In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
be smooth Riemannian manifolds of respective dimensions
be a smooth surjection such that the pushforward (differential) of
, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point
is a Riemannian submanifold of
, so the integrals in the right-hand side of the formulas above make sense.
This Riemannian geometry-related article is a stub.