The Jacobi-orthogonality in indefinite scalar product spaces

TL;DR

This paper extends the concept of Jacobi-orthogonality to indefinite scalar product spaces, exploring its relationships with various algebraic curvature tensors and establishing conditions for Jacobi-orthogonality in different dimensions.

Contribution

It introduces the generalization of Jacobi-orthogonality to indefinite spaces and analyzes its connections with Osserman, Jacobi-dual, and algebraic curvature tensors, providing new characterizations.

Findings

Every quasi-Clifford tensor is Jacobi-orthogonal.

A Jacobi-diagonalizable Jacobi-orthogonal tensor is Jacobi-dual if J_X has no null eigenvectors.

In 3D, Jacobi-orthogonality characterizes constant sectional curvature.

Abstract

We generalize the property of Jacobi-orthogonality to indefinite scalar product spaces. We compare various principles and investigate relations between Osserman, Jacobi-dual, and Jacobi-orthogonal algebraic curvature tensors. We show that every quasi-Clifford tensor is Jacobi-orthogonal. We prove that a Jacobi-diagonalizable Jacobi-orthogonal tensor is Jacobi-dual whenever J_X has no null eigenvectors for all nonnull X. We show that any algebraic curvature tensor of dimension 3 is Jacobi-orthogonal if and only if it is of constant sectional curvature. We prove that every 4-dimensional Jacobi-diagonalizable algebraic curvature tensor is Jacobi-orthogonal if and only if it is Osserman.