Non-Positive Curvature and Global Invertibility of Maps |
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Authors: | Gang Li Frederico Xavier |
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Institution: | 1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA 2. Department of Mathematics, Nanjing University, Nanjing, China
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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(? n ,?): 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(? n ,?) 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 ? n . |
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