  We provide several inequalities between eigenvalues of some classical
eigenvalue problems on domains with $C^2$ boundary in complete Riemannian
manifolds. A key tool in the proof is the generalized Rellich identity on a
Riemannian manifold. Our results in particular extend some inequalities due to
Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.
