differential geometry w notes from teacher_Part_74

differential geometry w notes from teacher_Part_74 - ± •...

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6.1. AFFINE CONNECTION 147 Theorem 6.1.2 The components of the curvature tensor have the form R i jkl = ω i jl | k - ω i jk | l + ω i mk ω m jl - ω i ml ω m jk - C m kl ω i jm . Proof : 1. In the coordinate frame the components of the curvature tensor are given by R i jkl = k ω i jl - l ω i jk + ω i mk ω m jl - ω i ml ω m jk . For a Riemannian manifold ( M , g ) the metric tensor g has the components g i j = g ( e i , e j ) = g μν e μ i e ν j . This metric is used to lower and raise the frame indices. Definition 6.1.5 Let ( M , g ) be a Riemannian manifold and be an a ffi ne connection on M. Then the connection is called compatible with the metric g if for any vector fields X , Y and Z it satisfies the condition Z ( g ( X , Y )) = g ( Z X , Y ) + g ( X , Z Y ) . An a ffi ne connection that is torsion-free and compatible with the metric is called the Levi-Civita connection . Wed define ω i jk = g im ω m jk C i jk = g im C m jk . The coe ffi cients of the Levi-Civita connection satisfy the equation g i j | k = ω i jk + ω jik . Theorem 6.1.3 Then ω i jk = 1 2 g i j | k + g ik | j - g jk | i + C ki j + C jik - C i jk . Proof : di ff geom.tex; April 12, 2006; 17:59; p. 146
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148 CHAPTER 6. CONNECTION AND CURVATURE 1. Direct calculation.
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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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