A Professor Level Topology Theorem

Topology usually lets us stretch and bend a shape while geometry keeps track of distances and angles. Mostow rigidity reveals a remarkable setting where the topology determines the geometry. Starting with the problem of telling knots apart, we explore knot complements, hyperbolic tilings, glued surfaces and universal covers. These pictures lead toward the theorem: for complete, finite-volume hyperbolic manifolds in dimension three and higher, the topology determines the hyperbolic geometry up to isometry, with curvature normalized to −1. This is a visual introduction to the theorem and why it matters, rather than a proof. 00:00 Topology versus geometry 01:05 Mostow's rigidity theorem 02:23 How can we tell knots apart? 05:05 The space around a knot 08:06 Hyperbolic surfaces and universal covers 12:35 Inside three-dimensional hyperbolic space Animated with Manim Community Edition. I used Codex for help programming but the script, storyboard and voice are all me. Credits Music: “Atmosphere Pulse” by leberch, via Pixabay. https://pixabay.com/music/pulses-atmo... Earth imagery: NASA Earth Observatory, Blue Marble: Next Generation; Reto Stöckli and Robert Simmon. https://science.nasa.gov/earth/earth-... Artwork references include M. C. Escher's Circle Limit IV (computer rendition by Douglas Dunham), Hilma af Klint's The Swan No. 16 and No. 17, Robert Delaunay's Simultaneous Disc, and stained-glass imagery from Notre-Dame and Sainte-Chapelle. Douglas Dunham's Circle Limit reference: https://www.d.umn.edu/~ddunham/dunbr0... Sainte-Chapelle west rose photograph: Didier B (Sam67fr), CC BY-SA 2.5. Cropped/warped artwork adaptation under the same license. https://commons.wikimedia.org/wiki/Fi... https://creativecommons.org/licenses/... Geometry reference: Giovanni Italiano, Bruno Martelli and Matteo Migliorini, “Hyperbolic manifolds that fiber algebraically up to dimension 8.” https://arxiv.org/abs/2010.10200