HW5 120909 - Differential Geometry Homework 5 19.06.08 Hava...

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Homework 5 - 19.06.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —hshabtai@campus.haifa.ac.il— 1. If X is a smooth manifold, let us remaind some notions from the last home works. 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 )) . In any smooth chart, a k -form ω can be written locally as ω = X I ω I dx i 1 ... dx i k where the coefficients ω I are continuous functions defined on the coordinate domain, determined by ω I = ω ± ∂x i 1 ,..., ∂x i k ² In order to prove d d = 0 we will first prove a short claim: If ω Ω k and η Ω l , then d ( ω η ) = η + ( - 1) k ω dη. By linearity it suffices to consider terms of the form ω = fdx i 1 ... dx i k and η = gdx j 1 ... dx j l . 1
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HW5 120909 - Differential Geometry Homework 5 19.06.08 Hava...

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