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
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 1form 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 1form is also called a Pfa ffi an . Let 1 , . . . , n k be ( n k ) linearly independent 1forms 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....
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.
 Spring '10
 Wong
 Geometry

Click to edit the document details