On SNS-Riemannian connections in sub-Riemannian manifolds

TL;DR

This paper introduces SNS-Riemannian connections on nearly sub-Riemannian manifolds, exploring their geometric properties, curvature relations, and conditions for projective flatness, advancing understanding of sub-Riemannian geometry structures.

Contribution

It defines SNS-Riemannian connections, analyzes their properties, and establishes criteria for projective flatness in nearly sub-Riemannian manifolds.

Findings

Characterization of horizontal curvature tensors

Relationship between horizontal and SNS-Riemannian connections

Necessary and sufficient condition for projective flatness

Abstract

The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate the geometric characteristics of the projective SNS-Riemannian connection, and obtain a necessary and sufficient condition for a nearly sub-Riemannian manifold being projectively flat.