Intrinsic measures and holomorphic retracts (Q1086714)

We examine the consequences of the equality of the Eisenman and Carathéodory norms on k-vectors, \(2\leq k\leq n-1,\) at a point p in an n- dimensional complex manifold M. We also investigate the consequences of the existence of a large number of two-dimensional holomorphic retracts of a complex manifold - one tangent to each 2-vector at p. If M is Carathéodory-hyperbolic then either assumption implies that the indicatrix of the Carathéodory metric at p is an ellipsoid. The first assumption also implies that the infinitesimal Kobayashi and Carathéodory metrics coincide at p. If M is hyperbolic then the second assumption implies that if the indicatrix of the Kobayashi-Royden metric at p is convex, then it must be an ellipsoid. Finally we give formulas and estimates for the Eisenman and Carathéodory norms on k-vectors at the origin of circular domains, exhibiting some differences between the \(k=1\) case and the \(k>1\) case.