# Orientability of a null submanifold

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1. Oct 9, 2015

### ifidamas

I have this question: Is it possible to define an orientation for a null submanifold with boundary?
In that case, is possible to use Stokes' theorem?
In particular, there is a way to define a volume form on that submanifold?

2. Oct 9, 2015

### bcrowell

Staff Emeritus
Orientability is topological. Volume forms only depend on affine structure. None of these notions are metrical, so it doesn't matter if the metric is degenerate on your submanifold. Stokes' theorem still holds. There is a good discussion of this sort of thing at the end of ch. 2 of the free online version of Carroll, http://arxiv.org/abs/gr-qc/?9712019 .

The only thing to worry about is that if there's curvature and the integrand is nonscalar (e.g., the flux of stress-energy), then this sort of thing fails, because we can't even define unambiguously what it means to add vectors that lie in different tangent spaces.

Last edited: Oct 9, 2015
3. Oct 9, 2015

### ifidamas

So I can't integrate a 2-form over a 2-submanifold which is a boundary of a nulla 3-submanifold?