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differential geometry w notes from teacher_Part_7

differential geometry w notes from teacher_Part_7 - 13 1.2...

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1.2. MANIFOLDS 13 Let the transition functions f αβ = ϕ α ϕ - 1 β : ϕ α ( U α U β ) C n , defined by z k α = f k αβ ( z 1 β , . . . , z n β ) , where z k α = x k α + iy k α and z k β = x k β + iy k β , k = 1 , . . . , n , be complex analytic. That is, they satisfy Cauchy-Riemann conditions x k α x j β = y k α y j β , x k α y j β = - y k α x j β , or z k α ¯ z j β = 0 , with j , k = 1 , . . . , n . Then M is called a n -dimensional complex manifold . The topological dimension of a n -dimensional complex manifold is 2 n . di geom.tex; April 12, 2006; 17:59; p. 17
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14 CHAPTER 1. MANIFOLDS 1.3 Tangent Vectors and Mappings A tangent vector to a submanifold M of R n at a point p 0 M is the velocity vector ˙ p at p 0 to some parametrized curve p = p ( t ) lying on M and passing through the point p 0 . Whitney theorem : Every n -dimensional manifold can be realized as a sub- manifold of R 2 n , or as a smooth submanifold of R 2 n + 1 . So, every manifold is a submanifold of a Euclidean space. However, the intrinsic geometry of M does not depend on its embedding in a Euclidean space. 1.3.1 Tangent Vectors
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