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Unformatted text preview: ± • The coe ﬃ cients of the Levi-Civita 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 Levi-Civita 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 Levi-Civita connection in an orthonormal frame have the form ω i jk = 1 2 ± C ki j + C jik-C 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 Levi-Civita connection. Proof : 1. By construction. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 147...
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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.
- Spring '10