Linear ${\mathbb Z}_2^n$-Manifolds and Linear Actions

Z-分次Q-流形的正规形

Normal forms of $\mathbb Z$-graded $Q$-manifolds

本文在Z-分次流形上研究Q-结构的局部与整体正规形，揭示其结构集中于曲率零点轨迹的机制，并给出局部分裂定理。

Hochschild cohomology of dg manifolds associated to integrable distributions

For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of $F$. In particular, for the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ associated with a complex manifold $X$, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomolo…

$\mathbb N$-流形的几何化

A geometrisation of $\mathbb N$-manifolds

本文通过证明对称 $n$ 重向量丛余循环与 $[n]$-流形余循环相同，建立了 $[n]$-流形范畴与带 $S_n$-对称性的 $n$ 重向量丛范畴之间的等价性，推广了已知的低阶情形。

On orbit dimensions under a simultaneous Lie group action on n copies of a manifold

We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudo-stabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open subset of a sufficiently big (but finite) number of copies of the manifold. The latter is the analogue for the Cartesian action to Ovsiannikov's theorem on jet spaces and is an important fact relative to the moving frame method and the computation of joint invariants. Some interesting corollaries are pr…

A Kastler-Kalau-Walze type theorem and the spectral action for perturbations of Dirac operators on manifolds with boundary

In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds.

$\mathbb{N}$-流形的几何刻画与Frobenius定理

A geometric characterization of $\mathbb{N}$-manifolds and the Frobenius theorem

本文通过经典几何数据给出$\mathbb{N}$-流形的规范刻画，并在此基础上证明分次流形上分布的Frobenius定理。

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion

We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Action-angle coordinates around a noncompact invariant manifold of a completely integrable Hamiltonian system

A trivial bundle of regular connected invariant manifolds of a completely integrable Hamiltonian system can be provided with action-angle coordinates.

Action-Angle variables on Dirac manifolds

The main purpose of this paper is to show the existence of action-angle variables for integrable Hamiltonian systems on Dirac manifolds under some natural regularity and compactness conditions, using the torus action approach. We show that the Liouville torus actions of general integrable dynamical systems have the structure-preserving property with respect to any underlying geometric structure of the system, and deduce the existence of action-angle variables from this property. We also discover co-affine structures on manifolds as a by-product of our study of action-angle variables. 22 pages.