On the existence of a curvature tensor for given Jacobi operators

TL;DR

This paper addresses the problem of determining when a set of Jacobi operators corresponds to a genuine curvature tensor, providing a complete proof and extending the results to indefinite scalar product spaces.

Contribution

It offers a complete proof for the existence of a curvature tensor given Jacobi operators and generalizes the main theorem to indefinite scalar product spaces.

Findings

Provided a complete proof for the existence of curvature tensors from Jacobi operators.

Extended the proportionality principle to indefinite scalar product spaces.

Generalized the main theorem to broader geometric contexts.

Abstract

It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work. Unfortunately, although the published theorem is correct, its proof is incomplete because it contains some omissions, and the aim of this paper is to present a complete and accurate proof. We also generalize the main theorem to the case of indefinite scalar product space. Accordingly, we generalize the proportionality principle for Osserman algebraic curvature tensors.