differential geometry w notes from teacher_Part_77

differential geometry w notes from teacher_Part_77 - 1 R i...

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6.2. TENSOR ANALYSIS 153 Proof : 1. 6.2.4 Properties of the Curvature Tensor Let ( M , g ) be an n -dimensional Riemannian manifold. We will restrict our- selves to the Levi-Civita connection below. We define some new curvature tensors. The Ricci tensor R i j = R k ik j . The scalar curvature R = g i j R i j = g i j R k ik j . The Einstein tensor G i j = R i j - 1 2 g i j R . The trace-free Ricci tensor E i j = R i j - 1 n g i j R . The Weyl tensor (for n > 2) C i j kl = R i j kl - 4 n - 2 R [ i [ k δ j ] l ] + 2 ( n - 1)( n - 2) R δ [ i [ k δ j ] l ] = R i j kl - 4 n - 2 E [ i [ k δ j ] l ] - 2 n ( n - 1) R δ [ i [ k δ j ] l ] di ff geom.tex; April 12, 2006; 17:59; p. 152

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154 CHAPTER 6. CONNECTION AND CURVATURE Theorem 6.2.2 The Riemann curvature tensor of the Levi-Civita con- nection has the following symmetry properties
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Unformatted text preview: 1. R i jkl =-R i jlk 2. R i jkl =-R jikl 3. R i jkl = R kli j 4. R i [ jkl ] = R i jkl + R i kl j + R i l jk = 5. R i j = R ji Proof : 1. ± • Theorem 6.2.3 The Weyl tensor has the same symmetry properties as the Riemann tensor and all its contractions vanish, that is, C i jik = . Proof : 1. ± • Theorem 6.2.4 The number of algebraically independent components of the Riemann tensor of the Levi-Civita connection is equal to n 2 ( n 2-1) 12 . Proof : 1. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 153...
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