Non-degenerate surfaces of revolution in Minkowski space that satisfy the relation $aH+bK=c$

TL;DR

This paper classifies spacelike and timelike surfaces of revolution in Minkowski space satisfying a linear relation between mean and Gauss curvature, providing explicit solutions based on the axis's causal character.

Contribution

It offers a comprehensive classification of such surfaces in Minkowski space, including explicit first integrals for the generating curves.

Findings

Classification depends on the causal character of the axis

Explicit first integrals of the generating curves obtained

Surfaces satisfy a linear curvature relation $aH+bK=c$

Abstract

In this work, we study spacelike and timelike surfaces of revolution in Minkowski space $\e_{1}^{3}$ that satisfy $aH+bK=c$, where $H$ and $K$ denote the mean curvature and the Gauss curvature of the surface and $a$, $b$ and $c$ are constants. The classification depends on the causal character of the axis of revolution and in all the cases, we obtain a first integral of the equation of the generating curve of the surface.