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Unformatted text preview: 5.3. FROBENIUS THEOREM 125 • Theorem 5.3.2 Frobenius Theorem. Let M be an ndimensional man ifold, Δ be a kdimensional 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 . . ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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