Nondegenerate solutions for constrained semilinear elliptic problems on Riemannian manifolds

Let M-n be a connected compact smooth manifold, where n >= 2. In this article, we prove that nondegeneracy of nonconstant solutions for a class of singularly perturbed semilinear elliptic problems on M is generic with respect to the pair (epsilon, g), where epsilon > 0 and g is a metric of class C-k, k >= 1. As applications, we show that under certain growth conditions, such result generalizes to nondegeneracy of any solution for the Allen-Cahn or nonlinear Schrodinger equations.