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

TL;DR

This paper establishes a new dynamical-topological obstruction preventing certain Riemannian manifolds with boundary from being smoothly isometrically embedded into Euclidean space, based on properties related to the incompressible Euler equations.

Contribution

It introduces a novel obstruction linked to dynamical and topological properties that restricts isometric embeddings of manifolds, inspired by hydrodynamics and Euler equations.

Findings

Obstruction applies when the first homology is nontrivial and the fundamental group has trivial center.

Embeddings into Euclidean space of dimension ≥3 must violate a kinetic energy-related condition.

The results connect topological invariants with dynamical constraints in geometric embedding problems.

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.