Invariant totally geodesic unit vector fields on three-dimensional Lie groups

2005/12/20 by Yampolsky, Alexander
#53B20 #53B25 #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.math/0512478

Abstract

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

Related

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