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9.2. Prove that the Riemann curvature tensor has the following symmetry properties: (a) Rijk = -R/ki, hence R = = 0; (b) RkRki + Rk²ij = 0; (c) if Rimjk =ΣR/jk81m then Rimjk (d) Rimjk=Rjkim = - Rmijk; and 9.3. Compute the Riemann curvature tensor for M (a) extrinsically; = S2 (b) intrinsically. (Hint: Use Problem 9.2 to cut down on the calcula- tions.)

9.2. Prove that the Riemann curvature tensor has the following symmetry properties: (a) Rijk = -R/ki, hence R = = 0; (b) RkRki + Rk²ij = 0; (c) if Rimjk =ΣR/jk81m then Rimjk (d) Rimjk=Rjkim = - Rmijk; and 9.3. Compute the Riemann curvature tensor for M (a) extrinsically; = S2 (b) intrinsically. (Hint: Use Problem 9.2 to cut down on the calcula- tions.)

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 18E
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9.2. Prove that the Riemann curvature tensor has the following symmetry properties: (a) Rijk = -R/ki, hence R = = 0; (b) RkRki + Rk²ij = 0; (c) if Rimjk =ΣR/jk81m then Rimjk (d) Rimjk=Rjkim = - Rmijk; and 9.3. Compute the Riemann curvature tensor for M (a) extrinsically; = S2 (b) intrinsically. (Hint: Use Problem 9.2 to cut down on the calcula- tions.)
Transcribed Image Text:9.2. Prove that the Riemann curvature tensor has the following symmetry properties: (a) Rijk = -R/ki, hence R = = 0; (b) RkRki + Rk²ij = 0; (c) if Rimjk =ΣR/jk81m then Rimjk (d) Rimjk=Rjkim = - Rmijk; and 9.3. Compute the Riemann curvature tensor for M (a) extrinsically; = S2 (b) intrinsically. (Hint: Use Problem 9.2 to cut down on the calcula- tions.)
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