Masahiko Kanai

This is a story about the cross ratio and its relatives which dwell in numbers of fields of mathematics, such as projective geometry, riemannian geometry, Hamiltonian dynamical systems, discrete differential geometry and so on and so forth. Among relatives of cross ratio are the Schwarzian derivative, geodesic currents and bipolarized symplectic structures.

The Schwarzian derivative, for instance, played an essential role in the proof of the theorem of Ghys on local rigidity of actions of surface groups. Geodesic current was initiated by Bonahon to reconstruct Thurston's compactification and the Weil-Peterson metric of the Teichmuller space, and was used by Otal to establish his marked length spectral rigidity. Also, one should remember a theorem of Navas on actions on the circles of groups with property (T), in which the notion of geodesic current plays an indispensable role. Another variant of cross ratio is what is called bipolarized symplectic structure (or equivalently, paraKaehler structures) which was employed by myself in the dynamics of geodesic flows of negatively curved manifolds.

In this talk, I would like to show a panoramic view of the lands on which those habitants are alive.