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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 $T_1S^n$. For the entire collection see [Zbl 1089.53004].
53C20Global Riemannian geometry, including pinching
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C43Differential geometric aspects of harmonic maps
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