HW3 070909 - Differential Geometry Homework 3 26.05.08 Hava...

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Differential Geometry Homework 3 - 26.05.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. Was proven in the previous home work. 2. Was proven in the previous home work. 3. We can use one-to-one correspondence in order to get an alternative definition of the Tangent Space. In that definition, for a smooth manifold X and a point x in X , the Tangent Space T x ( X ) is the set of all derivation of C ( X ) at x , where a linear map, F : C ( X ) -→ R is called a derivation at x if it satisfies F ( gh ) = g ( x ) F ( g ) + h ( x ) F ( h ) for all g,h C ( X ) . Now let ϕ : U X -→ D be a chart containing x (we toke ϕ := φ - 1 the inverse of the chart from the notion in class, since it is a diffeomorphism it is well defined, and it will be a great use in this question and the next) . Let us write a new sign ∂x i ± ± ± x that acts on a smooth function f : U -→ R by : ∂x i ± ± ± ± x f := ∂x i ( f ϕ
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This note was uploaded on 09/16/2009 for the course MATH 106723 taught by Professor Michaelpolyak during the Spring '09 term at Technion.

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HW3 070909 - Differential Geometry Homework 3 26.05.08 Hava...

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