On Translation Curves and Geodesics in <inline-formula><math display="inline"><semantics><msubsup><mrow><mi>Sol</mi></mrow><mn>1</mn><mn>4</mn></msubsup></semantics></math></inline-formula>

Abstract

A translation curve in a homogeneous space is a curve such that for a given unit vector at the origin, translation of this vector is tangent to the curve in its every point. Translation curves coincide with geodesics in most Thurston spaces, but not in twisted product Thurston spaces. Moreover, translation curves often seem more intuitive and simpler than geodesics. In this paper, we determine translation curves in Sol14 space. Their curvature properties are discussed and translation spheres are presented. Finally, characterization of geodesics in Sol14 space is given.

Keywords

• translation curve
• geodesic
• solvable Lie group
• Sol 1 4 space%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="inline-formula"> <mml:math id="mm1653468"> <mml:semantics> <mml:msubsup> <mml:mi>Sol</mml:mi> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </named-content> space