Flexibility and Rigidity of Lipschitz Riemannian Geometry

Abstract: Every smooth isometric embedding of the 2-sphere into $\mathbb{R}^3$ the standard one (upto rotations, translations, and reflections).

In contrast to this classical rigidity result we have flexibility: There are Lipschitz isometric embeddings of the 2-sphere in $\mathbb{R}^3$ whose image has arbitrarily small diameter.

The talk will present more of these surprising flexibility results for Lipschitz maps between Riemannian manifolds.

Eventually, our focus will be on the following rigidity result of Llarull:

let $f\colon M \to S^n$ be a smooth map between a compact Riemannian manifold M and S^n with the standard metric. If M is sufficiently curved (scalar curvature is everywhere >= the scalar curvature of $S^n$), if the map is non-expanding (Lipschitz constant <=1) and if it is far enough from a constant map (has non-zero degree) then f must be an isometry.

We will discuss the ideas of the proof, which involve the geometry of vector bundles, and Gromov's question whether rigidity prevails or flexibility occurs if just have Lipschitz continuity in the setup of the above theorem.

algebraic topologydifferential geometrygeometric topologyK-theory and homology

Audience: researchers in the topic

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Göttingen topology and geometry seminar

Series comments: Our seminar takes place via zoom every Tuesday afternoon (Central European Summer Time; the precise time slot varies—please always refer to the listing).

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