Carathéodory metric

Let (X, || ||) be a complex Banach space and let B be the open unit ball in X.

Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry.

Let the Poincaré metric ρ on Δ be given by (thus fixing the curvature to be −4).

There is an associated notion of Carathéodory length for tangent vectors to the ball B.

Let x be a point of B and let v be a tangent vector to B at x; since B is the open unit ball in the vector space X, the tangent space TxB can be identified with X in a natural way, and v can be thought of as an element of X.