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differential geometry w notes from teacher_Part_62

differential geometry w notes from teacher_Part_62 - 123...

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5.3. FROBENIUS THEOREM 123 Thus, span { v i } = Δ . 7. We have d α μ = μ<ν C μλν α λ α ν + i A μλ i β i α ν + i < j B μ i j β i β j = ν γ μν α ν + i < j B μ i j β i β j . 8. Thus, B μ i j = ( d α μ )( v i , v j ) = 0 . 9. (3) = (4). Suppose that d α μ ω = 0. 10. Then we have 0 = d α μ ω = ν γ μν α ν ω + i < j B μ i j β i β j ω . = i < j B μ i j β i β j α 1 ∧ · · · ∧ α n - k . 11. Thus, B μ i j = 0. Remarks. A k -dimensional distribution Δ can be described locally by either k linearly independent vector fields that span Δ or by ( n - k ) linearly independent 1-forms whose common null space is Δ . If d α μ = n - k ν = 1 γ μν α ν for some 1-forms γ μν , then we write d α μ = 0 (mod α ) . 5.3.2 Frobenius Theorem Definition 5.3.4 Let M be an n-dimensional manifold, W be a k- dimensional manifold and F : W M be a smooth map. Then F is an immersion if for each x W the di ff erential F * : T x W T F ( x ) M is injective, that is, Ker F * = 0 . The image F ( W ) of the manifold W is called an immersed submani- fold . di ff geom.tex; April 12, 2006; 17:59; p. 123
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124 CHAPTER 5. LIE DERIVATIVE Let M be an n -dimensional manifold. Let Δ be a k -dimensional distribution on
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