differential geometry w notes from teacher_Part_80

differential geometry w notes from teacher_Part_80 - 159...

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6.3. CARTAN’S STRUCTURAL EQUATIONS 159 Since the Levi-Civita connection is compatible with the metric, we have 0 = i g ( e j , e k ) = g ( i e j , e k ) + g ( e j , i e k ) . Thus, ω k ji + ω jki = 0 . Finally, we obtain, Proposition 6.3.3 The coe cients of the Levi-Civita connection in an orthonormal frame are given in terms of the commutation coe cients by ω i jk = 1 2 ± C ki j + C jik - C i jk ² . Proof : 1. Use the equations ω ki j + ω ik j = 0 ω i jk + ω jik = 0 ω jki + ω k ji = 0 and ω k ji - ω k i j = C k i j . ± Now we define the connection 1 -forms A i j = ω i jk σ k and the curvature 2 -forms F i j = 1 2 R i jkl σ k σ l . Then the equation d σ i = ω i jk σ j σ k can be written as d σ i + A i j σ j = 0 . This is called Cartan’s first structural equation . di geom.tex; April 12, 2006; 17:59; p. 158
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160 CHAPTER 6. CONNECTION AND CURVATURE The curvature 2-forms are obtained from the connection 1-forms by Car- tan’s second structural equation
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_80 - 159...

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