In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let
be smooth
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s of respective dimensions
. Let
be a smooth
surjection
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
such that the
pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x, ...
of
is surjective almost everywhere. Let
a measurable function. Then, the following two equalities hold:
:
:
where
is the Jacobian matrix and determinant">normal Jacobian of
, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point
is a regular point of
and hence the set
is a Riemannian submanifold of
, 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''.
Riemannian geometry
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