298 G. CORACH AND A.L. MAESTRIPIERI geodesics and the Finsler metric. Several additions to the literature are included, among them the variational characterization of horizontal lifts mentioned above. In Section 3 we introduce Thompson’s part metric and characterize the components of the cone A Dfa 2 A V a > 0g which carry a natural Banach homogeneous structure. In the case A D L.H/,where H is a Hilbert space, roughly speak- ing the smooth components correspond to closed subspaces of H.On them, we determine the Finsler metric and prove that it coincides with Thompson’s metric. The remaining component are parametrized by the non closed subspaces on H which are operator ranges. They are not Banach manifolds. However, we also calculate the Thompson’s metric on them. In Section 4 we present a brief survey of Uhlmann’s results and modify our constructions of Section 2 in order to compare both approaches. It is possible that this comparison may have some relevance to the study of the geometrical phase. 2. Differential Geometry of G This section contains a description of the differential geometry of G .Most of the results we mention are contained in the papers [10–12] but there is some new
Positivity – Springer Journals
Published: Oct 22, 2004
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