Spectrum of the Laplacian on vector bundles over \(C_{2\pi}\)-manifolds (Q1102556)

Let (M,g) be a \(C_{2\pi}\)-manifold, that is a Riemannian manifold all of whose geodesics are closed and have common length \(2\pi\). Let E be a \(C^{\infty}\) complex vector bundle over M with a Hermitian structure, and let \(\tilde d\) be a linear connection on E compatible with the Hermitian structure. The author considers a Laplace operator L on \(C^{\infty}\) sections of E. He shows that the asymptotic distribution of eigenvalues of L is described by the distribution of eigenvalues of holonomies of closed geodesics of (M,g). He considers the case where the spectrum consists of clusters of eigenvalues contained in a sequence of intervals of constant width.