Geometriae Dedicata

, Volume 168, Issue 1, pp 177–196 | Cite as

A topological splitting theorem for sub-Riemannian manifolds

  • Kazuki Itoh
Original Paper


We prove an analogue of the Cheeger–Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.


Sub-Riemannian manifold Measure contraction property  Splitting theorem 

Mathematics Subject Classification

53C21 (53C20 53C23) 



The author thanks Prof. Takashi Shioya for helpful advices especially giving him an idea and a method to study a splitting theorem.


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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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