Non-Positive Curvature and Global Invertibility of Maps

Abstract

Let F:M→N be a C 1 map between Riemannian manifolds of the same dimension, M complete, N Cartan–Hadamard. We show that F is a C 1 diffeomorphism if inf x∈M |d(B ζ ∘F)(x)|>0 for all ζ∈N(∞) and Busemann functions B ζ . This generalizes the Cartan–Hadamard theorem and the Hadamard invertibility criterion, which requires inf x∈M ∥DF(x)−1∥−1=inf ζ∈N(∞)inf x∈M |d(B ζ ∘F)(x)|>0. Our proofs use a version of the shooting method for two-point boundary value problems. These ideas lead to new results about the size of the critical set of a function f∈C 2(ℝ n ,ℝ): a) If \(\inf_{x\in \mathbb{R}^{n}}|\operatorname{Hess} f(x)v|>0\) for all v≠0 then the function f has precisely one critical point. (b) If g∈C 2(ℝ n ,ℝ) is the C 1 local uniform limit of functions as in a), and \(\operatorname{Hess} g(x)\) is nowhere singular, then g has at most one critical point. The totality of functions described in (b) properly contains the class consisting of all C 2 strictly convex functions defined on ℝ n .