On a type of semi-sub-Riemannian connection on a sub-Riemannian manifold
Abstract
The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla\rightarrow D$ is obtained. The authors then further deduce a sufficient and necessary condition that a sub-Riemannian manifold associated with a semi-sub-Riemannian connection is flat, and derive that a sub-Riemannian manifold with vanishing curvature with respect to semi-sub-Riemannnian connection $D$ is a group manifold if and only if it is of constant curvature.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2013
- DOI:
- 10.48550/arXiv.1306.4250
- arXiv:
- arXiv:1306.4250
- Bibcode:
- 2013arXiv1306.4250H
- Keywords:
-
- Mathematics - Differential Geometry
- E-Print:
- 18 pages