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(ℝⁿ,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(ℝⁿ,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 ℝⁿ.