Geometric structures in $Sol_3$

TL;DR

This paper explores the geometric structures of the $Sol_3$ space using Levi-Civita and semi-symmetric non-metric connections, analyzing special cases and providing comprehensive conclusions.

Contribution

It introduces new analyses of $Sol_3$ geometric structures with semi-symmetric non-metric connections and compares them to Levi-Civita connection cases.

Findings

Complete characterization of geometric structures in $Sol_3$ with Levi-Civita connection.

Analysis of semi-symmetric non-metric connections in two special cases.

Conclusions on the differences and properties of these connections in $Sol_3$.

Abstract

In this paper, we conclude the geometric structures in $Sol_3$ with the Levi-Civita connection and the semi-symmetric non-metric connection. We discuss two special cases to study the geometric structures in $Sol_3$ with the semi-symmetric non-metric connection, namely $\widehat{P}=a(X_3)\partial_3$ and $\widehat{P}=b(X_3)\partial_1$, also get relatively complete conclusions respectively. As a special case, we investigate the geometric structures in $Sol_3$ with the Levi-Civita connection.