By Oskar Perron
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Extra info for Die Lehre von den Kettenbruchen. Dritte Auflage.
Such a coordinate system P t , (xl,.. ,xn)may be called preferred and from these preferred systems one can pass to the Cartesian coordinate systems as we have done in the above discussion of the Euclidean metric space. A Riemann space is said to be locally flat if an arbitrary point of the space is contained in some allowable coordinate system relative to which the values are assumed by the components of the fundamental metric tensor. The well known condition for a Riemann space to be locally flat is the vanishing of its curvature tensor (defined in Remark 1 in Sect.
In the following remarks we have given a few examples of formulae of covariant differentiation which are of frequent occurrence in the geometrical and physical applications of this theory. Remark 1. As special formulae of covariant differentiation we have according as V is a covariant or a contravariant vector. 10) for comparison. 9. e. the components of the covariant derivative of the fundamental metric tensor of a Riemann space are equal to zero. 18) with respect to the coordinates yp and yq and then evaluating the resulting equations at the origin of the normal coordinate system.
12) can be applied to give the divergence of such a vector. 14) 44 9. e. 14) is a relative scalar of weight 1. It follows immediately from this result and the result in Remark 1 regarding the vanishing of the covariant derivative of the metric tensor and the covariant derivative of the relative scalar Vgof an oriented Riemann space that the quantity defines a relative scalar of weight 1 in the oriented space provided the V , are the components of a covariant vector. Remark 3. For the purpose of defining the normal coordinate system on which the above theory of covariant differentiation and extension of tensors is based we have assumed the analyticity of the components and of the allowable coordinate transformations.