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.

arXiv commentThe main Theorem 1.4 of this manuscript is flawed --- in fact, it is vacuously true, for the "rigid isotopy extension property'' can never hold for any isometric embedding. We shall correct this issue in a new preprint entitled "A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions'' (to be uploaded to ArXiv shortly)