Thanks! By chance, do you know anything about the local variant: if there is an open set of planes whose sections are equivalent, then these sections are ellipses?

@Deane: I mean that there are linear maps between 2-planes that send cross-sections one to another. The group of self-equivalences of a cross-section is indeed a subgroup of $SL(2,\mathbb R)$.

Igor, a manifold without conjugate points can have some amount of positive Ricci curvature. There are 2-dimensional examples. The paper you cite claims that it should have negative Ricci curvature somewhere in a certain set.

@Dominic van der Zypen: Yes we have just finished it. It is on arxiv.org/abs/1506.06781

@David: Because we tested it on live analysts. It turns out that the very language of graph theory is not as widely understood as one might hope.

@bof: I think we are going to thank you in acknowledgements. Do you prefer to be mentioned as an anonym or by some real name?

Thanks! I tracked it from Mirski & Perfect's paper down to Dulmage & Mendelsohn, Coverings of bipartite graphs, Canad. J. Math. 10 (1958) 517-534, Theorem 1. And there (unlike the other sources) it is exactly it, not something that one has to combine with Hall's Theorem.

Yes, we have the very same argument. It would be 5 lines long if we were aiming at combinatorialists. But it took more than a page to make the text consumable by analysts.

@darij: Yes, of course we will prove it in the paper if there is no reference. It just looks too natural to be unknown.