# 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

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• 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.).