In differential geometry, the integration along fibers of a k-form yields a
-form where m is the dimension of the fiber, via "integration".
It is also called the fiber integration.
be a fiber bundle over a manifold with compact oriented fibers.
is a k-form on E, then for tangent vectors wi's at b, let where
is the induced top-form on the fiber
is smooth, work it out in coordinates; cf.
is a linear map
By Stokes' formula, if the fibers have no boundaries(i.e.
), the map descends to de Rham cohomology: This is also called the fiber integration.
is a sphere bundle; i.e., the typical fiber is a sphere.
Then there is an exact sequence
, K the kernel, which leads to a long exact sequence, dropping the coefficient
: called the Gysin sequence.
be an obvious projection.
and consider a k-form: Then, at each point in M, From this local calculation, the next formula follows easily (see Poincaré_lemma#Direct_proof): if
As an application of this formula, let
be a smooth map (thought of as a homotopy).
is a homotopy operator (also called a chain homotopy): which implies
induce the same map on cohomology, the fact known as the homotopy invariance of de Rham cohomology.
As a corollary, for example, let U be an open ball in Rn with center at the origin and let
, the fact known as the Poincaré lemma.
Given a vector bundle π : E → B over a manifold, we say a differential form α on E has vertical-compact support if the restriction
has compact support for each b in B.
for the vector space of differential forms on E with vertical-compact support.
If E is oriented as a vector bundle, exactly as before, we can define the integration along the fiber: The following is known as the projection formula.
be an oriented vector bundle over a manifold and
Since the assertion is local, we can assume π is trivial: i.e.,
is a ring homomorphism, Similarly, both sides are zero if α does not contain dt.
The proof of 2. is similar.