Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

TL;DR

This paper presents a method to compute the Riemannian metric and curvature tensor of symmetric spaces directly from their Satake diagrams, facilitating computer algebra applications and classification of submanifolds.

Contribution

It introduces algorithms to derive geometric tensors from Satake diagrams and provides a Maple package implementation for practical computations.

Findings

Computed tensors for various symmetric spaces

Classified totally geodesic submanifolds of SU(3)/SO(3)

Provided a computational tool for geometric analysis

Abstract

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with computer algebra systems. As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified. The submission also contains an example implementation of the algorithms and formulas of the paper as a package for Maple 10, the technical documentation for this implementation, and a worksheet carrying out the computations for the space SU(3)/SO(3) used in the proof of Proposition 6.1 of the paper.