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http://dx.doi.org/10.4134/JKMS.2011.48.1.063

METRIC FOLIATIONS ON HYPERBOLIC SPACES  

Lee, Kyung-Bai (DEPARTMENT OF MATHEMATICS UNIVERSITY OF OKLAHOMA)
Yi, Seung-Hun (SCIENCES AND LIBERAL ARTS-MATHEMATICS DIVISION YOUNGDONG UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.1, 2011 , pp. 63-82 More about this Journal
Abstract
On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.
Keywords
Riemannian foliation; metric foliation; homogeneous foliation; totally geodesic foliation; hyperbolic space;
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