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

TL;DR

This paper develops a comprehensive geometric theory of nondegenerate R^n -actions on n-manifolds, linking dynamics and geometry, and providing classification, normal forms, and invariants for these actions.

Contribution

It introduces a systematic geometric framework for nondegenerate R^n -actions, including classification, normal forms, and invariants, connecting dynamics with geometric structures.

Findings

Derived local and semi-local normal forms.

Identified automorphism and twisting groups.

Established existence and classification theorems.

Abstract

This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-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.