Let D be the open disk of the
complex plane and T the unit circle. Let {Bei𝜃} be a family of Banach spaces
parametrized by the points ei𝜃 of T. The fundamental construction in the theory of
complex interpolation of Banach spaces produces from this data a family of Banach
spaces {Bz} which is parametrized by the points z of D and which has the given
{Bei𝜃} as boundary values. Basic facts about this construction are summarized in §2.
B =⋃z∈D{Bz} can be regarded as a complex vector bundle with base manifold D.
In this paper we study the differential geometry of B and related vector bundles. We
show relationships between interpolation theoretic inequalities for families of
Banach spaces and the signs of certain curvatures of the associated vector
bundles.
