METRIC FOLIATIONS ON HYPERBOLIC SPACES

Title & Authors
METRIC FOLIATIONS ON HYPERBOLIC SPACES
Lee, Kyung-Bai; Yi, Seung-Hun;

Abstract
On the hyperbolic space $\small{D^n}$, codimension-one totally geodesic foliations of class $\small{C^k}$ 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, $\small{\pi}$] $\small{\rightarrow}$ $\small{S^{n-1}}$ of class $\small{C^{k-1}}$ with z(0) = $\small{e_1}$ = z($\small{\pi}$) satisfying |z'(r)| $\small{{\leq}1}$ for all r, modulo an isometric action by O(n-1) $\small{{\times}\mathbb{R}{\times}\mathbb{Z}_2}$. Since 1-dimensional metric foliations on $\small{D^n}$ 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 $\small{D^2}$ (called "fifth-line") are found.
Keywords
Riemannian foliation;metric foliation;homogeneous foliation;totally geodesic foliation;hyperbolic space;
Language
English
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