Totally geodesic surfaces in the complex quadric

We provide explicit descriptions of all totally geodesic surfaces of a complex quadric of arbitrary dimension. Totally geodesic submanifolds of complex quadrics were first studied by Chen and Nagano in 1977 and fully classified by Klein in 2008. In particular, we interpret some of these surfaces as Gaussian images of surfaces in a unit three-sphere and all others as elements of the Veronese sequence introduced by Bolton, Jensen, Rigoli and Woodward. We also briefly discuss how the classification can be translated to the non-compact dual of the complex quadric, namely the hyperbolic complex quadric.