I was thinking about a possible use for the volume form on pseudo-Riemannian manifold. In Riemannian context we can use it to define minimal surfaces. Is there some physical or geometric interpretation in pseudo-Riemannian context?
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One can define the volume functional for a pseudo-Riemannian manifold, and prove that a non-degenerate submanifold is a critical point of the volume functional if and only if its mean curvature vector vanishes, $H = 0$. However, one loses the geometric interpretation from the Riemannian case, namely, that surfaces with $H=0$ locally minimize the volume, with respect to variations keeping the boundary fixed. For example, in Minkowski space $\Bbb R^n_1$, spacelike hypersurfaces (i.e., hypersurfaces for which the restriction of the standard Lorentzian metric is actually Riemannian) maximize volume.
That said, one can study:
- What happens to non-degenerate submanifolds with $H = 0$, finding conditions for when does it in fact maximize of minimize volume -- see for example Minimal Submanifolds in Pseudo-riemannian Geometry by Henri Anciaux.
- when do we obtain some Calabi-Bernstein like results in the pseudo-Riemannian case (there are some results about that in Minkowski space, e.g. here.
- What can one say about submanifolds satisfying the closest purely pseudo-Riemannian condition to $H = 0$: just having $H$ be a lightlike vector ($H \neq 0$ but $g(H,H)=0$). Such submanifolds are called marginally trapped, and can be used to represent surfaces of black holes in some $4d$ spacetime models, in physics (see relativity books, e.g. Hawking & Ellis, etc.).