Invariant Totally Geodesic Unit Vector Fields on Three-Dimensional Lie Groups

Анотація

We give a complete list of left-invariant unit vector fields on three- dimensional Lie groups equipped with a left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group equipped with the Sasaki metric. As a result we obtain that each three-dimensional Lie group admits totally geodesic unit vector field under some conditions on structural constants. From a geometrical viewpoint, the field is either parallel or a characteristic vector field of a natural almost contact structure on the group.

Mathematics Subject Classification: 53B20, 53B25 (primary); 53C25 (secondary).