differential geometry w notes from teacher_Part_64

# differential geometry w notes from teacher_Part_64 - 127...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern