differential geometry w notes from teacher_Part_64

differential geometry w notes from teacher_Part_64 - 127...

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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. An immersed submanifold does not have to be an embedded submanifold.
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