  We establish inequalities for the eigenvalues of Schr\"{o}dinger operators on
compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of
spheres, and of real, complex and quaternionic projective spaces, which are
related to inequalities for the Laplacian on Euclidean domains due to Payne,
P\'olya, and Weinberger and to Yang, but which depend in an explicit way on the
mean curvature. In later sections, we prove similar results for Schr\"{o}dinger
operators on homogeneous Riemannian spaces and, more generally, on any
Riemannian manifold that admits an eigenmap into a sphere, as well as for the
Kohn Laplacian on subdomains of the Heisenberg group.
  Among the consequences of this analysis are an extension of Reilly's
inequality, bounding any eigenvalue of the Laplacian in terms of the mean
curvature, and spectral criteria for the immersibility of manifolds in
homogeneous spaces.
