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Unformatted text preview: 2.2. TANGENT BUNDLE 37 A vector field is a map v : M T M , such that v = Id : M M . A vector field is a cross section of the tangent bundle. The image of the manifold under a vector field is a ndimensional subman ifold of the tangent bundle T M . The zero vector field defines the zero section of the tangent bundle. Definition 2.2.2 Let ( M , g ) be an ndimensional Riemannian mani fold. The unite tangent bundle of M is the set T M of all unit vectors to M, T M = { ( p , v )  p M , v T p M ,  v  = 1 } , where, locally,  v  2 = n i , j = 1 g ij ( p ) v i v j . The unit tangent bundle is a (2 n 1)dimensional submanifold of the tangent bundle T M . Theorem 2.2.1 Let S 2 be the unit 2sphere embedded in R 3 . The unit tangent bundle T S 2 is homeomorphic to the real projective space R P 3 and to the special orthogonal group S O (3) T S 2 R P 3 S O (3) ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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