g-natural metrics on tangent bundles and Jacobi operators

TL;DR

This paper explores the spectral properties of Jacobi operators on tangent bundles of Riemannian manifolds equipped with g-natural metrics, providing explicit computations and conditions for Osserman manifolds.

Contribution

It establishes a relation between Jacobi operators on (M,g) and (TM,G), and characterizes when (TM,G) is Osserman for surfaces.

Findings

Explicit spectrum of Jacobi operators on tangent bundles of surfaces

Necessary and sufficient conditions for (TM,G) to be Osserman

Relation between Jacobi operators on base manifold and tangent bundle

Abstract

Let (M,g) be a Riemannian manifold and G a nondegenerate g-natural metric on its tangent bundle T M . In this paper we establish a relation between the Jacobi operators of (M,g) and that of (T M,G). In the case of a Riemannian surface (M,g), we compute explicitly the spectrum of some Jacobi operators of (TM,G) and give necessary and sufficient conditions for (T M,G) to be an Osserman manifold.