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differential geometry w notes from teacher_Part_78

differential geometry w notes from teacher_Part_78 - These...

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6.2. TENSOR ANALYSIS 155 Corollary 6.2.2 In dimension n = 2 the Riemann tensor has only one independent component determined by the scalar curvature R 12 12 = 1 2 R . The trace-free Ricci tensor E i j vanishes, that is R i j kl = R δ [ i [ k δ j ] l ] R i j = 1 2 Rg i j . Proof : 1. Corollary 6.2.3 In dimension n = 3 the Riemann tensor has six in- dependent components determined by the Ricci tensor R i j . The Weyl tensor C i jkl vanishes, that is, R i j kl = 4 R [ i [ k δ j ] l ] + R δ [ i [ k δ j ] l ] . Proof : 1. 6.2.5 Bianci Identities Let ( M , g ) be an n -dimensional Riemannian manifold. We will restrict our- selves to the Levi-Civita connection below. Theorem 6.2.5 The Riemann tensor satisfies the following identities [ m R i j kl ] = m R i j kl + k R i j lm + l R i j mk
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Unformatted text preview: These identities are called the Bianci identities . Proof : 1. di ff geom.tex; April 12, 2006; 17:59; p. 154 156 CHAPTER 6. CONNECTION AND CURVATURE ± • Corollary 6.2.4 The divergences of the Riemann tensor and the Ricci tensor have the form ∇ i R i j kl = ∇ k R j l- ∇ l R j k , ∇ i R i j = 1 2 ∇ j R . The divergence of the Einstein tensor vanishes ∇ i G i j = . Proof : 1. ± • Problem. By using the Bianci identities simplify the Laplacian of the Rie-mann tensor, Δ R i j kl = ∇ m ∇ m R i j kl . di ff geom.tex; April 12, 2006; 17:59; p. 155...
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