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differential geometry w notes from teacher_Part_19

differential geometry w notes from teacher_Part_19 - 37 2.2...

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2.2. TANGENT BUNDLE 37 A vector field is a map v : M T M , such that π 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 n -dimensional 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 n-dimensional Riemannian mani- fold. The unite tangent bundle of M is the set T 0 M of all unit vectors to M, T 0 M = { ( p , v ) | p M , v T p M , || v || = 1 } , where, locally, || v || 2 = n i , j = 1 g i j ( 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 2 -sphere embedded in R 3 . The unit tangent bundle T 0 S 2 is homeomorphic to the real projective space R P 3 and to the special orthogonal group S O (3) T 0 S 2 R P 3 S O (3) . Proof : 1. di ff geom.tex; April 12, 2006; 17:59; p. 40
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38 CHAPTER 2. TENSORS 2.3 The Cotangent Bundle Definition 2.3.1 Let M be a smooth manifold. The cotangent bundle T * M to M is the collection of all covectors at all points of M T * M = { ( p , σ ) | p M , σ T * p M } Let dim M = n . Let p M be a point in the manifold M , ( U , x ) be a local chart and ( x
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