Remarks on the space of volume preserving embeddings

Let (N,g) be a Riemannian manifold. Given a compact, connected and oriented submanifold M of N, we define the space of volume preserving embeddings Embμ(M,N) as the set of smooth embeddings f:M↪N such that f⁎μf=μ, where μf (resp. μ) 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 Embμ(M,N) whose mean curvature is nowhere vanishing forms a tame Fréchet 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.