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Unformatted text preview: ± • The coe ﬃ cients of the LeviCivita connection in a coordinate frame are called Christo ﬀ el symbols and denoted by Γ i jk . • Corollary 6.1.1 The coe ﬃ cients of the LeviCivita connection in a coordinate frame (Christo ﬀ el symbols) have the form Γ i jk = 1 2 g im ± ∂ j g mk + ∂ k g jm∂ m g jk ² . Christo ﬀ el symbols have the following symmetry property Γ i jk = Γ i k j . Proof : 1. Direct calculation. ± • Corollary 6.1.2 The coe ﬃ cients of the LeviCivita connection in an orthonormal frame have the form ω i jk = 1 2 ± C ki j + C jikC i jk ² . They have the following symmetry properties ω i jk =ω jik . Proof : 1. Direct calculation. ± • Theorem 6.1.4 Each Riemannian manifold has a unique LeviCivita connection. Proof : 1. By construction. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 147...
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 Spring '10
 Wong
 Geometry, General Relativity, Riemannian geometry, Γi jk, ωi jk

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