Isometries of Spaces with Torsion

Abstract

In this paper, we study automorphisms (isometries) in Riemann–Cartan spaces (spaces with torsion) of positive definite and alternating Riemannian metrics. We prove that if the connection is semisymmetric, then the maximal dimension of the Lie group of isometries of an n-dimensional space is equal to \( \frac{n\left(n-1\right)}{2}+1 \). If n = 3, then the maximal dimension of the group is equal to 6 and the connection of the maximally movable space is skew symmetric. In this case, the space has a constant curvature k and a constant torsion s, while the Ricci quadratic form is positive (negative) definite if and only if k > s² (respectively, k < s²) and is equal to zero if k = s². We construct a maximally movable stationary de Sitter model of the Universe with torsion and propose a geometric interpretation of the torsion of spatial sections.