Gabriel Larotonda

Geometry of infinite dimensional Grassmannians: exponential map and conjugate points

For a $C^*$-algebra $\mathcal A$, we consider $Gr(P)$ the unitary orbit of an orthogonal projection $P\in \mathcal A$, which is one of the connected components of the Grassmanian manifold of $\mathcal A$. In this mini-course we will first present its differentiable structure, and then we will give a unified presentation of the canonical linear connection that appears in several sources of the literature in $Gr(P)$. We will describe in detail its geodesics, exponential map and curvature tensor. We will then give a concise description of the conjugate points along geodesics. These come in two flavours: conjugate points in the metric sense (cut locus for the Finsler metric induced by the norm of $\mathcal A$), and conjugate points in the tangent sense (the differential of the exponential map along $\gamma$ is not invertible). Along the course, several tools from operator theory will be introduced and applied to this specific framework, which generalizes the classical Grassmann manifolds over $\mathbb R$ or $\mathbb C$.