The Orthogonality Principle for Osserman Manifolds

TL;DR

This paper introduces the concept of Jacobi-orthogonality for algebraic curvature tensors and proves it characterizes Osserman tensors, providing a new potential criterion for identifying Osserman manifolds.

Contribution

It establishes Jacobi-orthogonality as a new characterization of Osserman algebraic curvature tensors, linking orthogonality conditions to the Osserman property.

Findings

Jacobi-orthogonal tensors are Osserman.

All known Osserman tensors are Jacobi-orthogonal.

Jacobi-orthogonality offers a new perspective on Osserman conditions.

Abstract

We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if $\mathcal{J}_XY\perp\mathcal{J}_YX$ holds for all $X\perp Y$, where $\mathcal{J}$ denotes the Jacobi operator. We prove that any Jacobi-orthogonal tensor is Osserman, while all known Osserman tensors are Jacobi-orthogonal.