On special types of minimal and totally geodesic unit vector fields

Details On special types of minimal and totally geodesic unit vector fields On special types of minimal and totally geodesic unit vector fields

2005-09-30

arxiv.org/abs/math/0510002v1

Mathematics

Differential Geometry

15 Pages

Scientific paper

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_1S^n$.

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