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Register a new userA dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations
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Siran Li
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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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