We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in R n (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:

Abstract. Existence and location of Nash equilibrium points are studied for a large class of a finite family of payoff functions whose domains are not necessarily convex in the usual sense. The geometric idea is to embed these non-convex domains into suitable Riemannian manifolds regaining certain geodesic convexity properties of them. By using recent non-smooth analysis on Riemannian manifolds and a variational inequality for acyclic sets, an efficient location result of Nash equilibrium points is given. Some examples show the applicability of our results.

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