Height functions on compact symmetric spaces

TL;DR

This paper studies height functions on compact symmetric spaces, analyzing their critical points, gradient flows, and providing explicit local charts, with a focus on the relationship between the space and its embedding in matrix Lie groups.

Contribution

It introduces a method to analyze height functions on symmetric spaces using a generalized Cayley transform and establishes a polar decomposition adapted to automorphisms.

Findings

Critical sets of height functions are characterized.

Gradient flows can be integrated explicitly via a generalized Cayley transform.

Reduction to a standard height function case is achieved through a specialized polar decomposition.

Abstract

We consider height functions on symmetric spaces $M\cong G/K$ embedded in the associated matrix Lie group $G$. In particular we study the relationship between the critical sets of the height function on $G$ and its restriction to $M$. Also we prove that the gradient flow on $M$ can be integrated by means of a generalized Cayley transform. This allows to obtain explicit local charts for the critical submanifolds. Finally, we discuss how to reduce the generic case to a height function whose ground hyperplane is orhogonal to a real diagonal matrix. This result requires to prove the existence of a polar decomposition adapted to the automorphism defining $M$. Detailed examples are given.