On the Kobayashi-Royden metric for ellipsoids (Q913042)

Let \({\mathcal E}\) be an ellipsoid in \({\mathbb{C}}^ n\), symmetric about 0. If the Kobayashi indicatrix of \({\mathcal E}\) at 0 is biholomorphic to the unit ball of \({\mathbb{C}}^ n\), then so is \({\mathcal E}\). Equivalently: If the infinitesimal Kobayashi-Royden metric of \({\mathcal E}\) at 0 is Hermitian, then \({\mathcal E}\) is biholomorphic to the ball.