Definitions

# 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 $scriptstylevarphi:M,longrightarrow, \left[0,infty\right]$ a measurable function. Then, the following two equalities hold:

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

$int_\left\{xin M\right\}varphi\left(x\right)N!J;F\left(x\right),dM = int_\left\{yin N\right\}int_\left\{xin F^\left\{-1\right\}\left(y\right)\right\} varphi\left(x\right),dF^\left\{-1\right\}\left(y\right),dN$

where $scriptstyle N!J;F\left(x\right)$ 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 $scriptstyle y,in, Y$ is a regular point of $scriptstyle F$ and hence the set $scriptstyle F^\left\{-1\right\}\left(y\right)$ 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.

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