Geometriae Dedicata

, Volume 154, Issue 1, pp 9–26 | Cite as

Horo-tight spheres in hyperbolic space

  • Marcelo Buosi
  • Shyuichi Izumiya
  • Maria Aparecida Soares Ruas
Original Paper


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 Cecil and 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.


Horo-tight immersion Sphere Hyperbolic space Horospherical geometry Totally absolute horospherical curvature 

Mathematics Subject Classification (2000)

53A35 53C99 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Marcelo Buosi
    • 1
  • Shyuichi Izumiya
    • 2
  • Maria Aparecida Soares Ruas
    • 3
  1. 1.Universidade Federal dos Vales do Jequitinhonha e MucuriDiamantinaBrazil
  2. 2.Department of MathematicsHokkaido UniversitySapporoJapan
  3. 3.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil

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