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.
Kyung Bai Lee and Seunghun Yi
University of Oklahoma, Youngdong University
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
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