An important problem in Riemannian geometry is to investigate the relation between topology an geometric structure on Riemannian manifolds. The celebrated theorem of S. B. Myers (Duke Math. J. 8 (1941)) guarantees the compactness of a complete Riemannian manifold under some positive lower bound on the Ricci curvature. This theorem may be considered as a topological obstruction for a complete Riemannian manifold to have a positive lower bound on the Ricci curvature. On the other hand, J. Lohkamp (Ann. Math. 140 (1944)) proved that in dimension al least three, any manifold admits a complete Riemannian metric of negative curvature. To give an interesting compactness criterion for complete Riemannian manifolds is one of the most important problems in Riemannian geometry, and the Myers theorem has been widely generalized in various directions by many authors.
The aim of this talk is to discuss the compactness of transverse Ricci solitons. The notion of Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where Ricci solitons played crucial roles in his afirmative resolution of the Poincaré conjecture.
In my talk, I would like to generalize the notion of Ricci solitons to the case of a sub-Riemann geometry, and define the notion of transverse Ricci solitons for Riemannian foliations on K-contact manifolds. Afther we have reviewed basic facts on Myers-type theorems for Ricci solitons, we shall establish some new Myers-type theorems due to W. Ambrose (Duke Math. J. 24 (1957)), J. Cheeger, M. Gromov, and M. Taylor (J. Diff. Geom. 17 (1982)), M. Fernández-López and E. García-Río (Math. Ann. 340 (2008)), M. Limoncu (Arch. Math. (Basel)) 95 (2010), Math. Z. 271 (2012)), Z. Qian (Q. J. math. 48 (1997)), and the author (Diff. Geom. Appl. 44 (2016), Pacific J. Math. 294 (2018)).
© Homare Tadano.