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

*(English)* Zbl 1095.53026
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2–10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 292-306 (2006).

Summary: We present a new equation with respect to a unit vector field on a Riemannian manifold ${M}^{n}$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasakian 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 ${T}_{1}{S}^{n}$.

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C43 | Differential geometric aspects of harmonic maps |