On the generic Simplicity of the spectrum for Connection Laplacian and $G$-simplicity on Principal Bundles

TL;DR

This paper demonstrates that for generic smooth connections on vector bundles, the spectrum of the connection Laplacian is simple, and extends this to show all eigenvalues on certain principal bundles are G-simple, revealing a broad spectral simplicity.

Contribution

It establishes the generic simplicity of eigenvalues for the connection Laplacian and G-simplicity on principal bundles, advancing spectral theory in geometric analysis.

Findings

Eigenvalues of the connection Laplacian are generically simple.

All eigenvalues of the Laplace-Beltrami operator on certain principal bundles are G-simple.

The results hold for a residual set of smooth connections.

Abstract

In this paper, we prove that, for a residual set of $C^{k}$ connections defined on a smooth vector bundle $E \to M$, all eigenvalues of the connection Laplacian operator $\mathscr{L}$, acting on the space of sections of $E$, are simple. As an application, we prove that all eigenvalues of the Laplace-Beltrami operator on a compact $G$-principal bundle $P \to M$ are $G$-simple.