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

TL;DR

This paper investigates conjugate points and the exponential map in the Grassmann manifold of a $C^*$-algebra, revealing nuanced behaviors of conjugacy and establishing the agreement of natural connections.

Contribution

It demonstrates the equivalence of various natural connections on the Grassmann manifold and analyzes conjugate points and the exponential map in this setting.

Findings

All natural connections agree under a finite trace and Riemannian metric.

Characterization of the cut locus and conjugate points in the manifold.

Examples showing classical conjugate points may not be conjugate in this context.

Abstract

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 in this manifold, and we first show that they all agree (in the presence of a finite trace in $\mathcal A$, 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 $P\in Gr$ for the spectral rectifiable distance, and also the conjugate tangent locus of $P\in Gr$ 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.