differential geometry w notes from teacher_Part_79

differential geometry w notes from teacher_Part_79 - 157...

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6.3. CARTAN’S STRUCTURAL EQUATIONS 157 6.3 Cartan’s Structural Equations Let μ be a coordinate basis for the tangent bundle T M and e i = e μ i μ be an orthonormal frame of vector fields. Then g i j = g ( e i , e k ) = g μν e μ i e ν j = δ i j . We use this metric to lower and raise the frame indices. Let dx μ be a coordinate basis for the cotangent bundle T M and σ i = σ i μ dx μ be an orthonormal frame of 1-forms dual to e i . Then σ i ( e j ) = σ i μ e μ j = δ i j and g μν σ i μ σ j ν = δ i j σ i μ e ν i = δ ν μ . The commutators of the frame vector fields define the commutation coe - cients C i jk by [ e i , e j ] = C k i j e k , or, in components, e ν j | i - e ν i | j = e μ i μ e ν j - e μ j μ e ν i = C k i j e ν k . That is, C k i j = σ k ([ e i , e j ]) or C k i j = σ k ν ± e μ i μ e ν j - e μ j μ e ν i ² . Proposition 6.3.1 There holds d σ i = - 1 2 C i jk σ j σ k . or ( d σ i )( e j , e k ) = - 1 2 C i jk . Proof : di geom.tex; April 12, 2006; 17:59; p. 156
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158 CHAPTER 6. CONNECTION AND CURVATURE 1. Direct calculation using the duality condition. ± Let ω i jk be the coe cients of the Levi-Civita connection in the orthonormal
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differential geometry w notes from teacher_Part_79 - 157...

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