Q-manifolds and Mackenzie theory: an overview

Graded manifolds and Drinfeld doubles for Lie bialgebroids

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, etc. I give a construction of \textit{doubles} for \textit{graded} $QS$- and \textit{graded $QP$-manifolds} (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded $QS…

爱因斯坦四维流形综述

An Overview of Einstein 4-Manifolds

该文是一篇说明性综述，基于作者2026年为巴黎数学科学基金会卓越教席开设的系列讲座，围绕爱因斯坦四维流形展开介绍。

具有Ricci下界的黎曼流形上Sobolev不等式稳定性综述

An overview of the stability of Sobolev inequalities on Riemannian manifolds with Ricci lower bounds

本文综述了具有Ricci曲率下界的黎曼流形上Sobolev不等式稳定性的最新进展，重点介绍了光滑与非光滑几何方法的结合，并针对非负Ricci曲率和欧氏体积增长情形给出了自包含的稳定性证明概述，最后讨论了相关结果和开放问题。

Ricci下有界流形上的等周问题：近期结果与方法综述

Isoperimetry on manifolds with Ricci bounded below: overview of recent results and methods

本文综述了Ricci下界流形上等周问题的最新进展，重点介绍等周轮廓二阶微分不等式及其证明方法，并展示了其在等周不等式和等周集存在性等方面的应用，同时指出了若干开放问题。

Euclidean volume growth for complete Riemannian manifolds

We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls.

A report on locally conformally Kähler manifolds

We present an overview of recent results in locally conformally Kähler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

三维流形的几何化

Geometrisation of 3-manifolds

本文概述了瑟斯顿在1980年代猜想、佩雷尔曼在2000年代证明的三维流形几何化定理，解释了定理内容及其在不同情境下的影响。

Sasaki-Einstein Manifolds

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.

Transitive double Lie algebroids via core diagrams

The core diagram of a double Lie algebroid consists in the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram. The comma double Lie algebroid associated to a morphism of Lie…

Metric geometries over the split quaternions

We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneous action of a semi-simple Lie group, and to construct distinct para-quaternionic Kahler metrics from indefinite real analytic conformal manifolds. We also indicate how the theory of toric varieties gives rise to constructions of hypersymplectic manifolds.

Manifolds of mappings for continuum mechanics

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.

Manifolds of mappings and shapes

This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of manifolds of smooth mappings. Smooth manifolds of immersions, diffeomorphisms, and shapes, and weak Riemannian metrics on them are treated, culminating in the surprising fact, that geodesic distance can vanish completely for them.

Warps and duality for double vector bundles

A linear section of a double vector bundle is a parallel pair of sections which form a vector bundle morphism; examples include the complete lifts of vector fields to tangent bundles and the horizontal lifts arising from a connection in a vector bundle. A grid in a double vector bundle consists of two linear sections, one in each structure, and thus provides two paths from the base manifold to the top space; the warp of the grid measures the lack of commutativity of the two paths. For a double vector bundle $D$, a linear section induces a linear map on the relevant dual of $D$ and thence a lin…

揭示 $\mathcal{S}-$ 空间形式中的 $f-$双调和 $\theta_{\alpha}-$斜曲线

Unveiling $f-$Biharmonic $θ_{α}-$Slant Curves in $\mathcal{S}-$ Space Forms

本文研究 $\mathcal{S}-$ 空间形式中的 $f-$ 双调和 $\theta_{\alpha}-$ 斜曲线，通过推导并分析关键方程，建立了该曲线为 $f-$ 双调和的充分必要条件，并给出实例验证。