Differential Geometry and its Applications最新文献

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Generalized Alexandrov theorems in spacetimes with integral conditions 具有积分条件的时空中的广义Alexandrov定理

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-05-14 DOI: 10.1016/j.difgeo.2025.102254

Kwok-Kun Kwong , Xianfeng Wang

We investigate integral conditions involving the mean curvature vector

\vec{H}

or mixed higher-order mean curvatures, to determine when a codimension-two submanifold Σ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in [18] by relaxing the curvature conditions on Σ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for Σ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.

我们研究了涉及平均曲率向量H→或混合高阶平均曲率的积分条件，以确定一个协维二子流形Σ何时位于时空中无剪切（脐带）零超曲面上。我们通过放宽Σ上的曲率条件，从几个方面推广了[18]中引入的时空中的alexandrov型定理。具体地说，我们用平均曲率积分不等式给出了Σ位于无剪切零超曲面的充分必要条件。我们方法的一个关键组成部分是使用具有任意权值的闵可夫斯基公式，它使我们能够推导出具有明显较弱积分曲率条件的子流形的刚度结果。

引用次数: 0

Time in classical and quantum mechanics 经典和量子力学中的时间

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-05-14 DOI: 10.1016/j.difgeo.2025.102253

J. Muñoz-Díaz, R.J. Alonso-Blanco

In this article we study the nature of time in Mechanics. The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation, implies the existence of an absolute time-duration in the sense of Newton. There is a second notion of time for conservative systems which makes the Hamiltonian action evolves at a constant rate. In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory. Then, we propose two different ways to reach the time dependent Schrödinger equation. One way consists of considering a “time constraint” on a free system. The other way is based on the point of view of Hertz, by considering the system as a projection of a free system. In the later manner, the “time” appearing in the Schrödinger equation is a linear combination of the time-duration with the “time” quotient of the action by the energy on each solution of the Hamilton-Jacobi equation. Both of them are based on a rule of quantization that we explain in Section 4.

在这篇文章中，我们研究力学中时间的性质。一个由二阶微分方程支配的机械系统的演化所依据的基本原理，意味着牛顿意义上的绝对时间的存在。对于保守系统，还有第二种时间概念，它使得哈密顿作用以恒定速率演化。在量子力学中，绝对时间失去了它的意义，就像它失去了轨迹的概念一样。然后，我们提出了两种不同的方法来达到与时间相关的Schrödinger方程。一种方法是考虑自由系统的“时间限制”。另一种方法是基于赫兹的观点，把这个系统看作一个自由系统的投影。在后一种方式中，Schrödinger方程中出现的“时间”是时间持续时间与哈密顿-雅可比方程每个解上能量作用的“时间”商的线性组合。它们都基于我们在第4节中解释的量子化规则。

引用次数: 0

Rigidity of closed minimal hypersurfaces in S5 S5中闭极小超曲面的刚性

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-05-09 DOI: 10.1016/j.difgeo.2025.102252

Pengpeng Cheng, Tongzhu Li

Let

M^{4} \to S^{5}

be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere

S^{5}

. In this paper, we prove that if the 3-mean curvature

H_{3}

and the number g of the distinct principal curvatures are constant, then

M^{4}

is an isoparametric hypersurface, and the value of S can only be

0, 4, 12

. This result supports Chern Conjecture.

设M4→S5为5维球面S5中第二基本形式S的平方长度为常数的封闭浸没极小超曲面。在本文中，我们证明了如果3-平均曲率H3和不同主曲率的个数g是常数，则M4是一个等参超曲面，且S的值只能为0,4,12。这一结果支持陈猜想。

引用次数: 0

The strong Diederich-Fornæss index on C2 domains in Hermitian manifolds 厄米流形中C2域上的强Diederich-Fornæss指标

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-24 DOI: 10.1016/j.difgeo.2025.102251

Phillip S. Harrington

For a relatively compact Stein domain Ω with

C^{2}

boundary in a Hermitian manifold M, we consider the strong Diederich-Fornæss index, denoted

D F (Ω)

: the supremum of all exponents

0 < η < 1

such that eigenvalues of the complex Hessian of

- {(- ρ)}^{η}

are bounded below by some positive multiple of

{(- ρ)}^{η}

on Ω for some

C^{2}

defining function ρ. We will show that

D F (Ω)

is completely characterized by the existence of a Hermitian metric with curvature terms satisfying a certain inequality when restricted to the null-space of the Levi-form.

对于厄米流形M中具有C2边界的相对紧凑的Stein定域Ω，我们考虑强diederlich - forn æss指标，记为DF（Ω）：所有指数0<；η<；1的极值，使得-(−ρ)η的复Hessian的特征值在Ω上被某个C2定义函数ρ的正倍数有界。我们将证明DF（Ω）是完全由曲率项满足一定不等式的厄米度规的存在所表征的，当它被限制在列维形式的零空间中时。

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引用次数: 0

The geometric Toda equations for noncompact symmetric spaces 非紧致对称空间的几何Toda方程

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-16 DOI: 10.1016/j.difgeo.2025.102249

Ian McIntosh

This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of τ-primitive harmonic maps from a surface into a homogeneous space

G / T

for which G is a noncomplex noncompact simple real Lie group, τ is the Coxeter automorphism which Drinfel'd & Sokolov assigned to each affine Dynkin diagram, and T is the compact torus fixed pointwise by τ. Here τ may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface Σ as equations for a metric on a holomorphic principal

T^{C}

-bundle

Q^{C}

over Σ whose Chern connection, when combined with a holomorphic field φ, produces a G-connection which is flat precisely when the Toda equations hold. The second purpose is to establish when stability criteria for the pair

(Q^{C}, φ)

can be used to prove the existence of solutions. We classify those real forms of the Toda equations for which this pair is a principal pair and we call these totally noncompact Toda pairs: stability theory then gives algebraic conditions for the existence of solutions. Every solution to the geometric Toda equations has a corresponding G-Higgs bundle. We explain how to construct this G-Higgs bundle directly from the Toda pair and show that Baraglia's cyclic Higgs bundles arise from a very special case of totally noncompact cyclic Toda pairs.

本文有两个目的。第一个是对Toda方程的所有版本进行分类，这些版本控制了从曲面到齐次空间G/T的τ-原始调和映射的存在性，其中G为非复非紧单实李群，τ为Coxeter自同构，其中Drinfel'd &；Sokolov分配给每个仿射动力学图，T是紧化环面，由τ点固定。这里τ可以是内自同构也可以是外自同构。我们将紧致黎曼曲面Σ上的Toda方程解释为Σ上全纯主tc束QC上的度规方程，当其Chern连接与全纯场φ结合时，产生一个g连接，当Toda方程成立时，该连接恰好是平的。第二个目的是确定何时可以用（QC,φ）对的稳定性判据来证明解的存在性。我们对这些实形式的Toda方程进行分类其中这对是主对我们称它们为完全非紧Toda对稳定性理论给出了解存在的代数条件。几何Toda方程的每个解都有对应的g -希格斯束。我们解释了如何直接从Toda对构造g -希格斯束，并证明了Baraglia的循环希格斯束是由一个非常特殊的完全非紧化的循环Toda对产生的。

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引用次数: 0

New classes of Finsler metrics: The birth of new projective invariant 芬斯勒度量的新类别：新射影不变量的诞生

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-14 DOI: 10.1016/j.difgeo.2025.102250

Nasrin Sadeghzadeh

This paper presents a pioneering projective invariant in Finsler geometry, introducing a new class of Finsler metrics that are preserved under projective transformations. The newly formulated weakly generalized Douglas-Weyl

(W - G D W)

equation facilitates the generalization of generalized Douglas-Weyl

(G D W)

-metrics into the broader category of

W - G D W

-metrics, which encompasses all GDW-metrics. Within this class, there are also two additional subclasses: generalized weakly-Weyl metrics, characterized by a milder form of Weyl curvature, and generalized

\tilde{D}

-metrics, defined by a less strict version of Douglas curvature. The paper provides a comprehensive overview of these generalized class of Finsler metrics and elucidates their properties, supported by detailed examples.

本文提出了芬斯勒几何中的一个开创性的射影不变量，引入了一类新的在射影变换下保持的芬斯勒度量。新建立的弱广义Douglas-Weyl （W−GDW）方程有助于将广义Douglas-Weyl (GDW)-度量推广到更广泛的W−GDW-度量范畴，其中包括所有GDW-度量。在这类中，还有两个额外的子类：广义弱Weyl度量，以温和形式的Weyl曲率为特征，广义D ~度量，由不严格版本的道格拉斯曲率定义。本文全面概述了这类广义的芬斯勒度量，并举例说明了它们的性质。

引用次数: 0

On the integration of Manin pairs 关于Manin对的积分

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-04 DOI: 10.1016/j.difgeo.2025.102246

David Li-Bland, Eckhard Meinrenken

It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid A: The source-simply connected Lie groupoid

G ⇉ M

integrating A automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the ‘quasi’ settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs

(E, A)

, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid G integrating A is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.

泊松流形对辛群似体的可积性仅取决于其余切李代数似体a的可积性，这是一个值得注意的事实：对a积分的源单连通李群似体G M自动获得了一个乘法辛2型。更一般地说，对于李双代数群与泊松群的积分，以及在狄拉克结构和拟李双代数群的“拟”设置中，也有类似的结果。在本文中，我们将把这些结果放在Manin对（E, a）的一般上下文中，从而获得这些积分结果的简单几何方法。我们还澄清了对A积分的群形G不是源单连通的情况。在此基础上，得到了泊松群和拟辛群的哈密顿空间的描述。

引用次数: 0

Bour's theorem for helicoidal surfaces with singularities 带奇点的螺旋曲面的小时定理

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-03 DOI: 10.1016/j.difgeo.2025.102248

Yuki Hattori , Atsufumi Honda , Tatsuya Morimoto

In this paper, by generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge and, more generally, every generic n-type edge, which is invariant under a helicoidal motion in Euclidean 3-space admits non-trivial isometric deformations. As a corollary, several geometric invariants, such as the limiting normal curvature, the cusp-directional torsion, the higher order cuspidal curvature and the bias, are proved to be extrinsic invariants.

本文通过推广Bour定理的技巧，证明了欧几里得三维空间中在螺旋运动下不变的每一个一般的尖头边，更一般地说，每一个一般的n型边都允许非平凡等距变形。作为推论，证明了极限法向曲率、尖向扭转、高阶尖向曲率和偏置等几何不变量为外在不变量。

引用次数: 0

The manifold of polygons degenerated to segments 多边形的流形退化为线段

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-04-01 DOI: 10.1016/j.difgeo.2025.102247

Manuel A. Espinosa-García , Ahtziri González , Yesenia Villicaña-Molina

In this paper we study the space

L (n)

of n-gons in the plane degenerated to segments. We prove that this space is a smooth real submanifold of

C^{n}

, and describe its topology in terms of the manifold

M (n)

of n-gons degenerated to segments and with the first vertex at 0. We show that

M (n)

and

L (n)

contain straight lines that form a basis of directions in each one of their tangent spaces, and we compute the geodesic equations in these manifolds. Finally, the quotient of

L (n)

by the diagonal action of the affine complex group and the re-enumeration of the vertices is described.

本文研究了简并成节平面上n-gon的空间L(n)。我们证明了这个空间是Cn的光滑实子流形，并且用n-gon的流形M(n)描述了它的拓扑结构，该流形退化为线段，第一个顶点为0。我们证明M(n)和L(n)包含直线，这些直线在它们的每个切线空间中构成了方向的基础，并且我们计算了这些流形中的测地线方程。最后，描述了仿射复群对角作用下L(n)的商和顶点的重新枚举。

引用次数: 0

Regulated curves on a Banach manifold and singularities of endpoint map. I. Banach manifold structure Banach流形上的调节曲线及端点映射的奇异性。1 .巴拿赫流形结构

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2025-03-27 DOI: 10.1016/j.difgeo.2025.102245

Tomasz Goliński , Fernand Pelletier

We consider regulated curves in a Banach bundle whose projection on the basis is continuous with regulated derivative. We build a Banach manifold structure on the set of such curves. This result was previously obtained for the case of strong Riemannian Banach manifold and absolutely continuous curves in [16]. The essential argument used was the existence of a “local addition” on such a manifold. Our proof is true for any Banach manifold. In the second part of the paper the problems of controllability will be discussed.

考虑Banach束上的可调曲线，其在基上的投影是连续的，导数是可调的。我们在这些曲线的集合上建立了一个Banach流形结构。这一结果已在强黎曼巴拿赫流形和[16]中绝对连续曲线的情况下得到。所用的基本论证是在这种流形上存在一个“局部加法”。我们的证明对任何巴拿赫流形都成立。论文的第二部分将讨论可控性问题。

引用次数: 0

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