On the differential structure of matrix-valued rational inner functions

The present work is devoted to a differential-geometric study of rather classical objects in analysis namely inner matrix-valued functions. More precisely, we prove that the collection of such matrices with prescribed size and McMillan degree is an embedded submanifold in the Hardy space, two parametrizations of which are derived. We do this with an eye on linear control systems not only because several methods from system-theory are relevant to such a study but also because the geometry of inner matrix-valued functions in itself impinges on the parametrization of systems via the Douglas-Shapiro-Shields factorization of transfer-functions introduced in [29].