Equivalence between the Osserman condition and the Raki\'c duality principle in dimension four

TL;DR

This paper proves that in four-dimensional Riemannian manifolds, satisfying the Rakić duality principle is equivalent to being Osserman, linking two important geometric conditions through their eigenvalue properties.

Contribution

The paper establishes the equivalence between the Rakić duality principle and the Osserman condition specifically in four-dimensional Riemannian manifolds, clarifying their relationship.

Findings

Rakić duality principle implies Osserman condition in 4D

Eigenvalues of the Jacobi operator are constant under these conditions

Both conditions are shown to be equivalent in dimension four

Abstract

We show that 4-dimensional Riemannian manifolds which satisfy the Raki\'c duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant), thus both conditions are equivalent.