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.
Categories: math.DG
