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

Abstract

Let

be a component of the Grassmann manifold of a $C^{⁎}$-algebra, presented as the unitary orbit of a given orthogonal projection

. 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

the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of

for the spectral rectifiable distance, and also the conjugate tangent locus of

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.