In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.

be smooth Riemannian manifolds be a smooth surjection such that the pushforward (differential) a measurable function. Then, the following two equalities hold:

is the normal Jacobian, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

Note that from Sard's lemma, so the integrals in the right-hand side of the formulas above make sense.

References

• Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.