Automorphisms of Riemann–Cartan manifolds

Abstract

It is proved that the maximal dimension of the Lie group of automorphisms of an n-dimensional Riemann–Cartan manifold (space) (Mn, g, \(\tilde \nabla \)) equals n(n − 1)/2+ 1 for n > 4 and, if the connection \(\tilde \nabla \) is semisymmetric, for n ≥ 2. If n = 3, then the maximal dimension of the group equals 6. Three-dimensional Riemann–Cartan spaces (M3, g, \(\tilde \nabla \)) with automorphism group of maximal dimension are studied: the torsion s and the curvature \(\tilde k\) are introduced, and it is proved that s and \(\tilde k\) are characteristic constants of the space and \(\tilde k\) = k − s2, where k is the sectional curvature of the Riemannian space (M3, g); a geometric interpretation of torsion is given. For Riemann–Cartan spaces with antisymmetric connection, the notion of torsion at a given point in a given three-dimensional direction is introduced.