Height functions on compact symmetric spaces

We consider height functions on symmetric spaces $M\cong G\mathrm{/}K$ 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.