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On hyperbolic manifolds neither totally geodesic (many authors) nor
totally umbilical (Langevin-Walczak) foliations of codimension \(1\)
exist. Totally geodesic foliations of the real hyperbolic space
\(\mathbb H^n\) were classified by Ferus using curvature of orthogonal
transversal. The next geometric step towards it was done by Browne
who weakened these conditions. Lastly, Lee and Yi gave a boundary
classification concerning centers of leaf ideal boundaries.
Using the Sitter space \(\Lambda^{n+1}\) in the Lorentz space
understood as the space of \((n-1)\)-spheres in \(S^n\) we give a
conformal classification of totally umbilical codimension \(1\)
foliations of \(\mathbb H^n\) (in particular, for totally geodesic).
We prove that a curve in \(\Lambda^{n+1}\) represents a foliations of
(a domain in) \(\mathbb H^n\) iff its tangent vector belongs
everywhere to a boosted time cone (Shadok cone) and give geometric
interpretation in terms of the mean curvature of leaves and geodesic
curvature of an orthogonal transversal.
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References
- H. Browne, Codimension one totally geodesic foliations of \(H^n\) , Tohoku Math. Journ. 36 (1984), 315-340.
- D. Ferus, On isometric immersions between hyperbolic spaces , Math. Ann. 205 (1973), 193-200.
- R. Langevin, P. G. Walczak, Conformal geometry of foliations , Geom. Dedicata 132 (2008), 659-682.
- K. B. Lee, S. Yi, Metric foliations on hyperbolic spaces , J. Korean Math. Soc. 48(1) (2011), 63-82.
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