Journal of the Mathematical Society of Japan

Geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds

Nguyen VAN MINH and Nguyen Tien ZUNG

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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.

Article information

J. Math. Soc. Japan, Volume 66, Number 3 (2014), 839-894.

First available in Project Euclid: 24 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 58K50: Normal forms 37J35: Completely integrable systems, topological structure of phase space, integration methods 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 58K45: Singularities of vector fields, topological aspects 37J15: Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]

integrable system toric manifold group action reflection principle normal form complete fan hyperbolic elbolic monodromy


ZUNG, Nguyen Tien; VAN MINH, Nguyen. Geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. J. Math. Soc. Japan 66 (2014), no. 3, 839--894. doi:10.2969/jmsj/06630839.

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