Geometriae Dedicata

, Volume 136, Issue 1, pp 123–131 | Cite as

On convexity of hypersurfaces in the hyperbolic space

  • Konstantin RybnikovEmail author
Original Paper


In the Hyperbolic space \({\mathbb{H}^n}\) (n ≥ 3) there are uncountably many topological types of convex hypersurfaces. When is a locally convex hypersurface in \({\mathbb{H}^n}\) globally convex, that is, when does it bound a convex set? We prove that any locally convex proper embedding of an (n − 1)-dimensional connected manifold is the boundary of a convex set whenever the complement of (n − 1)-flats of the resulting hypersurface is connected.


Hyperbolic Convex Local convexity Hypersurface 

Mathematics Subject Classification (2000)

51M 52A 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.University of Massachusetts at LowellLowellUSA

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