LectureNotes4G

# LectureNotes4G - Math 598 1 Geometry and Topology II Spring...

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

Unformatted text preview: Math 598 Jan 21, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 4 2 Differentiable manifolds 2.1 Differential structures and maps Recall that a mapping f : R n → R m is differentiable of class C r provided that all of its partial derivatives exist and are continuous up to and including order r . If all partial derivatives of f exist up to any order we say that f is C ∞ or smooth . A collection { ( U i , φ i ) } i ∈ I is called an atlas of a manifold M , if U i cover M and φ : U i → R n are homeomorphisms. We say that this atlas is C r if for every i , j ∈ I , φ i ◦ φ − 1 j : φ j ( U i ∩ U j ) → φ i ( U i ∩ U j ) is C r . We say that M is a C r manifold if it admits a C r atlas. Exercise 1. Show that S n is a smooth ( C ∞ ) manifold. Exercise 2. Show that RP n is a smooth manifold ( Hint : Define RP n as the quotient space ( R n +1 − { o } ) / ∼ , where x ∼ y iff x = λy for some λ 6 = 0....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

LectureNotes4G - Math 598 1 Geometry and Topology II Spring...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online