# Osserman conjecture

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Let be the Riemann curvature tensor of a Riemannian manifold . Let be the Jacobi operator. If is a unit tangent vector at a point of , then is a self-adjoint endomorphism of the tangent bundle at . If is flat or is locally a rank- symmetric space (cf. also Symmetric space), then the set of local isometries acts transitively on the sphere bundle of unit tangent vectors, so has constant eigenvalues on . R. Osserman [a6] wondered if the converse implication was valid; the following conjecture has become known as the Osserman conjecture: If has constant eigenvalues, then is flat or is locally a rank- symmetric space.

Let be the dimension of . If is odd, if modulo , or if , then C.S. Chi [a3] has established this conjecture using a blend of tools from algebraic topology and differential geometry. There is a corresponding purely algebraic problem. Let be a -tensor on which defines a corresponding curvature operator . If satisfies the identities,  then is said to be an algebraic curvature tensor. The Riemann curvature tensor of a Riemannian metric is an algebraic curvature tensor. Conversely, given an algebraic curvature tensor at a point of , there always exists a Riemannian metric whose curvature tensor at is . Let ; this is a self-adjoint endomorphism of the tangent bundle at . One says that is Osserman if the eigenvalues of are constant on the unit sphere in . C.S. Chi classified the Osserman algebraic curvature tensors for odd or modulo ; he then used the second Bianchi identity to complete the proof. However, if modulo , it is known [a4] that there are Osserman algebraic curvature tensors which are not the curvature tensors of rank- symmetric spaces and the classification promises to be considerably more complicated in these dimensions.

There is a generalization of this conjecture to metrics of higher signature. In the Lorentzian setting, one can show that any algebraic curvature tensor which is Osserman is the algebraic curvature tensor of a metric of constant sectional curvature; it then follows that any Osserman Lorentzian metric has constant sectional curvature [a2]. For metrics of higher signature, the Jordan normal form of the Jacobi operator enters; the Jacobi operator need not be diagonalizable. There exist indefinite metrics which are not locally homogeneous, so that is nilpotent for all tangent vectors , see, for example, [a5].

If is an orthonormal basis for an -plane , one can define a higher-order Jacobi operator One says that an algebraic curvature tensor or Riemannian metric is -Osserman if the eigenvalues of are constant on the Grassmannian of non-oriented -planes in the tangent bundle. I. Stavrov [a8] and G. Stanilov and V. Videv [a7] have obtained some results in this setting.

In the Riemannian setting, if I. Dotti, M. Druetta and P. Gilkey [a1] have recently classified the -Osserman algebraic curvature tensors and showed that the only -Osserman metrics are the metrics of constant sectional curvature.

How to Cite This Entry:
Osserman conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Osserman_conjecture&oldid=15029
This article was adapted from an original article by P.B. Gilkey (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article