Seminar Analysis: Esther Cabezas-Rivas (Goethe University Frankfurt)

A generalization of Gromov’s almost flat manifold theorem

Almost flat manifolds are the solutions of bounded size perturbations of the equation Sec=0 (Sec is the sectional curvature). In a celebrated theorem, Gromov proved that the presence of an almost flat metric implies a precise topological description of the underlying manifold.

Integral pinching theorems express curvature assumptions in terms of certain L^p-norms and try to deduce topological conclusions. But typically one needs to require p >n2, where n is the dimension of the manifold, to prove such rigidity theorems.

During this talk we will explain how, under lower sectional curvature bounds, to imposeanL1-pinching condition on the curvature is surprisingly rigid, leading indeed to the same conclusion as in Gromov’s theorem under more relaxed curvature conditions.

This is a joint work with B. Wilking.

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