Abstract: We generalize known results for periodic Schr\"odinger operators in a purely geometric setting. Let $X$ be a non-compact Riemannian manifold with compact quotient given by a discrete abelian subgroup of the isometries on $X$ (e.g. the lattice group $\Z^d$ acting on $X=\R^d$). By Floquet theory the spectrum of the Laplacian on $X$ is purely essential and consists of a locally finite union of intervals, called bands. We shortly discuss examples where non-trivial gaps between two adjacent bands occur.
The second part deals with a periodic manifold $X$ with a non-trivial spectral gap $(a,b)$. A continuous local perturbation $g(\tau)$ of the metric $g=g(0)$ on $X$ leads to bound states in the gap. We give a lower asymptotic bound on the number of parameters $\tau$ such that a given energy $\lambda \in (a,b)$ is an eigenvalue of the corresponding Laplacian of the perturbed manifold.