2005/03/31 by Alexander Yampolsky
Mathematics · #math.DG #msc:53B25 #msc:53C25
published as Publ. Math. Debrecen 63/4 (2003), 565 -- 567 · 13 pages
arxiv created 2005/05/10 · arxiv updated 2009/12/01
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.