Toda Flows and Isospectral Manifolds

Abstract

Abstract.We apply Bott's method to the calculation of Betti numbers of isospectral manifolds. Necessary properties of Toda flows, including a description of the phase portrait, are given with complete proofs.

The Full Symmetric Toda Flow and Intersections of Bruhat Cells

“…On the other hand, most part of the statements from the previous section remain true: the points w ∈ W (g, k) are invariant points of Toda system, their Bruhat cells and dual Bruhat cells (or their shifted versions) are unstable and stable manifolds of the Toda system (see [9]) and the paths γ α,w are invariant with respect to Toda flow.…”

“…In addition to these invariant curves we shall need the following result, which one can find, for instance in [7,9]: Bruhat cells are invariant manifolds of this system. In fact, one can make this statement more precise, bringing it to the following form.…”

Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies

“…Isospectral manifolds in connection with Toda flows (including non-Abelian generalizations) have attracted a lot of interest (see, e.g., [9], [14], [33], [68], [69], [70] and the references therein). In the present finite-gap case the situation is analogous to the (m)KdV case and briefly summarized below.…”