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

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

2005-12-20

arxiv.org/abs/math/0512478v1

Mathematics

Differential Geometry

38 pages

Scientific paper

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.

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