locimthm_132sp17.pdf

locimthm_132sp17.pdf - Math 132 Topology II Smooth...

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

Math 132 - Topology II: Smooth Manifolds. Spring 2017. Local Immersion Theorem George Melvin In this note we will state and prove the Local Immersion Theorem, following Guillemin- Pollack’s Differential Topology . Lemma (Linear algebra exercise) . Let A be a l × k matrix, where k l , and rank A = k . Then, there exists invertible k × k matrix P and an invertible l × l matrix Q such that I k 0 = QAP. Show that the existence of Q,P as above is equivalent to the following: if L A R k R l is the linear map L A ( x ) = Ax , then there exists isomorphisms T R k R k and S R l R l such that the matrix of S L A T R k R l with respect to the standard bases is I k 0 . Theorem (Local Immersion Theorem) . Let X R N , Y R M be smooth manifolds, f X Y a smooth map. Suppose that f is a local immersion at x X . Then, there exists local parameterisations R k V 1 U 1 X φ 1 ψ 1 R l V 2 U 2 Y φ 2 ψ 2 with x U 1 , and y = f ( x ) U 2 , such that ( ψ 1 f φ 1 )( x 1 ,...,x k ) = ( x

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