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

TL;DR

The paper proves that a residual set of $C^{k}$ connections on a smooth vector bundle over a compact Riemannian manifold yields a simple spectrum for the connection Laplacian, addressing a long-standing question about generic spectral simplicity beyond scalar Laplacians. A key tool is an explicit first-order eigenvalue variation formula under one-parameter perturbations $ abla(t)= abla+toldsymbol{A}$, which together with a Splitting Eigenvalue Lemma shows degeneracies are non-generic. These results imply that, for fixed base metrics, generic connections produce simple spectra, and, via metric perturbations, yield residual sets of metrics with simple eigenvalues as well. As an application, the authors establish a residual (hence generic) $G$-simple spectrum for the Laplace-Beltrami operator on total spaces of compact $G$-principal bundles, resolving a Yau open problem in that setting. Overall, the work extends Uhlenbeck-type genericity to connection perturbations and provides a robust framework for $G$-equivariant spectral simplicity on principal bundles with broad implications for geometric analysis and mathematical physics.

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.

Theorems & Definitions (31)

• Theorem 1
• Corollary 2
• Theorem 3
• Proposition 4
• proof
• Proposition 5
• proof
• Lemma 1
• proof
• Lemma 2