Riemannian geometry in infinite dimensional spaces

On holomorphic Riemannian geometry and submanifolds of Wick-related spaces

In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first rephrase and extend some background theory on holomorphic Riemannian manifolds, which is essential for the later application of the presented method.

修正黎曼扩张的几何

The geometry of modified Riemannian extensions

证明了仿复空间型与修正的黎曼扩张局部等距，并给出了修正的黎曼扩张是Einstein扩张的充要条件.本文证明了具有非平凡Jordan标准形的Jacobi算子的非幂零的签名为（3，3）的黎曼扩张Osserman流形。我们提出了新的四维Osserman几何的结果。

Invariants of contact sub-pseudo-Riemannian structures and Einstein-Weyl geometry

We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.

关于Riemannian geometry中的A-张量

On A-tensors in Riemannian geometry

我们给出了局部非齐次真A-流形的例子，包括紧的和非紧完备的。

Spectral geometry of Riemannian Legendre foliations

We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant $ϕ$-sectional curvature.

表示论与黎曼几何中的若干主题

Topics in representation theory and Riemannian geometry

该博士课程讲义回顾黎曼几何中李群相关已知主题，讨论紧连通李群理论及表示（侧重低维例子），介绍和乐群分类等应用，提及领域最新进展。

Riemannian Geometry and the Fundamental Theorem of Algebra

If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.

Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator

We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure. The L$^*$-algebra that we construct is a complete local isomorphism invariant and allows us to classify Riemannian symmetric spaces with fixed-sign curvature operator. The case of nonpositive curvature is emphasized.

Riemannian Geometry

These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, torsion and curvature, the exponential map, and the Riemannian connection follow quickly. The Taylor series for of the metric in normal coordinates is an unusual feature. The last third covers, first and second variation of energy, completeness, cut points, the Hadamard-Cartan theore…

Riemannian geometry for shape analysis and computational anatomy

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and functional analysis interact in a given problem. This paper aims to provide a roadmap to the unitiated to the world of infinite-dimensional Riemannian manifolds, spaces of mappings and Sobolev metrics: all tools used in computational anatomy and shape analysis.

关于Infinite Dimensional Riemannian Geometry的一些结果

Some Results on Infinite Dimensional Riemannian Geometry

本文研究了具有上下界截面曲率的完备Hilbert流形的整体性质。我们将证明焦点指数引理，我们将允许我们扩展有限维黎曼几何的一些经典结果，如Rauch和Berger定理和Topogonov定理在流形类上的Hopf-Rinow定理。

The geometry of embedded pseudo-Riemannian surfaces in terms of Poisson brackets

Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian setting and derive explicit formulas for the case of surfaces embedded in $\R^m$ with indefinite metric.

Sub-Riemannian Geometry and Geodesics in Banach Manifolds

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.

（准）八元数射影平面的（半）Riemannian geometry

(Semi-)Riemannian geometry of (para-)octonionic projective planes

本文用约化的齐次坐标研究八元数（或凯莱）射影平面的黎曼几何。我们的方法扩展到para-octonionic（或分裂octonionic）的射影平面，八元射影平面的不定签名，双曲对偶的八元射影平面，我们讨论这些流形在后面的章节。

Shape stability of a quadrature surface problem in infinite Riemannian manifolds

In this paper, we give a simple control on how an optimal shape can be characterized. The framework of Riemannian manifold of infinite dimension is essential. And the covariant derivative plays a key role in the computation and in the analysis of qualitative properties from the shape hessian. The control depends only on the mean curvature of the domain which is a minimum or a critical point.

Riemannian Geometry of Two Families of Tangent Lie Groups

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated.

金属黎曼几何的最新进展：综合综述

Recent Advances in Metallic Riemannian Geometry: A Comprehensive Review

本文综述了自2002年金属结构提出以来金属黎曼几何的最新研究进展，系统梳理该快速发展领域的理论基础与应用成果。

A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$. The framework is that of the tangent sphere bundle of $M$. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler-Lagrange differential systems.

紧致类空Cauchy超曲面空间的黎曼几何

The Riemannian geometry of the space of compact spacelike Cauchy hypersurfaces

研究全局双曲时空中紧致类空Cauchy超曲面空间上的弱黎曼度量，证明测地距离为正且截面曲率非正。

Shortest and Straightest Geodesics in Sub-Riemannian Geometry

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Rie…

Sub-semi-Riemannian geometry on $H$-type groups

We consider $H$(eisenberg)-type groups whose law of left translation gives rise to a bracket generating distribution of step 2. In the contrast with sub-Riemannian studies we furnish the horizontal distribution with a nondegenerate indefinite metric of arbitrary index and investigate the problem concerning causal geodesics on underlying manifolds. The exact formulae for geodesics are obtained.

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.

Riemannian Geometry on Contact Lie Groups

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups

关于Clairaut warped product Riemannian maps的几何

On the geometry of Clairaut warped product Riemannian maps

本文引入Clairaut warped product Riemannian maps，通过推导正则曲线的测地线条件，得出该类映射的成立条件并求出其曲率张量。

通过Maslov指数比较半Riemannian geometry中共轭点和焦点的结果

Comparison results for conjugate and focal points in semi-Riemannian geometry via Maslov index

我们证明了一个估计的差异马斯洛夫指数相对于选择两个不同的参考拉格朗日连续路径的拉格朗日格拉斯曼辛空间。本文讨论了半黎曼流形中沿测地线的共轭点和焦点的一些应用。

Uniqueness of curvature measures in pseudo-Riemannian geometry

The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

On the generalization of theorems from Riemannian to Finsler Geometry I: Metric Theorems

A method to generalize results from Riemannian Geometry to Finsler geometry is presented. We use the method to generalize several results that involve only metric conditions. Between them we show that the topology induced by the Finsler structure is equivalent to the manifold topology, we provide a new proof of the Hopf-Rinow theorem in Finsler geometry and we prove the existence of the center of mass of a convex body when the non-symmetric distance function comes from a non-reversible Finsler function.

子Riemannian geometry中周长测量的变化

Variation of Perimeter Measure in sub-Riemannian geometry

在特征点存在的条件下，给出了完全一般的次黎曼流形的$C^2$超曲面的水平周长测度的一阶变分公式.对于垂直刚性子黎曼流形中的$C^2$超曲面，我们也给出了远离特征轨迹的变分的二阶变分公式.这个变分公式被用来说明\hn{2}中的泡集在保体积变分下是稳定的。

Quantitative comparison theorems in Riemannian and Kähler geometry

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general Bonnet-Myers theorem and Cheng's eigenvalue estimate under weak integral curvature assumptions.

Infinitesimal Hilbertianity of weighted Riemannian manifolds

The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,μ)$ - i.e. endowed with a generic non-negative Radon measure $μ$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,μ)$ is a Hilbert space. We actually prove a stronger result: the abstract tangent module (à la Gigli) associated to any weighted reversible Finsler manifold $(M,F,μ)$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $μ$.

黎曼曲率半径的几何

The geometry of Riemannian curvature radii

该研究聚焦黎曼流形上曲线曲率半径的几何结构，证明了关联次黎曼流形的存在性并探究其性质，通过一对全局向量场编码流形几何的内蕴信息。

Riemannian几何中的一些广义不等式

Some generalized inequalities in Riemannian Geometry

本文在完备非紧Riemann流形上进一步扩展了Hardy和Rellich型不等式，通过引入权函数提供更锐利的几何估计，并以定理和证明形式展示。

On one class of holonomy groups in pseudo-Riemannian geometry

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, we prove the following theorem. Let g be a nondegenerate bilinear form on a vector space V, and L:V -> V a g-symmetric operator. Then the identity component of the centraliser of L in SO(g) is a holonomy group for a suitable Levi-Civita connection.

具有交换曲率算子的黎曼流形的整体几何

The global geometry of Riemannian manifolds with commuting curvature operators

我们给流形在黎曼和在更高的签名设置的黎曼曲率算子交换，即满足R（a，b）R（c，d）=R（c，d）R（a，b）的所有切向量。这些流形具有全局几何现象，这些现象对于高签名流形与对于黎曼流形是完全不同的。我们的重点是全局性质;研究测地完整性和指数映射的行为问题。

The Metric Geometry of the Manifold of Riemannian Metrics over a Closed Manifold

We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Frechet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.

Note on the conjecture of D.Blair in contact Riemannian geometry

The conjecture of D.Blair says that there are no nonflat Riemannian metrics of nonpositive curvature compatible with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample. We also prove that a hyperbolic metric on $\mathbb{R}^3$ cannot be compatible with any contact structure.

On the Spherical Hausdorff Measure in Step 2 Corank 2 sub-Riemannian Geometry

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a previous work by the auhors.

Riemannian geometry of noncommutative super surfaces

In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the connections are metric-compatible and have zero torsion when the super metric is symmetric, giving rise to the corresponding super Riemann curvature. The latter also satisfies the noncommutative super analogue of the first and second Bianchi identities. We also give some examples and study them in details.

关于黎曼扭曲乘积映射的几何

On the geometry of Riemannian warped product maps

本文引入Clairaut黎曼扭曲乘积映射，基于Ricci曲率建立正则曲线成为测地线的条件，利用曲率张量研究Ricci soliton结构，推导Bochner型公式与标量曲率积分公式，并扩展至conformal映射的几何性质，构造了相关实例。

Some inequalities and applications of Simons' type formulas in Riemannian, affine and statistical geometry

A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.

Affine connections and curvature in sub-Riemannian geometry

We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a compatible affine connection and induced tensors, without any restriction on our sub-Riemannian manifold or the choice of connection. In particular, we obtain a universal Bonnet-Myers theorem of Riemannian type. We also give universal formulas for the canonical horizontal frame along each geodesic and an algorithm for computing the curvature in general. Seve…

The Geometry of Loop Spaces I: $H^s$-Riemannian Metrics

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms, which take values in pseudodifferential operators. These calculations are used in a followup paper "The Geometry of Loop Spaces II: Characteristic Classes" to construct Chern-Simons classes on the tangent bundle TLM which detect nontrivial elements in the diffeomorphism group of certain Sasakian 5-manifolds associated to Kaehler surfaces.

On the geometry of almost Golden Riemannian manifolds

An almost Golden Riemannian structure $(φ,g)$ on a manifold is given by a tensor field $φ$ of type (1,1) satisfying the Golden section relation $φ^{2}=φ+1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(φX,Y)=g(X,φY)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $φ$ and the integrability of the $G$-structure corresponding to $(φ,g)$.

近似Kaehler几何与黎曼叶理

Nearly Kaehler geometry and Riemannian foliations

考虑了具有正则Hermite联络的严格完备的近Kaehler流形。研究了正则厄米联络的完整表示。证明了严格完备的近似Kaehler是齐次近似Kaehler空间、四元数Kaehler流形上的扭量空间和6维近似Kaehler流形的局部黎曼积.作为应用，我们得到了容许相容Kaehler结构的全测地黎曼叶理的结构结果。最后，我们得到了一个分类结果的同质的情况下，减少了一个猜想的Wolf和Gray的6维形式。

Riemannian-Finsler几何的全局研究

The global study of Riemannian-Finsler geometry

本文比较研究Riemannian与Finsler几何，探讨切点和共轭区域的差异，分析凸函数在Riemannian流形上的拓扑与微分结构及Finsler流形中的类比结果。

三间隙定理与Riemannian Geometry

The Three Gap Theorem and Riemannian Geometry

经典的三间隙定理认为，对于一个自然数n和一个实数p，在由p的前n个倍数的模1约简组成的子集[0，1）中，相邻元素之间最多有三个不同的距离。我们发现了与紧黎曼流形的等距和沿其测地线的点的分布有关的类似的结果。

关于黎曼超叶的单位法向量场的外几何

On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations

考虑给定黎曼流形上的（局部）超叶的单位法向量场作为具有Sasaki度量的单位切丛中的子流形。给出了该子流形的第二基本型的一个显式表达式，并给出了全脐超叶状子流形的全测地性的一个相当简单的条件。一个相应的例子显示了这个条件的非平凡性。在二维情形下，我们给出了允许测地单位向量场具有全测地性质的黎曼流形的完整描述。

Liouville-type theorems and applications to geometry on complete Riemannian manifolds

On a complete Riemannian manifold M with Ricci curvature satisfying $$\textrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2...(\log^{k}r)^2$$ for $r\gg 1$, where A>0 is a constant, and r is the distance from an arbitrarily fixed point in M. we prove some Liouville-type theorems for a C^2 function $f:M\rightarrow \Bbb R$ satisfying $Δf\geq F(f)$ for a function $F:\Bbb R\rightarrow \Bbb R$.

黎曼流形的退化

Degenerations of Riemannian manifolds

这是一篇为《现代数学物理百科全书》（MEMPys）撰写的关于黎曼几何中坍缩理论的论述性文章。本文主要讨论了在各种曲率假设下，黎曼流形中塌陷/非塌陷区域的几何和拓扑结构。许多应用崩溃理论黎曼几何没有讨论在这一调查。

The relative heat content for submanifolds in sub-Riemannian geometry

We study the small-time asymptotics of the relative heat content for submanifolds in sub-Riemannian geometry. First, we prove the existence of a smooth tubular neighborhood for submanifolds of any codimension, assuming they do not have characteristic points. Next, we propose a definition of relative heat content for submanifolds of codimension $k\geq 1$ and we build an approximation of this quantity, via smooth tubular neighborhoods. Finally, we show that this approximation fails to recover the asymptotic expansion of the relative heat content of the submanifold, by studying an explicit exampl…

李代数胚的Riemannian Geometry

Riemannian Geometry of Lie Algebroids

我们引入黎曼李代数胚作为推广的黎曼流形，我们表明，大多数的经典工具和结果已知的黎曼几何可以说在这个设置。我们也给出了关于黎曼李代数胚可积性的一些新结果。

黎曼叶理的非对易谱几何

Noncommutative spectral geometry of Riemannian foliations

在非交换微分几何的意义下，我们构造了紧流形上的黎曼叶状结构的谱三元组，并描述了它的维数谱。

关于Infinite Dimensional黎曼流形中指数映射的奇性

On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

利用辛技巧和自伴算子光滑路径的谱分析，刻画了无穷维Riemann Hilbert流形中沿测地线的共轭瞬时集.

Connections in sub-Riemannian geometry of parallelizable distributions

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable linear connections have been constructed on a sub-Riemannian parallelizable distribution, namely, the Weitzenböck connection and the sub-Riemannian connection. The obtained results have been applied to two concrete examples: the spheres $S^3$ and $S^7$.