Osserman Conjecture in dimension n ? 8, 16

Abstract

Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ TpM n , the Jacobi operator is defined by RX = R(X, · )X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that globally Osserman manifolds are twopoint homogeneous. We prove the Osserman Conjecture for n = 8, 16, and its pointwise version for n = 2, 4, 8, 16. Partial result in the case n = 16 is also given.

The spectral geometry of the Weyl conformal tensor

“…timelike) Jordan Osserman if the Jordan normal form of J R on the appropriate pseudo-sphere bundle is independent of P . It is known that any global Riemannian (p = 0) Osserman manifold is locally isometric to a rank 1 symmetric space if m = 8, 16 [3,17,18] and that any local Lorentzian (p = 1) Jordan Osserman manifold has constant sectional curvature [1,5]. In the higher signature setting, there exist spacelike and timelike Jordan Osserman manifolds which are not locally homogeneous [2,7].…”

The Structure of Algebraic Covariant Derivative Curvature Tensors

“…Osserman [22] wondered if the converse held; this question has been called the Osserman conjecture by subsequent authors. The conjecture has been answered in the affirmative if m = 16 by work of Chi [3] and Nikolayevsky [18,19,20].…”

Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

“…Moreover due to the connection between Osserman and (anti-) self-dual metrics many special features occur like the existence of pointwise Osserman metrics which are not Osserman, i.e., the eigenvalues of the Jacobi operators are still independent of the direction, but they may change from point to point. (See for example [12], [16], [19] and the references therein for more information on pointwise Osserman 4-manifolds, and [5], [17], [18], [22], [23] for the higher dimensional Riemannian and pseudo-Riemannian cases).…”