Painleve-Calogero Correspondence Revisited

We extend the work of Fuchs, Painlev\'e and Manin on a Calogero-like expression of the sixth Painlev\'e equation (the ``Painlev\'e-Calogero correspondence'') to the other five Painlev\'e equations. The Calogero side of the sixth Painlev\'e equation is known to be a non-autonomous version of the (rank one) elliptic model of Inozemtsev's extended Calogero systems. The fifth and fourth Painlev\'e equations correspond to the hyperbolic and rational models in Inozemtsev's classification. Those corresponding to the third, second and first are seemingly new. We further extend the correspondence to the higher rank models, and obtain a ``multi-component'' version of the Painlev\'e equations.