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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)
  • Received : 2009.04.06
  • Accepted : 2010.06.29
  • Published : 2011.01.01

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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