Open Access
July, 2014 Geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds
Nguyen VAN MINH, Nguyen Tien ZUNG
J. Math. Soc. Japan 66(3): 839-894 (July, 2014). DOI: 10.2969/jmsj/06630839


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.


Download Citation

Nguyen VAN MINH. Nguyen Tien ZUNG. "Geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds." J. Math. Soc. Japan 66 (3) 839 - 894, July, 2014.


Published: July, 2014
First available in Project Euclid: 24 July 2014

zbMATH: 1339.37043
MathSciNet: MR3238320
Digital Object Identifier: 10.2969/jmsj/06630839

Primary: 53C15
Secondary: 37C85 , 37J15 , 37J35 , 58K45 , 58K50

Keywords: complete fan , elbolic , group action , Hyperbolic , integrable system , Monodromy , normal form , reflection principle , toric manifold

Rights: Copyright © 2014 Mathematical Society of Japan

Vol.66 • No. 3 • July, 2014
Back to Top