Symmetric spaces with convex metrics

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We develop the basic theory of a general class of symmetric spaces, called lineated symmetric spaces, that satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry. Our primary interest is the case that these symmetric spaces are Banach manifolds, in which case they exhibit an interesting geometric structure, and particularly in the metric case, where it is assumed the symmetric space carries a convex metric, an invariant complete metric contracting the square root function. One major result is that the distance function between points evolving over time on two geodesics is a convex function. Primary examples arise from involutive Banach-Lie groups (G,σ) admitting a polar decomposition G = P · K, where K is the subgroup fixed by σ and P is the associated symmetric set. We consider an appropriate notion of seminegative curvature for such symmetric spaces endowed with an invariant Finsler metric and prove that the corresponding length metric must be a convex metric. The preceding results provide a general framework for the interesting Finsler geometry of the space of positive Hermitian elements of a C*-algebra that has emerged in recent years. © Walter de Gruyter 2007.

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Forum Mathematicum

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