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On special types of minimal and totally geodesic unit vector fields

Alexander Yampolsky
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Abstract

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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Alexander Yampolsky. 2005-09-30. On special types of minimal and totally geodesic unit vector fields. https://arxiv.org/abs/math/0510002

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