| Differentiable Structures With Zero Entropy On Simply Connected 4-Manifolds (2007) | |||||||||||||
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| . We show that a compact 4-dimensional simply connected topological manifold M admits a differentiable structure with a C 1 Riemannian metric whose geodesic flow has zero topological entropy if and only if M is homeomorphic to S 4 , C P 2 , S 2 \Theta S 2 , C P 2 #C P 2 or C P 2 #CP 2 . 1. Introduction The Riemannian metric of constant Gaussian curvature one on S 2 and the flat metric on T 2 have geodesic flows with zero topological entropy. On the other hand since the fundamental group of a closed oriented surface of genus 2 has exponential growth, it follows from a result of E. Dinaburg [3] that any Riemannian metric on a closed oriented surface of genus 2 will have a geodesic flow with positive topological entropy. Hence a closed oriented surface M admits a Riemannian metric whose geodesic flow has zero topological entropy if and only if M is diffeomorphic to S 2 or T 2 . Here we propose a version of this fact for closed simply connected 4-manifolds. L... | |||||||||||||
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