  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.
