Non-Positive Curvature and Global Invertibility of Maps

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Abstract

Let F:MN 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 xM |d(B ζ F)(x)|>0 for all ζN(∞) and Busemann functions B ζ . This generalizes the Cartan–Hadamard theorem and the Hadamard invertibility criterion, which requires inf xM DF(x)−1−1=inf ζN(∞)inf xM |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 fC 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 gC 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 .

To Professor Jianguo Cao, in memoriam.
Communicated by Jiaping Wang.
Work partially supported by CAPES and IMPA (Brazil).