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Unformatted text preview: 5.3. FROBENIUS THEOREM 127 20. We define a transversal to F ( B ) ( n- k )-dimensional submanifold W with local coordinates y 1 , . . . , y n- k . 21. If the integral balls are su ffi ciently small, then for di ff erent points of W the integral balls at those points are disjoint. 5.3.3 Foliations • Let M be an n-dimensional manifold and Δ be a k-dimensional distribution on M . • Let Δ be in involution, and, therefore, integrable. • Then, at each point x of M there exists an integral manifold of Δ . • The integral manifold may return to the Frobenius coordinate patch around the point x infinitely many times. • Definition 5.3.5 The integral manifolds of an integrable distribution define a foliation of M. Each connected integral manifold is called a leaf of the foliation. A leaf that is not properly contained in another leaf is called a maximal leaf . • A maximal leaf is not necessarily an embedded submanifold....
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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