Horo-tight spheres in Hyperbolic space
Preprint Series # 932
Buosi, Marcelo and Izumiya, Shyuichi and Maria Aparecida, Soares Ruas Horo-tight spheres in Hyperbolic space. (14 January 2009); (Submitted)
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.