Two theorems on Osserman manifolds.
Let \(M^n\) be a Riemannian manifold. For a point \(p\in M^n\) and a unit vector \(X\in T_pM^n\), the Jacobi operator is defined by \(R_X=R(X,\cdot )X\), where \(R\) is the curvature tensor. An algebraic curvature tensor \(R\) is Osserman if the eigenvalues of the Jacobi operator \(R_X\) do not depend on the choice of a unit vector \(X\in \mathbb{R}^n\). A Riemannian manifold \(M^n\) is called pointwise Osserman if its curvature tensor is Osserman. If, in addition, the eigenvalues of the Jacobi operator are constant on \(M^n\), the manifold \(M^n\) is called a globally Osserman. Osserman conjectured that globally Osserman manifold \(M^n\) is two-point homogeneous. Based on this conjecture, the author proves the following theorems: Theorem 1. A pointwise Osserman manifold \(M^n\) of dimension \(n=8k+4\), \(k\in \mathbb{N}\), is two-point homogeneous. Theorem 2. Let \(M^n\) be a globally Osserman manifold with the Jacobi operator having exactly two eigenvalues. In the case \(n=16\), assume in addition that the multiplicities of the eigenvalues are not 7 or 8. Then \(M^n\) is two-point homogeneous.
scientific article; zbMATH DE number 1970800
scientific article