open access

https://doi.org/10.14943/84080

https://hdl.handle.net/2115/69740

Horo-tight spheres in Hyperbolic space

英語

2009-01-14

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

932

1-17

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by T. Cecil and P. Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight if only one of its total absolute curvature attends its minimum. We prove a characterization theorem for weak horo-tight hyperspheres.

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