Uses for volume form on a pseudo-Riemannian manifold [closed]

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?

closed as unclear what you're asking by uniquesolution, Yanior Weg, Ernie060, Shailesh, Adrian KeisterMay 2 at 14:31

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Why don't you pick your favorite pseudo-Riemannian manifold, and analyse the volume form there? – uniquesolution May 1 at 14:43

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.
• 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.
• 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.).