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arXiv · 2006.15423

A dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations

Siran Li
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Abstract

We obtain a dynamical--topological obstruction for the existence of isometric embedding of a Riemannian manifold-with-boundary $(M,g)$: if the first real homology of $M$ is nontrivial, if the centre of the fundamental group is trivial, and if $M$ is isometrically embedded into a Euclidean space of dimension at least $3$, then the isometric embedding must violate a certain dynamical, kinetic energy-related condition (the "rigid isotopy extension property" in Definition 1.1). The arguments are motivated by the incompressible Euler equations with prescribed initial and terminal configurations in hydrodynamics.

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Siran Li. 2020-06-27. A dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations. https://arxiv.org/abs/2006.15423

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