Darboux-Egorov system, bi-flat \(F\)-manifolds and Painlevé. VI. (Q2878729)

The author generalizes here the construction of semisimple bi-flat \(F\)-manifolds, initiated in [\textit{A. Arsie} and the author, J. Geom. Phys. 70, 98--116 (2013; Zbl 1283.53077)], using homogeneous solutions of degree 1 of the Darboux-Egorov system (see [\textit{G. Darboux}, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Deuxième édition, complétée. Paris: Gauthier-Villars (1910; JFM 41.0674.04)] and [\textit{D. Th. Egorov}, Collected Papers on Differential Geometry, Moscow: Nauka (1970) (in Russian)]). The Lamé coefficients \(H_i\) remain homogeneous functions of degree \(d_i\), but in this paper the general case is considered, i.e., the case when \(d_i\neq d_j\). Consequently, the rotation coefficients \(\beta_{ij}\) are homogeneous functions of degree \(d_i-d_j-1\). It follows that in this manner one can obtain any semisimple bi-flat \(F\)-manifold, provided that a supplementary assumption is satisfied. The author shows that, starting from generic solutions of Painlevé VI, one can construct three-dimensional, semisimple, bi-flat \(F\)-manifolds. More exactly, he discusses how the solutions of the Darboux-Egorov system are related to the sigma form of Painlevé VI. From Darboux-Egorov to Painlevé VI the proof is based on \textit{H. Aratyn} and \textit{J. Van De Leur} [Int. Math. Res. Not. 2008, Article ID rnn080, 41 p. (2008; Zbl 1195.34136)], and in the other direction, the author extends the proof of Arsie and Lorenzoni [loc. cit.] to the case when \(d_i=d_j\).