY} such that λ(0) = λ_{0}, and a differentiable unit vector function e(ϕ), a section of the λ(ϕ)-eigenspace of R_{cos ϕX+sin ϕY} , such that e(0) = e_{0}. Differentiating the equation

R(e(ϕ), cosϕX + sinϕY, cosϕX + sinϕY, e(ϕ)) = λ(ϕ)

at ϕ = 0 we obtain

2R(e_{0}, Y, X, e_{0}) + 2R(e_{0}, X, X, e′(0)) = λ′(0).

But R(e_{0}, X, X, e′(0)) = λ_{0}⟨e_{0}, e′(0)⟩ = 0 and R(e_{0}, Y, X, e_{0}) = R(X, e_{0}, e_{0}, Y) = λ_{0}⟨X, Y ⟩ = 0, by duality. It follows that the eigenvalues of R_{X} are constant on every connected component of S′. Then the coefficients of χ_{X} (t) are constant on the whole unit sphere S, which implies that R is Osserman. □

References

1. Z. Rakić, V. Andrejić, On the duality principle in pseudo-Riemannian Osserman manifolds, J. Geom. Phys. 57 (2007), 2158–2166.

2. M. Brozos-Vázquez, E. Merino, Equivalence between the Osserman condition and the Rakić duality principle in dimension 4, J. Geom. Phys. 62 (2012), 2346–2352.

3. Q. S. Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Diff. Geom. 28 (1988), 187–202.

4. T. Kato, Perturbation theory for linear operators, Grundlehren 132, 1976, Springer-Verlag, Berlin, New York.

5. Y. Nikolayevsky, Osserman manifolds of dimension 8, Manuscr. Math. 115 (2004), 31–53.

6. , Osserman conjecture in dimension n ≠ 8, 16, Math. Ann. 331 (2005), 505–522.

7. , On Osserman manifolds of dimension 16; in: Contemporary Geometry and Related Topics, Proc. Conf., Belgrade (2006), 379–398.

8. R. Osserman, Curvature in the eighties, Am. Math. Monthly 97 (1990), 731–756.

9. Z. Rakić, On duality principle in Osserman manifolds, Linear Alg. Appl. 296 (1999), 183–189.

10. F. Rellich, Perturbation Theory of Eigenvalue Problems, New York Univ. Lect. Notes, 1953; reprinted by Gordon and Breach, 1968.

Department of Mathematics and Statistics La Trobe University Melbourne, 3086, Victoria Australia

y.nikolayevsky@latrobe.edu.au

Faculty of Mathematics University of Belgrade Belgrade Serbia

zrakic@matf.bg.ac.rs