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differential geometry w notes from teacher_Part_61

# differential geometry w notes from teacher_Part_61 - 121...

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5.3. FROBENIUS THEOREM 121 Definition 5.3.2 Let Δ be a distribution on M. It is said to be in involution if it is closed under Lie brackets, that is, for any two vector field X and Y in Δ the Lie bracket [ X , Y ] is also in the distribution, or [ Δ , Δ ] Δ . Let Δ be an integrable distribution. Let X and Y be two vector fields in Δ . Then X and Y are tangent to the integral manifold of Δ . Therefore, the Lie bracket [ X , Y ] is also tangent to the integral manifold and is in Δ . Proposition 5.3.1 Every integral distribution is in involution. Definition 5.3.3 Let α be a 1 -form on M. Let x M be a point such that α x 0 . The null space of the form α at x is the ( n - 1) -dimensional subspace of T x M spanned by the vectors X T x M such that α ( X ) = 0 . Remark. A 1-form is also called a Pfa ffi an . Let α 1 , . . . , α n - k be ( n - k ) linearly independent 1-forms such that α 1 ∧ · · · ∧ α n - k 0 Let N 1 , . . . N n - k be their null spaces. Then the intersection of the null spaces forms a k -dimensional distribution

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differential geometry w notes from teacher_Part_61 - 121...

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