Aleksey Zinger
This note collects a number of standard statements in Riemannian geometry and in Sobolev-space theory that play a prominent role in analytic approaches to symplectic topology. These include relations between connections and complex structures, estimates on exponential-like maps, and dependence of constants in Sobolev and elliptic estimates.
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In this paper we determine conditions of existence of an induced Riemannian structure on the symplectic quotient of a symplectic and Riemannian manifold following the action of a Lie group acting upon it in a hamiltonian way with equivariant momentum mapping.
We introduce a new class of Poisson structures on a Riemannian manifold. A Poisson structure in this class will be called a Killing-Poisson structure. The class of Killing-Poisson structures contains the class of symplectic structures, the class of Poisson structures studied in{\it (Differential Geometry and its Applications, {\bf Vol. 20, Issue 3} (2004), 279--291)} and the class of Poisson structures induced by some infinitesimal Lie algebras actions on Riemannian manifolds. We show that some classical results on symplectic manifolds (the integrability of the Lie algebroid structure associat…
The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost everything can be done on a general Carnot-Caratheodory group. Such generalisations will be the subject of a forthcoming note.
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and the Viterbo conjecture. This will give us a perfect excuse to introduce the reader to some important ideas in Riemannian and symplectic geometry.
The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expr…
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricc…
Riemannian distance and symplectic embeddings in cotangent bundle
本文利用标准球到余切丛邻域的辛嵌入定义类距离函数，证明其在单位圆盘余切丛情形恢复黎曼度量，并证明强 Viterbo 猜想及两球辛填充数界。
Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichmüller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.
We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a non-linear version of the classical Banach-Mazur distance, called symplectic Banach-Mazur distance and denoted by $d_{SBM}.$ The relevant persistence modules come from filtered symplectic homology and are stable with respect to $d_{SBM}.$ Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped w…
The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorph…
Injectivity radius lower bound of convex sum of tame Riemannian metrics and applications to symplectic topology
该研究证明了一对曲率有界、单射半径有正下界的完备驯黎曼度量的凸和仍保持曲率有界且单射半径有一致下界，进而证明驯度量拟等距类的凸性与可缩性，以及对应$C^3$-驯殆复结构集合的可缩性，为辛拓扑研究提供了黎曼几何工具。
If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(ω, ω)> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $ω$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.
This article details a construction of symplectic foliations on 3-dimensional orientable riemannian manifolds from harmonic forms; and how it suggests a topological approach to Poisson's equation and newtonian gravity.
Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton's methods
该研究针对辛约束优化问题，在辛Stiefel流形上构建易处理度量的黎曼优化几何要素，开发显式二阶几何与多种黎曼牛顿法，提出兼具全局与局部快速收敛性的混合算法，数值实验验证了方法有效性。
In this paper we produce a lower bound for the number of periodic orbits of certain Hamiltonian vector fields near Bott-nondegenerate symplectic critical submanifolds. This result is then related to the problem of finding closed orbits of the motion of a charged low energy particle on a Riemannian manifold under the influence of a magnetic field
A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and S…
Cohomological localization for transverse Lie algebra actions on Riemannian foliations
本文证明了Riemannian叶状结构等变基本上同调的局部化与积分公式，并以此推导出横截辛叶状结构的Duistermaat-Heckman定理。
On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $π_J$ and a deRham class $α\in H^2(M, \mathbb R)$, we study the space of generalized Kähler (GK) structures defined by a symplectic form $F\in α$ and whose holomorphic Poisson tensor is $π_J$. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger and Goto to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki--Mabuchi extremal vector field and Calabi--Lichnerowicz--Matsushima result for the Lie algebra of the group of automorp…
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