Parallel submanifolds of the real 2-Grassmannian

TL;DR

This paper classifies complete parallel submanifolds of the oriented 2-plane Grassmannian, showing they are contained in totally geodesic submanifolds, extending to non-compact duals.

Contribution

It provides a classification of parallel submanifolds in the Grassmannian, revealing their structure as symmetric submanifolds within totally geodesic subspaces.

Findings

Complete parallel submanifolds are contained in totally geodesic submanifolds.

The classification applies to both compact and non-compact duals.

Non-curve submanifolds are characterized as symmetric submanifolds.

Abstract

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,. Our main result states that every complete parallel submanifold of $\rmG^+_2(\R^{n+2})$\,, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of $\rmG^+_2(\R^{n+2})$\,.