Osserman manifolds of dimension 8

Abstract

For a Riemannian manifold $M^n$ with the curvature tensor $R$, the Jacobi operator $R_X$ is defined by $R_XY = R(X,Y)X$. The manifold $M^n$ is called {\it pointwise Osserman} if, for every $p \in M^n$, the eigenvalues of the Jacobi operator $R_X$ do not depend of a unit vector $X \in T_pM^n$, and is called {\it globally Osserman} if they do not depend of the point $p$ either. R. Osserman conjectured that globally Osserman manifolds are flat or rank-one symmetric. This Conjecture is true for manifolds of dimension $n \ne 8, 16$. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.