Smooth coarea formula

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

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