Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on $S^n$

TL;DR

This paper calculates the eigenvalues and eigentensors of the Lichnerowicz Laplacian on symmetric covariant tensor fields over the Euclidean sphere, providing explicit descriptions of these spectral components.

Contribution

It explicitly determines the eigenvalues, multiplicities, and eigentensors of the Lichnerowicz Laplacian on symmetric tensors on $S^n$, a novel spectral analysis.

Findings

Eigenvalues with multiplicities are computed.

Explicit forms of symmetric eigentensors are provided.

Spectral properties of the Lichnerowicz Laplacian on $S^n$ are characterized.

Abstract

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^n$. The spaces of symmetric eigentensors are explicitly given.