Conjugate points in the Grassmann manifold of a C*-algebra

Let Gr be a component of the Grassmann manifold of a C*-algebra, presented as the unitary orbit of a given orthogonal projection Gr=Gr(P). There are several natural connections on this manifold, and we first show that they all agree (in the presence of a finite trace in A, when we give Gr the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of in Gr for the spectral rectifiable distance, and also the conjugate tangent locus of in Gr along a geodesic. Furthermore, for each tangent vector V at P, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all. Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Área de Matemática; Argentina