differential geometry w notes from teacher_Part_63

# differential geometry w notes from teacher_Part_63 - 5.3....

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5.3. FROBENIUS THEOREM 125 • Theorem 5.3.2 Frobenius Theorem. Let M be an n-dimensional man- ifold, Δ be a k-dimensional distribution in involution on M, X 1 , . . . , X k be vector fields that span Δ and ϕ 1 t , . . . , ϕ k t be the corresponding flows. Let x ∈ M, B be a su ffi ciently small ball around the origin in R k and F : B → M be defined by F ( t ) = ϕ k t k ◦ ··· ◦ ϕ 1 t 1 ( x ) . Then: 1. F ( B ) is an immersed submanifold of M, 2. F ( B ) is an integral manifold of Δ , 3. the distribution Δ is integrable. Proof : 1. (I) done earlier. 2. (II). We need to show that Δ is tangent to F ( B ) at each point of F ( B ). 3. We have for μ = 1 , . . . , k F * ∂ ∂ t μ = ϕ k t k * ◦ ··· ◦ ϕ μ t μ * X μ h ϕ k- 1 t k- 1 ◦ ··· ◦ ϕ 1 t 1 ( x ) i , 4. Thus, the tangent space T F ( t ) F ( B ) has a basis ϕ k t k * ◦ ··· ◦ ϕ 2 t 2 * X 1 ϕ 1 t 1 ( x ) ϕ k t k * ◦ ··· ◦ ϕ 3 t 3 * X 2 h ϕ 2 t 2 ◦ ϕ 1 t 1 ( x ) i . . ....
View Full Document

## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

### Page1 / 2

differential geometry w notes from teacher_Part_63 - 5.3....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online