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Hadamard foliations of Hn
by
Maciej Czarnecki
Uniwersytet Lodzki, Poland
Poster
We introduce the notion of Hadamard foliation as a foliation of simply connected complete Riemannian manifold of nonpositive curvature (Hadamard manifold) which leaves are Hadamard.
We prove that
(i) Any foliation of hyperbolic space Hn is Hadamard if its norm of the second fundamental form is less than 1.
(ii) For such a foliation there is the canonical mapping from the union of ideal boundaries of leaves into the ideal boundary of Hnn.
(iii) This mapping is continous and homeomorphism on the boundary of a unique leaf.
Some techniques of the hyperbolic geometry was developed. We obtain simple proofs of two facts:
1. Any curve in Hn defined on the ray and parametrized by arc length has a limit on the boundary provided that it has bounded curvature by the number less than 1.
2. Any complete and connected submanifold of Hn with the norm of the second fundamental form less than 1 is simply connected.
Date received: June 29, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-36.