chapter  3
12 Pages

Factorization Method

Definition. Let F : X → Y be a proper smooth map of Riemannian manifolds of equal dimensions, Σ = {σ} be a family of smooth subvarieties of X. We say that F is factorable for this family, if for any σ ∈ Σ and arbitrary point x ∈ σ,

dVY (F (x) , F (σ))

dVX (x, σ) = j (x) J (σ) ,

where dVX and dVY are the corresponding Riemannian volume forms. We call j : X → R+ and J : Σ → R+ Jacobian factors of the mapping F. The corresponding metric integral operators RX and RY are related by the following equation for any σ ∈ Σ and a function f on Y :

RY f (F (σ)) =

fdVY = J (σ)

∫ σ

jgdVX = J (σ) RX (jg) (σ) ,

where g (x) = f (F (x)) . If there is an inversion operator IX for RX |Σ, then the operator IY (g) = j

−1IX ( J−1 (σ) g (σ)

) provides inversion for RY |F (Σ) .

If F is a factorable diffeomorphism, the map F−1 is also factorable for the family F (Σ). Moreover, the transitivity property holds: if F : X → Y and G : Y → Z are factorable maps of Riemannian manifolds for the family Σ and F (Σ) , respectively, then the composition GF : X → Z is factorable for Σ with the Jacobian factors