Integrable Gradient Flows and Morse Theory

TL;DR

This paper explores Morse functions with integrable gradient flows on classical Riemannian manifolds, providing explicit descriptions of their topology and applications to Grassmannians.

Contribution

It demonstrates that generic height functions on symmetric embeddings of Lie groups are perfect Morse functions with explicitly describable gradient flows.

Findings

Generic height functions are perfect Morse functions on certain manifolds.

Explicit cell decompositions of manifolds are constructed.

Elementary proof of Vassiljev's theorem on flag joins of Grassmannians.

Abstract

Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the homology requires, and the corresponding gradient flow can be described explicitly. This gives an explicit cell decomposition and geometric realization of the homology for such a manifold. As another application of the integrable Morse functions we give an elementary proof of Vassiljev's theorem on the flag join of Grassmannians.