Smooth coarea formula

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

Let \scriptstyle M,\,N be smooth Riemannian manifolds of respective dimensions \scriptstyle m\,\geq\, n. Let \scriptstyle F:M\,\longrightarrow\, N be a smooth surjection such that the pushforward (differential) of \scriptstyle F is surjective almost everywhere. Let \scriptstyle\varphi:M\,\longrightarrow\, [0,\infty] a measurable function. Then, the following two equalities hold:

\int_{x\in M}\varphi(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)}\varphi(x)\frac{1}{N\!J\;F(x)}\,dF^{-1}(y)\,dN
\int_{x\in M}\varphi(x)N\!J\;F(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)} \varphi(x)\,dF^{-1}(y)\,dN

where \scriptstyle N\!J\;F(x) is the normal Jacobian of \scriptstyle F, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

Note that from Sard's lemma almost every point \scriptstyle y\,\in\, N is a regular point of \scriptstyle F and hence the set \scriptstyle F^{-1}(y) is a Riemannian submanifold of \scriptstyle M, so the integrals in the right-hand side of the formulas above make sense.

References[edit]

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