Summary: Integration of a scalar function over a submanifold.
Suppose n is a positive integer, V is an n-dimensional inner product space, 0 m < n and M Mm(V ).
Definition. We say (T, ) is a smooth local parameter for M if
(1) T is an open subset of Rm
(2) : T V is smooth;
(3) rng M;
(4) is univalent;
(5) dim rng (t) = m whenever t T.
If (T, ) is a smooth local parameter for M we define the smooth function
Jm : T (0, ),
the m-Jacobian of , by letting
Jm(t) = det (t) (t) whenever t T.
We say a function f on M with values in a topological space is Lebesgue measurable if f is
Lebesgue measurable whenever (T, ) is a local coordinate for M.
be the family of Lebesgue measurable functions.
Remark. Suppose a M. Then there are an open subset U of V and (U, , Un