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In this paper we study the asymptotic behavior of second-order uniformlyelliptic operators on weighted Riemannian manifolds. They naturally emerge whenstudying spectral properties of the Laplace-Beltrami operator on families ofmanifolds with rapidly oscillating metrics. We appeal to the notion ofH-convergence introduced by Murat and Tartar. In our main result we establishan H-compactness result that applies to elliptic operators with measurable,uniformly elliptic coefficients on weighted Riemannian manifolds. We furtherdiscuss the special case of ``locally periodic'' coefficients and study theasymptotic spectral behavior of compact submanifolds of $\mathbb R^n$ withrapidly oscillating geometry.

H-compactness of elliptic operators on weighted Riemannian Manifolds