**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.