# Circles in self dual symmetric R-spaces

## TL;DR

This paper characterizes the transformation group of self dual symmetric R-spaces by their circle-preserving diffeomorphisms and describes these special curves using Riemannian geometry, with examples from complex quadrics and Grassmannians.

## Contribution

It provides a characterization of the big transformation group of self dual symmetric R-spaces through circle-preserving diffeomorphisms and relates circles to geodesics in Riemannian metrics.

## Key findings

- Transformation group elements send circles to circles.
- Circles correspond to diametrical geodesics in a canonical metric.
- Examples include complex quadric and Grassmannian manifolds.

## 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.