Remarks on the space of volume preserving embeddings

TL;DR

This paper studies the geometric structure of volume-preserving embeddings of submanifolds in Riemannian manifolds, showing they form a tame Fréchet manifold and extending fluid dynamics equations to higher-dimensional membranes.

Contribution

It proves that certain volume-preserving embeddings form a tame Fréchet manifold and derives Euler-Lagrange equations, extending fluid mechanics to membranes of arbitrary dimension.

Findings

Volume-preserving embeddings form a tame Fréchet manifold.

Explicit Euler-Lagrange equations for natural Lagrangians.

Generalization of Euler equations to higher-dimensional membranes.

Abstract

Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_{\mu}(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*\mu^{f}=\mu, where \mu^{f} (resp. \mu) is the Riemannian volume form on f(M) (resp. M) induced by the ambient metric g (the orientation on f(M) being induced by f). In this article, we use the Nash-Moser inverse function Theorem to show that the set of volume preserving embeddings in Emb_{\mu}(M,N) whose mean curvature is nowhere vanishing forms a tame Fr\'echet manifold, and determine explicitly the Euler-Lagrange equations of a natural class of Lagrangians. As an application, we generalize the Euler equations of an incompressible fluid to the case of an "incompressible membrane" of arbitrary dimension moving in N.