# A course of mathematical analysis by Khinchin A. By Khinchin A.

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Example text

Since the vector eo and the curve p(t) were arbitrarily chosen, the leaves of the foliation Al> passing through the points p(O) and p(I), are parallel, and subsequently each of them is autoparallel. Using the same method, one easily prove the theorem for the leaves of the foliation A2. 59) imply that e;, = e (eDlt=t= d(e; + e~) + (e{ + e4)w~ = 0, and a vector tangent to a leaf F3 of the third foliation is defined by the condition e~ + e~ = 0, then the previous statement also valid for the third foliation A3 .

101 ) 36 1. , a ",J°kl . = a",J,°kl - . 101), satisfy the following conditions: *i ~[jkll _ - 0 , *i ~jkl *i *i + ~jkl + gjkl = 0. 103) The tensors ~;kl are covariant derivatives of the torsion tensor a~k with respect to the middle connection f (see Problem 19). Consider the following identity: which can be easily verified. 104) This is the desired decomposition of the curvature tensor b~kl. Thus, the curvature tensor b~kl is represented as the sum of the summands of three types: the first is expressed in terms of the torsion tensor, the second is expressed in terms of the covariant derivatives of the torsion tensor with respect to the middle * and the third does not depend on the torsion tensor.

A",J,°kl - . 101), satisfy the following conditions: *i ~[jkll _ - 0 , *i ~jkl *i *i + ~jkl + gjkl = 0. 103) The tensors ~;kl are covariant derivatives of the torsion tensor a~k with respect to the middle connection f (see Problem 19). Consider the following identity: which can be easily verified. 104) This is the desired decomposition of the curvature tensor b~kl. Thus, the curvature tensor b~kl is represented as the sum of the summands of three types: the first is expressed in terms of the torsion tensor, the second is expressed in terms of the covariant derivatives of the torsion tensor with respect to the middle * and the third does not depend on the torsion tensor.