Affine connections of non integrable distributions

TL;DR

This paper investigates the geometry of non integrable distributions in Riemannian manifolds under various affine connections, deriving fundamental equations and inequalities, and providing new examples of Einstein and constant scalar curvature distributions.

Contribution

It derives Gauss, Codazzi, and Ricci equations for non integrable distributions with semi-symmetric and statistical connections, and introduces new examples of Einstein and constant scalar curvature distributions.

Findings

Derived fundamental equations for non integrable distributions with various connections.

Established Chen's inequalities for these distributions in space forms.

Presented new examples of Einstein distributions and distributions with constant scalar curvature.

Abstract

In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen's inequalities for non integrable distributions of real space forms endowed with a semi-symmetric metric connection and a semi-symmetric non-metric connection. We give some examples of non integrable distributions in a Riemannian manifold with affine connections. We find some new examples of Einstein distributions and distributions with constant scalar curvature.