Constant sectional curvature surfaces with a semi-symmetric non-metric connection

TL;DR

This paper classifies surfaces in Euclidean space with a semi-symmetric non-metric connection that have constant sectional curvature, providing full classifications for cylindrical and rotational surfaces under various conditions.

Contribution

It offers the first complete classification of constant sectional curvature surfaces in Euclidean space with a semi-symmetric non-metric connection, including cylindrical and rotational cases.

Findings

Cylindrical surfaces with orthogonal rulings are classified.

Rotational surfaces have axes parallel to the vector field  and are classified.

Existence results for rotational surfaces with curvature /2 and orthogonal intersection.

Abstract

Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to this connection is constant. In case that the surface is cylindrical, we obtain full classification when the rulings are orthogonal or parallel to $\mathsf{C}$. If the surface is rotational, we prove that the rotation axis is parallel to $\mathsf{C}$ and we classify all conical rotational surfaces with constant sectional curvature. Finally, for the particular case $\frac12$ of the sectional curvature, the existence of rotational surfaces orthogonally intersecting the rotation axis is also obtained.