RESEARCH METHODOLOGY SEMINAR
10:30 am - 11:30 am, Lecture Hall 5
Cartan Hadamard Theorem & Synge’s Theorem
Pritthijit Biswas
Chennai Mathematical Institute.
10-04-19

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Abstract

One of the modern aspects of research in Riemannian geometry is to understand how the curvature of a Riemannian manifold affects it's topology. The purpose of this talk is to explore some of this.

Cartan Hadamard theorem (1928) states that the universal cover of a complete Riemannian manifold of non positive sectional curvatures is diffeomorphic to a Eucledian space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881 and independently by Jacques Hadamard in1898. E'lie Cartan generalised it to Riemannian manifolds in 1928. Mikhail Gromov further generalised this to metric spaces of non positive curvature in 1987

Synge's Theorem (1936) asserts that compact orientable even dimensional Riemannian manifold with positive sectional curvatures is simply connected.