Special Riemannian geometries and the Magic Square of Lie algebras

从黎曼几何视角看四维瑟斯顿几何

Thurston geometries in dimension four from a Riemannian perspective

本文从黎曼几何视角综述四维瑟斯顿几何，重点描述非乘积且等距群为4维或5维的 $\mathrm{Sol}^4_0$、$\mathrm{Sol}^4_1$、$\mathrm{Sol}^4_{m,n}$ 和 $\mathrm{Nil}^4$ 空间，并给出其全部不变黎曼度量。

Constant Curvature Models in Sub-Riemannian Geometry

Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.

Distinguished dimensions for special Riemannian geometries

The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups H_k\subset SO(n_k), with H_1=SO(3), H_2=SU(3), H_4=Sp(3) and H_8=F_4. This enables studies of special Riemannian geometries with structure groups H_k in dimensions n_k. The neccessary and sufficient conditions for the H_k geometries to admit the characteristic co…

Notes on symmetric conformal geometries

In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or covered by a pseudo-Riemannian symmetric space, where the covering is a conformal map. We construct examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian symmetric spaces.

Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors

We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geomet…

Géométries Lorentziennes de dimension 3 : classification et complétude

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the metric has constant sectionnal curvature, or is a left invariant metric on the Heisenberg group or the Sol group. These geometries, on each of the latter two groups are characterized by having a non-compact isotropy without being flat. Recall, for the need of his formulation of …

Theoretically and computationally convenient geometries on full-rank correlation matrices

In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemannian homogeneous space. In contrast, we view in this work SPD matrices as a Lie group and the affine-invariant metric as a left-invariant metric. This unexpected new viewpoint allows us to generalize the construction of the quotient-affine metric and t…

The Projective Parabolic Geometry of Riemannian, Kähler and Quaternion-Kähler Metrics

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned class…

关于约化嘉当几何的一些初步观察

Some elementary observations regarding reductive Cartan geometries

本文将黎曼几何中的协变导数与测地线概念推广到约化嘉当几何，并证明了一系列类比结果，特别是给出了霍普夫-里诺定理的推广及其简洁证明。

Digital finite quantum Riemannian geometries

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $Δ=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector.

Coordinate-free classic geometries

This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features o…

固定秩矩阵流形上的两种牛顿方法（带黎曼商几何）

Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries

本文考虑固定秩矩阵流形上的两种黎曼几何，通过黎曼淹没理论推导几何对象，并给出相应的黎曼牛顿方法，其收敛性分析优于欧几里得方法，计算开销小，适用于低秩矩阵补全等问题。

黎曼深度学习：模块、网络与几何

Riemannian Deep Learning: Modules, Networks, and Geometries

研究为黎曼深度学习构建统一框架，从可复用神经模块等三个角度出发，推广批量归一化等，开发相关神经网络及度量，经理论分析、数值实验和多领域应用验证方法的有效性。

拖拉机丛的嵌入定理及其在共形伪黎曼几何中的应用

An Embedding Theorem for Tractor Bundles, and an Application in Conformal Pseudo-Riemannian Geometry

将Cartan几何的Gromov-Zimmer嵌入定理推广到携带不变连接的拖拉机丛，并证明特殊伪酉群在闭单连通解析伪黎曼流形上共形作用的刚性结果。

On the class of pseudo-Riemannian geometries which can not be locally described using curvature scalars solely

A classic problem with intriguing implications at the level of both applied differential geometry and theoretical physics is dealt with in this short work: Is there any criterion in order to decide whether a pseudo-Riemannian space can be locally described using curvature scalars solely? Surprisingly enough, this question is susceptible of a very simple and elegant answer. In order to avoid unnecessary complexity, the analysis is restricted to local rather than global considerations, without any loss of not only the generality but also the insights to the initial problem.

On symmetric CR geometries of hypersurface type

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with th…

非欧几里得几何中的内接问题

Inscriptions in non-Euclidean Geometries

本文将平面内接问题推广至常曲率 Riemannian 曲面，利用 symplectic 与 Riemannian 几何思想，证明了双曲平面中光滑 Jordan 曲线的广义内接定理，以及两球面上不与对跖点相交的光滑 Jordan 曲线的矩形内接定理。

On types of non-integrable geometries

We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique connection with totally skew-symmetric torsion. Under weak conditions on the structure group we prove that this geometry is the only one with this property. Finally, we discuss the automorphism group of a Riemannian manifold with a fixed non-integrable $\mathrm{G}$-structure.

Metric connections with parallel skew-symmetric torsion

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…

Riemannian geometries on spaces of plane curves

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: $$ G^A_c(h,k) := \int_{S^1}(1+A\ka_c(þ)^2)< h(þ),k(þ) > |c'(þ)| dþ$$ where $\ka_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. The term $A\ka^2$ is a sort of geometric Tikhonov regularization because, for A=0, the geodesic distance between any 2 distinct curves is 0, while for $A>0$ …

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.

关于 Finsler 几何与 Lorentz 几何之间联系的一个说明（面向 Riemannian 几何学家）

An account on links between Finsler and Lorentz Geometries for Riemannian Geometers

本文面向 Riemannian 几何学家介绍 Lorentz 与 Finsler 几何间的联系，以 Zermelo 导航问题为例，阐述了利用 Finsler 元素描述 Lorentz 因果关系、处理 Finsler 奇点及因果边界，并应用于波与火线传播。

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, the geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is …

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

Riemannian metrics on differentiable stacks

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

Curvature, integrability, and the six sphere

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not he…

Pseudo-Riemannian Ricci-flat and Flat Warped Geometries and New Coordinates for the Minkowski metric

It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann cu…

具有平行、斜对称和闭合 torsion 的几何结构

Geometries with parallel, skew-symmetric and closed torsion

研究具有平行、斜对称和闭合 torsion 的黎曼流形，证明其局部可分解为已知因子，实现完整局部分类，并探讨多种 PSCT 类型的 G 结构，特别是几乎 Hermitian 结构及其可能的 Gray-Hervella 类别。

Projective planes over quadratic 2-dimensional algebras

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2 \times A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization o…

BGG序列——黎曼视角

BGG Sequences -- A Riemannian perspective

本文从黎曼视角探讨BGG序列，提出在Riemannian流形和具有体积保持连接的流形上构造BGG序列的方法，为应用数学中的数值方法提供理论支持。

度量 Lie 群：齐性视角下的 Carnot-Carathéodory 空间

Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint

本书探讨 Lie 群上左不变距离定义的非光滑几何，重点阐述 Carnot 群等度量 Lie 群在幂零群渐近锥、对称空间抛物边界及流形极限等数学语境中的核心作用。

Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories

In this article we introduce $A$-valued Einstein-Hilbert-Palatini functional ($A$-EHP) over a n-manifold $M$, where $A$ is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if $A$ is weak $(k,s)$-solvable, then $A$-EHP is non-null only if $n<k+s+3$. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for $k=1$ and $s=2$, including Hitchin's generalized complex geometry, Pantilie's generalized quaternio…

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

The degenerate special Lagrangian equation on Riemannian manifolds

We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…

拉回对称黎曼几何用于数据分析

Pulling back symmetric Riemannian geometry for data analysis

本文研究如何为非线性数据构造对称黎曼几何，刻画并通过深度学习生成满足测地子空间映射与局部等距条件的微分同胚，实验验证了其稳定性及正曲率拉回的风险。

基于几何的相似度度量用于Riemannian和统计流形上的神经表示

Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds

本文提出MSA方法，利用Riemannian几何工具比较神经表示的内在几何，用于解纠缠深度网络特征、比较非线性动态和研究扩散模型。

Riemann-Cartan 几何中的对称性

Symmetries in Riemann-Cartan Geometries

本文回顾 Riemann-Cartan 几何中对称性的研究方法，并给出所有静态与稳态球对称 Riemann-Cartan 几何以及允许七维对称群的球对称子类的新结果。