On a type of semi-sub-Riemannian connection on a sub-Riemannian manifold

TL;DR

This paper introduces a new type of semi-symmetric metric connection on sub-Riemannian manifolds, explores their properties, and establishes conditions for flatness and constant curvature, linking geometric structures to group manifolds.

Contribution

It defines semi-sub-Riemannian connections, studies their relations with existing connections, and provides criteria for flatness and constant curvature in sub-Riemannian geometry.

Findings

An invariant under connection transformation is obtained.

A necessary and sufficient condition for flatness is established.

Sub-Riemannian manifolds with zero curvature are group manifolds if of constant curvature.

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.