Transverse totally geodesic submanifolds of the tangent bundle

2005/03/24 by Mohamed Tahar Kadaoui Abbassi, Alexander Yampolsky
Mathematics · #math.DG #msc:53B25 #msc:53C42

paper · pdf

published as Publ. Math. Debrecen 64/1-2 (2004), 129-154

arxiv created 2005/03/24 · arxiv updated 2009/12/01

Abstract

It is well-known that if ξ is a smooth vector field on a given Riemannian manifold Mⁿ then ξ naturally defines a submanifold ξ(Mⁿ) transverse to the fibers of the tangent bundle TMⁿ with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We show that a transverse submanifold N^l of TMⁿ (1 ≤ l ≤ n) can be realized locally as the image of a submanifold F^l of Mⁿ under some vector field ξ defined along F^l. For such images ξ(F^l), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l=n, and we treat several special cases (ξ of constant length, ξ normal to F^l, Mⁿ of constant curvature, Mⁿ a Lie group and ξ a left invariant vector field)

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