Classification of translation surfaces in $\mathbb{R}^3$ with constant sectional curvature

TL;DR

This paper classifies translation surfaces with constant sectional curvature in Euclidean space equipped with a semi-symmetric non-metric connection, revealing they are generalized cylinders, similar to the Levi-Civita case but with broader curvature properties.

Contribution

It provides a complete classification of such surfaces under a semi-symmetric non-metric connection, extending known results from Levi-Civita geometry.

Findings

Translation surfaces of constant curvature are generalized cylinders.

In this setting, generalized cylinders can have constant or non-constant sectional curvature.

The classification aligns with Levi-Civita results but allows more curvature variation.

Abstract

In this paper, we study translation surfaces in the Euclidean space endowed with a canonical semi-symmetric non-metric connection. We completely classify the translation surfaces of constant sectional curvature with respect to this connection, proving that they are generalized cylinders. This consequence is the same as in the case of the Levi-Civita connection, but in this new setting, there are also generalized cylinders whose sectional curvature can be constant or non-constant, in contrast to the Levi-Civita connection, where the Gaussian curvature is zero.