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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 &gt; such that for every &lt; &lt; 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 &gt; 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 &gt; . 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 &gt; such that for every p V and &lt; &lt; 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....
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