Local minimal curves in homogeneous reductive spaces of the unitary group of a finite von Neumann algebra
Let \({\mathcal M}\) be a finite von Neumann algebra, \(U_{\mathcal M}\) its unitary group and \({\mathcal P}\) an (eventually infinite-dimensional) reductive space of \(U_{\mathcal M}\) endowed with a non complete Riemannian metric. This means that the theory of Riemann-Hilbert manifolds does not apply to \({\mathcal P}\). The author introduces a condition on \({\mathcal P}\) called orthogonality that is sufficient for to prove the following result (\({\mathcal V}_R(\rho)= \exp_{\rho}(B_R(0))\) for a ball \(B_R(0)\) and \(\rho \in{\mathcal P}\)): Theorem 3.8. Let \({\mathcal P}\) be an orthogonal homogeneous reductive space and \(\rho \in {\mathcal P}\). Given any \(\rho_1 \in{\mathcal V}_R(\rho)\), there exist a unique geodesic inside \({\mathcal V}_R(\rho)\) which has minimal length among all the piecewise smooth curves inside \({\mathcal V}_R(\rho)\), joining the points \(\rho\) and \(\rho_1\). The result is discussed for several examples.
scientific article; zbMATH DE number 5573974