LectureNotes5G - Math 598 Feb 2, 2005 1 Geometry and...

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Unformatted text preview: Math 598 Feb 2, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 5 2.2 Definition of Tangent Space If M is a smooth n-dimensional manifold, then to each point p of M we may associate an n-dimensional vector space T p M which is defined as follows. Let Curves p M := { : ( , ) M | (0) = p } be the space of smooth curves on M centered at p . We say that a pair of curve , Curves p M are tangent at p , and we write , provided that there exists a local chart ( U, ) of M centered at p such that ( ) (0) = ( ) (0) . Note that if ( V, ) is any other local chart of M centered at p , then, by the chain rule, ( ) (0) = ( 1 ) (0) = h ( 1 ) ( ( (0)) i h ( ) (0) i = h ( 1 ) ( ( (0)) i h ( ) (0) i = ( ) (0) . Thus is well-defined, i.e., it is indpendent of the choice of local coordinates. Further, one may easily check that is an equivalence relation. The set of tangent vectors of M at p is defined by T p M := Curves p M/ . Next we describe how T p M may be given the structure of a vector space. Let ( U, ) denote, as always, a local chart of M centered at p , and recall that n = dim( M ). Then we define a mapping f : T p M R n by ([ ]) := ( ) (0) . 1 Last revised: February 6, 2005 1 Exercise 1. Show that the above mapping is well-defined and is a bijection. Since is a bijection, we may use it to identify T p M with R n and, in particular, define a vector space structure on T p M . More explicitly, we set [ ] + [ ] := 1 ( ([ ]) + ([ ]) ) , and [ ] := 1 ( ([ ]) ) ....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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LectureNotes5G - Math 598 Feb 2, 2005 1 Geometry and...

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