On the Kobayashi-Royden metric for ellipsoids

The Kobayashi indicatrix (infinitesimal unit ball) of a domain in IE n is known to be a biholomorphic invariant. In particular, if a domain is biholomorphic to a ball, then the indicatrix is the ball. Until the recent deep results of Lempert [4], it was not known to what extent the indicatrix characterizes the domain. Sibony had shown earlier that the indicatrix of any pseudoconvex circular domain is the domain itself [10]; hence the indicatrix determines such domains up to biholomorphism. This is not true in general. Lempert has shown that, for smoothly bounded strictly convex domains, three invariants determine the domain up to biholomorphism and a linear change of variables at a base point. These invariants consist of not only the Kobayashi indicatrix but also a quadratic form and a hermitian form on a certain vector bundle. In two dimensions, there is an open set in a Frechet space, corresponding to domains not biholomorphic to the ball but with indicatrix the ball. The result of this paper is that, for ellipsoids, if the Kobayashi indicatrix is the ball then the ellipsoid is biholomorphic to the ball. This is proved in the following equivalent form: if an ellipsoid symmetric about 0 is not biholomorphic to the ball, then its infinitesimal Kobayashi metric at 0 is not hermitian. In this language, our result answers a question posed by Reiffen in [7] about the differential-geometric nature of the Carath6odory metric for ellipsoids. By two results of Lempert ([3] and [2]), for strictly convex domains the Carathe6dory and Kobayashi metrics coincide and determine a Finsler metric. One now sees that, for ellipsoids, the metric is not hermitian and hence, by a theorem of Reiffen [7], not Riemannian. We prove that the infinitesimal Kobayashi metric at 0 is not hermitian by proving that the parallelogram law fails to hold. An ellipsoid ~ may be written (via a linear change of coordinates) as )-], ] zj] 2 ..~ ,~j Re (z 2) < , 0 < Aj < 1. By a