Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(1): 63-82

Printed January 1, 2011

https://doi.org/10.4134/JKMS.2011.48.1.63

Copyright © The Korean Mathematical Society.

Metric foliations on hyperbolic spaces

Kyung Bai Lee and Seunghun Yi

University of Oklahoma, Youngdong University

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 $z: [0,\pi]\to S^{n-1}$ 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

MSC numbers: 53C12, 53C20, 57R30