Preserve one, preserve all.

Abstract: Let $(\mathbb{E}^n,d)$ denote $n$-dimensional Euclidean space.

A striking theorem due to Beckman and Quarles asserts that if $n \geq 2$ and if $f:\mathbb{E}^n \rightarrow \mathbb{E}^n$ is a function satisfying $d(f(x),f(y))=1$ whenever $d(x,y)=1$, then $f$ is necessarily an isometry. I will discuss a conjecture, formulated in collaborative work with Meera Mainkar, that motions of Riemannian manifolds preserving a sufficiently small distance are necessarily isometries. I will present examples and supporting results to highlight the role of convexity in this rigidity phenomenon.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

The seminar meets every other Wednesday. To accommodate most time zones, the time rotates. The Zoom link is sent to the mailing list around 24 hours before each talk. To subscribe to the mailing list, send a message to geometrywithsymmetries@gmail.com requiring it.

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