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HW2 070909

# HW2 070909 - X be a chart containing x ∈ X since this is...

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Differential Geometry Homework 2 - 22.05.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. We will show an example that prove the claim is not true. Let G the projection function: G : R 2 -→ R ± x y ² 7-→ x The jacobian matrix: (1 0) is of rank 1, so by deﬁnition: X := ³± x y ² R 2 ´ ´ ´ ´ G ±± x y ²² = 0 µ = ³± 0 y ² R 2 ´ ´ ´ ´ y R µ = R is a submanifold of dimention one, moreover for every x X T x ( X ) = R . Let F be another function: F : R 2 -→ R ± x y ² 7-→ x 2 By the way we choose F , X := ³± x y ² R 2 ´ ´ ´ ´ G ±± x y ²² = 0 µ = ³± x y ² R 2 ´ ´ ´ ´ F ±± x y ²² = 0 µ , 1

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yet for x = ( 0 0 ) , a point in X calculating Ker ( DF x ) Ker ( DF x ) = ±² x y ³ R 2 ´ ´ ´ ´ DF x ²² x y ³³ = 0 µ which is Ker ( DF x ) = ±² x y ³ R 2 ´ ´ ´ ´ (0 , 0) ² x y ³ = 0 µ = R 2 2. Let v T x ( X ) and let f C ( X ) , and v is the class of curves γ : ( - ±,± ) -→ X such that γ (0) = x , withe ﬁrst deﬁnition: h f,v i = ( f γ ) 0 (0) = d dt ( f γ ) ( t ) ´ ´ ´ ´ t =0 . Let φ : D -→ U
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Unformatted text preview: X be a chart containing x ∈ X , since this is a diffeomorphism: ( f ◦ γ ) (0) = ( f ◦ φ ◦ φ-1 ◦ γ ) (0) = ( ( f ◦ φ ) ◦ ( φ-1 ◦ γ )) (0) , hence by the chain rule ( ( f ◦ φ ) ◦ ( φ-1 ◦ γ )) (0) = ( f ◦ φ ) ( φ-1 ( γ (0))) D ( φ-1 ◦ γ ) (0) . 3. Using the terms of the last question, choosing the deﬁnition: h f,v i = ( f ◦ γ ) (0) = d dt ( f ◦ γ ) ( t ) ´ ´ ´ ´ t =0 , we can apply Leibnitz rule on the designated expression: h fg,v i = ( fg ◦ γ ) (0) = (( f ◦ γ ) ( g ◦ γ )) (0) = = g ( γ (0)) ( f ◦ γ ) (0) + f ( γ (0)) ( g ◦ γ ) (0) = = g ( x ) h f,v i + f ( x ) h g,v i 2...
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HW2 070909 - X be a chart containing x ∈ X since this is...

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