On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations

Alexander Yampolsky

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On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations

Alexander Yampolsky

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Abstract

We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a totally umbilic hyperfoliation. A corresponding example shows the non-triviality of this condition. In the 2-dimensional case, we give a complete description of Riemannian manifolds admitting a geodesic unit vector field with totally geodesic property.

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BibTeX RIS

Alexander Yampolsky. 2005-05-10. On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations. https://arxiv.org/abs/math/0503566

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