Geometriae Dedicata

, Volume 203, Issue 1, pp 347–351 | Cite as

On open flat sets in spaces with bipolar comparison

  • Nina LebedevaEmail author
Original Paper


We show that if a Riemannian manifold satisfies (3,3)-bipolar comparisons and has an open flat subset then it is flat. The same holds for a version of MTW where the perpendicularity is dropped. In particular we get that the (3,3)-bipolar comparison is strictly stronger than the Alexandrov comparison.


Metric geometry Optimal transport Differential geometry Rigidity Comparison geometry 



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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Steklov InstitutePetersburgRussia

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