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

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.