Euler Characteristic of Epi-Lipschitz Subsets of Riemannian Manifolds

We present some properties of tangent and normal cones of epi-Lipschitz subsets of complete Riemannian manifolds. The fact that epi-Lipschitz subsets of complete Riemannian manifolds are absolute neighborhood retracts is proved. A notion of Euler characteristic of epi-Lipschitz subsets of complete Riemannian manifolds is introduced. Moreover, we provide a sufficient condition which ensures that the Euler characteristic of this class of sets is equal to one. Then, these results are applied to equilibrium theory on complete parallelizable Riemannian manifolds.