Circles in self dual symmetric $R$-spaces

Abstract

Self dual symmetric $R$-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group $G$ of a self dual symmetric $R$-space $M$ as those diffeomorphisms of $M$ sending circles in circles. Besides, although these curves belong to the realm of the invariants by $G$, we manage to describe them in Riemannian geometric terms: Given a circle $c$ in $M$, there is a maximal compact subgroup $K$ of $G$ such that $c$, except for a projective reparametrization, is a diametrical geodesic in $M$ (or equivalently, a diagonal geodesic in a maximal totally geodesic flat torus of $M$), provided that $M$ carries the canonical symmetric $K$-invariant metric. We include examples for the complex quadric and the split standard or isotropic Grassmannians.