Automorphisms of Riemann--Cartan Manifolds

V. I. Panzhenskij

Matematičeskie zametki, Tome 98 (2015) no. 4, pp. 544-556

Résumé

It is proved that the maximal dimension of the Lie group of automorphisms of an $n$-dimensional Riemann–Cartan manifold (space) $(M^{n},g,\widetilde{\nabla})$ equals ${n(n-1)}/2+1$ for $n>4$ and, if the connection $\widetilde{\nabla}$ is semisymmetric, for $n\geqslant2$. If $n=3$, then the maximal dimension of the group equals 6. Three-dimensional Riemann–Cartan spaces $(M^{3},g,\widetilde{\nabla})$ with automorphism group of maximal dimension are studied: the torsion $s$ and the curvature $\widetilde{k}$ are introduced, and it is proved that $s$ and $\widetilde{k}$ are characteristic constants of the space and $\widetilde{k}=k-s^{2}$, where $k$ is the sectional curvature of the Riemannian space $(M^{3},g)$; 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.

Détail

Keywords: Riemann–Cartan manifold, Lie group of automorphisms, curvature.
Mots-clés : automorphism group of maximal dimension, torsion