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Unformatted text preview: 5.3. FROBENIUS THEOREM 121 Definition 5.3.2 Let 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 the Lie bracket [ X , Y ] is also in the distribution, or [ , ] . Let be an integrable distribution. Let X and Y be two vector fields in . Then X and Y are tangent to the integral manifold of . Therefore, the Lie bracket [ X , Y ] is also tangent to the integral manifold and is in . Proposition 5.3.1 Every integral distribution is in involution. Definition 5.3.3 Let be a 1-form on M. Let x M be a point such that x , . The null space of the form at x is the ( n- 1)-dimensional subspace of T x M spanned by the vectors X T x M such that ( X ) = . Remark. A 1-form is also called a Pfa ffi an . Let 1 , . . . , n- k be ( n- k ) linearly independent 1-forms such that 1 n- k , Let N 1 , . . . N n- k be their null spaces. Then the intersection of the null spacesbe their null spaces....
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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