differential geometry w notes from teacher_Part_76

# differential geometry w notes from teacher_Part_76 - 151...

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6.2. TENSOR ANALYSIS 151 The covariant derivative of a 1-form has the form ( i α ) k = e i ( α k ) - ω l ki α l In the coordinate frame this simplifies to ( i α ) k = i α k - ω l ki α l . The covariant derivative of a tensor of type ( p , q ) in a coordinate basis has the form i T j 1 ... j p k 1 ... k q = i T j 1 ... j p k 1 ... k q + p m = 1 ω j m li T j 1 ... j m - 1 l j m + 1 ... j p k 1 ... k q - q n = 1 ω l k n i T j 1 ... j p k 1 ... k n - 1 lk n k q The parallel transport of tensor fields is defined similarly to vector fields. Let C be a smooth curve on a manifold M described locally by x i = x i ( t ), where t [0 , 1], with the tangent vector X = ˙ x ( t ). Let T be a tensor field on M . We say that T is parallel transported along C if X T = 0 . 6.2.2 Ricci Identities The commutators of covariant derivatives of tensors are expressed in terms of the curvature and the torsion. In a coordinate basis for a torsion-free connection we have the following identities (called the Ricci identities): [ i , j ] Y k = R k li j Y l [ i , j ]

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