Geodesics and Translation Curves in <inline-formula><math display="inline"><semantics><msubsup><mi>Sol</mi><mn>0</mn><mn>4</mn></msubsup></semantics></math></inline-formula>

Abstract

A translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces, translation curves seem more intuitive and simpler than geodesics. In this paper, geodesics and translation curves in Sol04 space are classified and the curvature properties of translation curves are investigated.

Keywords

• geodesic
• translation curve
• solvable Lie group
• Sol 0 4 space%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm310000"> <mml:semantics> <mml:msubsup> <mml:mi>Sol</mml:mi> <mml:mn>0</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content> space