The spectral geometry of the Riemann curvature tensor

TL;DR

This survey explores the spectral properties of various curvature-related operators in pseudo-Riemannian geometry, focusing on conditions for their spectra or Jordan forms to remain constant across the manifold.

Contribution

It provides a comprehensive overview of when the spectrum or Jordan normal form of natural curvature operators remains invariant, including new results in the complex setting.

Findings

Spectral invariance of the Jacobi operator under certain conditions

Constant Jordan normal form for higher order Jacobi operators

Results on spectral properties of the skew-symmetric curvature operator in complex geometry

Abstract

Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. It deals with results for the Jacobi operator, the higher order Jacobi operator, the Szabo operator, and the skew-symmetric curvature operator. Some results in the complex setting are presented as well for the skew-symmetric curvature operator.