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HW4 130909

# HW4 130909 - j ∂x i ∂ ∂ ˜ x j and therefore the same...

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Differential Geometry Homework 4 - 05.06.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. If X is a smooth manifold, let ϕ : U X -→ D be a chart (like in HW 3, we toke ϕ := φ - 1 the inverse of the chart from the notion in class). Define the component functions ( x 1 , . . . , x n ) of ϕ by ϕ ( p ) = ( x 1 ( p ) , . . . , x n ( p )) , as we explained in HW 3, any tangent vector V T x ( X ) can be expressed in terms of the coordinate basis as V = n i =1 v i ∂x i x for some n -tuple v = ( v 1 , . . . , v n ) . Define a bijection η : π - 1 ( U ) X -→ U × R n by sending V T x ( X ) to ( x, v ) . Where two coordinate charts ( x 1 , . . . , x n ) and x 1 , . . . , ˜ x n ) overlap, the respective coordinate basis vectors are related by ∂x i =

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Unformatted text preview: j ∂x i ∂ ∂ ˜ x j , and therefore the same vector V is represented by V = n X i =1 v i ∂ ∂x i = n X j =1 ˜ v j ∂ ∂ ˜ x j = n X i =1 n X j =1 v i ∂ ˜ x j ∂x i ∂ ∂ ˜ x j . This means that ˜ v j = n X j =1 v i ∂ ˜ x j ∂x i , so the corresponding local trivializations η and ˜ η are related by ˜ η ◦ η-1 ( x,v ) = ˜ η ( V ) = ( x, ˜ v ) = ( x,τ ( x ) v ) , where τ ( x ) is the GL ( n, R )- valued function ∂ ˜ x j /∂x i , hence receiving that the reqeusted maps are just the Jacobian matrix of the coordinate transition map. 1 2. Was 3. We 2...
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