Smooth coarea formula

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

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

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

${\displaystyle \int _{x\in M}\varphi (x)N\!J\;F(x)\,dM=\int _{y\in N}\int _{x\in F^{-1}(y)}\varphi (x)\,dF^{-1}(y)\,dN}$

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

Note that from Sard's lemma almost every point ${\displaystyle \scriptstyle y\,\in \,N}$ is a regular point of ${\displaystyle \scriptstyle F}$ and hence the set ${\displaystyle \scriptstyle F^{-1}(y)}$ is a Riemannian submanifold of ${\displaystyle \scriptstyle M}$, 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.