Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

TL;DR

This paper constructs higher-dimensional Osserman metrics with non-nilpotent Jacobi operators in neutral signature, featuring non-trivial Jordan forms and almost para-Hermitian structures, expanding the class of known examples in differential geometry.

Contribution

It provides explicit examples of higher-dimensional Osserman metrics with non-nilpotent Jacobi operators and non-trivial Jordan forms in neutral signature, which were previously unknown.

Findings

Existence of Osserman metrics with non-nilpotent Jacobi operators in (n,n) signature for n ≥ 3

Examples admit almost para-Hermitian structures and are semi para-complex Osserman

These metrics do not satisfy the third Gray identity nor are they integrable

Abstract

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.