Affine projective Osserman structures

TL;DR

This paper introduces the concept of projective Osserman manifolds in affine and pseudo-Riemannian geometry, explores their properties, constructs examples, and analyzes the eigenvalues of the Jacobi operator using algebraic topology.

Contribution

It defines affine projective Osserman manifolds, studies their properties, and provides new examples, including cases with non-symmetric Ricci tensors and odd-dimensional eigenvalue analysis.

Findings

Modified Riemannian extension metric on cotangent bundles is projective Osserman.

Rank 1 symmetric spaces are affine projective Osserman.

Jacobi operator in odd dimensions has a single real non-zero eigenvalue.

Abstract

By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection is not the Levi-Civita connection of any metric. If M is an affine projective Osserman manifold of odd dimension, we use methods of algebraic topology to show the Jacobi operator has only one non-zero eigenvalue and that eigenvalue is real.