HW04 - Differential Geometry 1 Homework 04 1. Let ζ ∈...

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Unformatted text preview: Differential Geometry 1 Homework 04 1. Let ζ ∈ Vec(S 1 ) be the standard vector field given by (a, b) ∈ S 1 ⇒ ζ(a,b) = τ(a,b) (−b, a) and let Θ ∈ Ω1 (S 1 ) be the standard one-form given by Θ(ζ ) ≡ 1. (i) Find explicitly the integral curve γ of ζ through (a, b) ∈ S 1 . (ii) Show that if g ∈ C ∞ (S 1 ) and g Θ = df for some f ∈ C ∞ (S 1 ) then 2π g (γ (t))dt = 0. 0 2. Let ω ∈ Ωm (Rm+1 ) be defined by ωa (τa (v1 ), . . . , τa (vm )) = Det[a|v1 | · · · |vm ] for a, v1 , . . . , vm ∈ Rm+1 . Show that m (−1)i xi dx0 ∧ · · · ∧ dxi ∧ · · · ∧ dxm ω= i=0 in terms of the standard coordinates (x0 , · · · , xm ) on Rm+1 . 1 ...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.

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