Nonlinear Equations - UFRJ

[SEC. 4.4: THE CO-AREA FORMULA 49
E
π −1 (b) ≃ F
π −1 (U) ≃ U × F
π
U
b
B
Figure 4.1:
Fiber bundle.
5. The local triviality condition: for every p ∈ E, there is an
open neighborhood U ∋ π(p) in B and a diffeomorphism Φ :
π −1 (U) → U × F . (the local trivialization).
6. Moreover, Φ |π −1 ◦π(p) → F is a diffeomorphism.
(See figure 4.1).
Familiar examples of fiber bundles are the tangent bundle of a
manifold, the normal bundle of an embedded manifold, etc... In those
case the fiber is a vector space, so we speak of a vector bundle. The
fiber may be endowed of another structure (say a group) which is
immaterial here.
Here is a less familiar example of a vector bundle. Recall that P d
is the space of complex univariate polynomials of degree ≤ d. Let
V = {(f, x) ∈ P d × C : f(x) = 0}. This set is known as the solution
variety. Let π 2 : V → C be the projection into the second set of
coordinates, namely π 2 (f, x) = x. Then π 2 : V → C is a vector
bundle.
The co-area formula is a Fubini-type theorem for fiber bundles:
Theorem 4.7 (co-area formula). Let (E, B, π, F ) be a real smooth
fiber bundle. Assume that B is finite dimensional. Let f : E → R ≥0