On maximal mobility of linearly connected spaces \(L_n\)
Let \(L\) be the space of linear elements \((x,l)\), where \(x=(x^1, \dots, x^n)\) and \(l=(l^1, \dots, l^n)\) subjected to the change of variables \(\overline x^i=f^i(x)\), \(\overline l^k=\sum f_s^kl^s\) (abbreviating \(f^i_k= \partial f^i/ \partial x^k)\). The space \(L\) is equiped with linear connections \(\Gamma=(\Gamma^1_1, \dots, \Gamma^n_n)\) which are homogeneous of order one with respect to the coordinates \(l^1,\dots,l^n\), and subjected to the transformation rule \(\Gamma^i_j(\overline x,\overline l)=\sum f^i_kg^r_j \Gamma^k_r (x,l)-\sum f^i_{ks}g^k_j l^s\) where \(x^k=g^k (\overline x)\) is the inversion. The author deals with infinitesimal symmetries \(x=x+v(x)\delta t\) satisfying the Lie derivative condition \(L_v\Gamma=0\) which is made more explicit in terms of the covariant derivatives and the curvature tensor. He is interested in connections which admit the largest possible symmetry groups.
scientific article; zbMATH DE number 1262946