LectureNotes16G

LectureNotes16G - Math 6455 Nov 22, 2006 1 Differential...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 6455 Nov 22, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 16 Exponential Map 0.1 ODEs revisited; Local flows of vector fields Recall that earlier we proved: Theorem 0.1. Let U R n be an open set and F : U R n be C 1 , then for every p U , there exists an > such that for every < < there exists a unique curve : (- , ) U with (0) = p and ( t ) = F ( ( t )) . Further recall that in the proof of the above theorem we showed that we may set := min r sup B r ( p ) k F k , 1 n sup B r ( p ) | D j F i | , where r is any number which is chosen so small that B r ( p ) U . Note that depends continuously on r and p . Now let V be an open neighborhood of p such that V U and B r ( p ) U for all p V and some fixed r > 0. Define f : V R by f ( p ) := min r sup B r ( p ) k F k , 1 n sup B r ( p ) | D j F i | . Then f is continuous and positive. Thus := inf V f > . This shows that the above theorem may be restated in somewhat more general terms: Theorem 0.2. Let U R n be an open set and F : U R n be C 1 , then for every p U , there is an open neighborhood V U , p V , and an > such that for every p V and < < there exists a unique curve p : (- , ) U with p (0) = p and p ( t ) = F ( p ( t )) . The above theorem allows us to define, for every p U , a mapping : (- , ) V U by ( t,p ) := p ( t ) 1 Last revised: December 6, 2006 1 where V is some open neighborhood of p . This mapping is called a local flow of the vector field F at p . The previous theorem states then that C 1 vector fields have a local flow at each point. Then next result shows that this flow is continuous:local flow at each point....
View Full Document

Page1 / 4

LectureNotes16G - Math 6455 Nov 22, 2006 1 Differential...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online