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By Luther Pfahler Eisenhart

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Let us suppose that the general linear connection is regular. We denote by V the corresponding covariant derivation in the fibre bundle p'TM -+ V(M). Let X, Y, Z be three vector fields on V(M) over X, Y, Z belonging to T pz. 2) T(X, Y)= Vx Y-VfX-p[X,Y] Q(X, Y)Z= VxVf Z- V^ ViZ- V ^ Z t and Q being respectively the torsion and the curvature of V. They determine two torsion tensors, denoted by S and T, and three curvature tensors R, P, and Q. following the decomposition of vector fields into horizontal and vertical parts.

6). This is called the Berwald connection. We denote by H and G the corresponding curvature tensors. 5) is automatically satisfied, and D is none other than the Riemannian connection. 2) and on deriving twice vertically Sj G1 = G'J; 5$ G1 = G jk = Y jk. [this method was suggested to Berwald by Emmy Noether (see E. Cartan Oeuvres Completes [14] p. 1393)]. 8). In this case G jk is related to Y jk by Gjk = F jk + t'jk where tjk is a symmetric tensor, homogeneous of degree zero in v and satisfies t'ok = t'ko = 0- The choice of this tensor determines the connection F .

8) 5-, f 'Jk = Vk TV + VjTV. 9) P'jkl = V1 Tkj, - Vj fk, + TVr Vo Trj, - Trkj Vo T1,,. 14) Qijki=eijki = T i ri T r j k -T i rk T r j 1 Qojk, = Qioki = Qijoi = Quko = 0. Thus we have established the proposition. b. In paragraph 10 of chapter I we have defined the complete list of Bianchi identities for a general regular linear connection. These five identities become in the case of a Finslerian connection [1]. 19) V^R'k, + T r km RVi + Trlm Rijkr+ Vk P ^ - V, Pjjkm - (P jlr V0 T fkm - P'jkr V0 TV) + Qjrm R'okl = 0 V m Pjjkl - y , PVm + Vk Qjjlm + P'jrl T rkm - P ^ Trk, + Q'jrlV0Trkm - Q'jrm V0 Trkl = 0 SVmQV^O where we have denoted by Vm and V m the two covariant derivations in the Finslerian connection and by S the sum of terms obtained by permuting cyclically the indices (k, 1, m).

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An Introduction To Differential Geometry With Use Of Tensor Calculus by Luther Pfahler Eisenhart

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