Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications

TL;DR

This paper investigates the geometric properties of the kernel of the Lichnerowicz Laplacian on covariant tensors, using Bochner's analytical methods, with applications to Einstein deformations and manifold stability.

Contribution

It provides new geometric insights into the kernel of the Lichnerowicz Laplacian on symmetric spaces and extends vanishing theorems to broader tensor classes.

Findings

Characterization of the kernel on symmetric spaces

Vanishing theorems for null spaces of the Laplacian

Applications to Einstein deformation theory

Abstract

There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of proving vanishing theorems for the null space of Laplace operator. In particular, we pay special attention to the kernel of the Lichnerowicz Laplacian on Riemannian symmetric spaces of compact and noncompact types. In conclusion, we give some applications to the theories of infinitesimal Einstein deformations and the stability of Einstein manifolds.