An Atlas Adapted to the Toda Flow

Abstract We describe an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra and on its Hessenberg-type submanifolds. We show that in the local coordinates of the atlas the Toda flow becomes linear. The local coordinates are also used to show that the Toda flow on the manifold of full flags is Morse–Smale, which generalizes the result for traceless matrices in [27] to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory.