Geometry of R^n actions on n-manifolds (2012)

By NTZung, on March 10th, 2012

Updated 17/03/2013: this paper has been accepted for publications in J. Math. Soc. Japan, after a small revision.

Apparently it has also been cited in a recent work by Ishida: arxiv.org/pdf/1302.0633

Updated 18/03/2012: 2nd version, which corrects a series of misprints and imprecisions in the 1st version.

After two months of intensive work, we have finished the paper on the geometry of nondegenerate Rn actions on n-manifolds. The pdf file of the article is available here: Rn_Actions_2012.pdf

Arxiv: http://arxiv.org/abs/1203.2765

This is a joint article with Nguyen Van Minh, who is a PhD student of mine.

Abstract: This paper is devoted to a systematic study of the geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

58 pages, 16 figures

Submitted to the preprint server arxiv on 10/March/2012

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The paper, though already 58 pages long and contains more than 20 theorems, still leaves a lot of interesting open questions and problems:

1) Show that nondegenerate Rn actions on n-manifolds are topologically structurally stable under small perturbations. (We believe that this fact is true and not too difficult to establish, but it needs some additional work).

2) Find necessary and sufficient conditions for a configuration on a 2-dimensional surface (a configuration here means a decomposition of the surface into polygones by a family of simple closed curves with transerval intersections) to be the configuration of a hyperbolic R2-actions.

3) Find onstructions (if any) for a manifold $M$ to admit a totally hyperbolic action. Does it have anything to do with the torsion of $H_1(M,{\mathbb Z})$ ? In particular, do 3-dimensional lens spaces other than ${\mathbb S}^3$ admit a totally hyperbolic action of ${\mathbb R}^3$ ?