Differential Geometry and its Applications最新文献

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Left-invariant pseudo-Riemannian metrics on Lie groups: The null cone 李群上的左变伪黎曼度量：空锥

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102205

Sigbjørn Hervik

We study left-invariant pseudo-Riemannian metrics on Lie groups using the moving bracket approach of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the

G = O (p, q)

-action; i.e., Lie algebras μ where zero is in the closure of the orbits:

0 \in \overline{G \cdot μ}

. We provide examples of such Lie groups in various signatures and give some general results. For signatures

(1, q)

and

(2, q)

we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some

O (p, q)

action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given.

我们利用相应李代数的移动括号方法研究李群上的左不变伪黎曼度量。我们的研究重点是李代数在 G=O(p,q) 作用的空锥中的度量，即零值在轨道闭合中的李代数 μ：0∈G⋅μ‾。我们举例说明了不同符号下的此类李群，并给出了一些一般结果。对于符号 (1,q) 和 (2,q)，我们对属于空锥的所有情况进行了分类。更一般地说，我们证明了所有零能和完全可解的李代数都在某个 O(p,q) 作用的空锥中。此外，我们还给出了空锥中的几个非三维列维可分解李代数的例子。

引用次数: 0

Singularities of discrete indefinite affine minimal surfaces 离散不定仿射极小曲面的奇点

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102206

Marcos Craizer

A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and swallowtails. By discretizing the initial curves, one can obtain by the discrete Lelieuvre's formulas a discrete affine minimal surface with indefinite metric. The aim of this paper is to define the singular edges and vertices of the corresponding discrete asymptotic net in such a way that the most relevant properties of the singular set of the smooth version remain valid.

根据勒里厄尔公式，可以从一对平滑的非相交空间曲线得到具有不定度量的平滑仿射极小曲面。这些曲面可能会出现奇点，一般是尖顶边缘和燕尾形。通过将初始曲线离散化，可以用离散的勒里厄尔公式得到具有不定度量的离散仿射极小曲面。本文的目的是定义相应离散渐近网的奇异边和顶点，从而使光滑版本奇异集的最相关特性保持有效。

引用次数: 0

Mean curvature flows of graphs sliding off to infinity in warped product manifolds 扭曲积流形中滑向无穷远的图的平均曲率流

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-11-12 DOI: 10.1016/j.difgeo.2024.102207

Naotoshi Fujihara

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and

R

. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.

我们研究由封闭黎曼流形和 R 定义的翘曲积流形中的平均曲率流。在这样的翘曲积流形中，我们可以定义一个图的概念，称为大地图。我们证明，对于任何翘曲函数，曲线缩短流都会保留大地图，而对于某些单调凸翘曲函数，超曲面的平均曲率流也会保留大地图。特别是，我们考虑了一些在无穷远处归零的翘曲函数，这意味着曲线或超曲面沿着流动在无穷远处归于一点。在这种情况下，我们证明了流的长期存在性，以及曲率及其高阶导数沿流归零。

引用次数: 0

On geodesics in the spaces of constrained curves 关于受约束曲线空间中的大地线

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-11-11 DOI: 10.1016/j.difgeo.2024.102209

Esfandiar Nava-Yazdani

In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.

在这项工作中，我们研究了沉浸曲线空间的某些几何和物理子空间的大地线，这些子空间被赋予了一阶索波列夫度量。这包括弹性曲线，以及将平面同心圆的一些结果扩展到曲面。工作重点是内在和构造方法。

引用次数: 0

Generalized almost-Kähler–Ricci solitons 广义的近凯勒-里奇孤子

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-10-23 DOI: 10.1016/j.difgeo.2024.102193

Michael Albanese , Giuseppe Barbaro , Mehdi Lejmi

We generalize Kähler–Ricci solitons to the almost-Kähler setting as the zeros of Inoue's moment map [25], and show that their existence is an obstruction to the existence of first-Chern–Einstein almost-Kähler metrics on compact symplectic Fano manifolds. We prove deformation results of such metrics in the 4-dimensional case. Moreover, we study the Lie algebra of holomorphic vector fields on 2n-dimensional compact symplectic Fano manifolds admitting generalized almost-Kähler–Ricci solitons. In particular, we partially extend Matsushima's theorem [41] to compact first-Chern–Einstein almost-Kähler manifolds.

我们将 Kähler-Ricci 孤子概括为近 Kähler 设定的井上矩图[25]的零点，并证明它们的存在阻碍了紧凑交映法诺流形上第一切恩-爱因斯坦近 Kähler 度量的存在。我们证明了 4 维情况下此类度量的变形结果。此外，我们还研究了 2n 维紧凑交折射法诺流形上全形向量场的李代数，它允许广义的近凯勒-里奇孤子。特别是，我们将松岛定理[41]部分扩展到了紧凑的第一切恩-爱因斯坦近凯勒流形。

引用次数: 0

Deforming locally convex curves into curves of constant k-order width 将局部凸曲线变形为 k 阶宽度不变的曲线

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-10-22 DOI: 10.1016/j.difgeo.2024.102192

Laiyuan Gao , Horst Martini , Deyan Zhang

A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant k-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is k-symmetric.

引入了一种非局部曲率流来演化平面中的局部凸曲线。研究证明，任何初始局部凸曲线的非局部曲率流都有一个全局解，它保持了演化曲线的局部凸性和弹性能量，而且随着时间的无穷大，曲线会收敛到一条具有恒定 k 阶宽度的光滑局部凸曲线。特别是，当且仅当初始局部凸曲线是 k 对称曲线时，极限曲线是一个多重圆。

引用次数: 0

Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds 黎曼流形上的协变薛定谔算子和 L2- 消失特性

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-09-13 DOI: 10.1016/j.difgeo.2024.102191

Ognjen Milatovic

Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $E$ be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator $H_{X, V} = \nabla^{†} \nabla + \nabla_{X} + V$ , where $\nabla^{†}$ is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of $E$ , X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle $End E$ . We give a sufficient condition for the triviality of the $L^{2}$ -kernel of $H_{X, V}$ . As a corollary, putting $X \equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the $L^{2}$ -kernel of $D^{2}$ , where D is the Dirac operator corresponding to ∇. In particular, when $E = Λ_{C}^{k} T^{⁎} M$ and $D^{2}$ is the Hodge–deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for $L^{2}$ -harmonic (complex-valued) k-forms.

假设 M 是满足加权波恩卡列不等式的完整黎曼流形，假设 E 是 M 上的赫尔墨斯向量束，并配有度量协变导数∇。我们考虑算子 HX,V=∇†∇+∇X+V，其中∇† 是∇关于 E 的平方可积分截面空间内积的形式邻接，X 是 M 上的光滑（实）向量场，V 是内形束 EndE 的纤维自交光滑截面。我们给出了 HX,V 的 L2 内核三性的充分条件。作为推论，假设 X≡0 并在配备了克利福德连接∇的克利福德模块的环境中工作，我们会得到 D2 的 L2 内核的三性，其中 D 是对应于∇的狄拉克算子。特别是，当 E=ΛCkT⁎M 和 D2 是（复值）k 形式上的霍奇-德拉姆拉普拉卡时，我们恢复了 L2 谐波（复值）k 形式的一些最新消失结果。

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引用次数: 0

The Sasakian statistical structures of constant ϕ-sectional curvature on Sasakian space forms 萨萨基空间形式上恒定ϕ截面曲率的萨萨基统计结构

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-09-12 DOI: 10.1016/j.difgeo.2024.102190

Xinlei Wu, Yanyan Sheng, Liang Zhang

In this paper, we investigate the Sasakian statistical structures of constant ϕ-sectional curvature based on Sasakian space forms. We obtain the classification of this kind of Sasakian statistical structures. Our classification results show that the Sasakian statistical structures of constant ϕ-sectional curvature on a Sasakian space form with dimension higher than 3 must be almost-trivial; on a 3-dimensional Sasakian space form, in addition to the almost-trivial Sasakian statistical structure, there exist other Sasakian statistical structures which satisfy the constant ϕ-sectional curvature condition. We also point out that a rigidity result for cosymplectic statistical structures of constant ϕ-sectional curvature on 3-dimensional cosymplectic space forms in [11] can be improved to the corresponding classification result.

本文以萨萨空间形式为基础，研究了恒定ϕ截面曲率的萨萨统计结构。我们获得了这类 Sasakian 统计结构的分类。我们的分类结果表明，在维数大于 3 的 Sasakian 空间形式上的ϕ截面曲率恒定的 Sasakian 统计结构必须是几乎三维的；在三维 Sasakian 空间形式上，除了几乎三维的 Sasakian 统计结构之外，还存在其他满足ϕ截面曲率恒定条件的 Sasakian 统计结构。我们还指出，[11]中关于三维折射空间形式上恒定ϕ截面曲率的折射统计结构的刚度结果可以改进为相应的分类结果。

引用次数: 0

Nearly half-flat SU(3) structures on S3 × S3 S3 × S3 上的近半平面 SU(3) 结构

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-09-12 DOI: 10.1016/j.difgeo.2024.102187

Ragini Singhal

We study the $SU (3)$ -structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel $G_{2}$ -structure. Such $SU (3)$ -structures are called nearly half-flat. We characterise the left invariant nearly half-flat structures on $S^{3} \times S^{3}$ . This characterisation then helps us to systematically analyse nearly parallel $G_{2}$ -structures on an interval times $S^{3} \times S^{3}$ .

我们研究了在一个 7 维流形的定向超曲面上诱导出的近乎平行 G2 结构的 SU(3)- 结构。这种 SU(3) 结构被称为近半平面结构。我们描述了 S3×S3 上的左不变近半平面结构。这一特征有助于我们系统地分析S3×S3间隔上的近平行G2结构。

引用次数: 0

Vector bundle automorphisms preserving Morse-Bott foliations 保持莫尔斯-波特叶形的矢量束自形变

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-09-12 DOI: 10.1016/j.difgeo.2024.102189

Sergiy Maksymenko

Let M be a smooth manifold and $F$ a Morse-Bott foliation with a compact critical manifold $Σ \subset M$ . Denote by $D (F)$ the group of diffeomorphisms of M leaving invariant each leaf of $F$ . Under certain assumptions on $F$ it is shown that the computation of the homotopy type of $D (F)$ reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of $D (F)$ consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups $D (F)$ for such foliations are also presented.

假设 M 是光滑流形，F 是莫尔斯-波特流形，且有一个紧凑临界流形 Σ⊂M。在 F 的某些假设条件下，D(F) 的同调类型的计算可以简化为三个独立的群：Σ 的差分变形群、Σ 的某个规则邻域的向量束自动变形群以及由固定在 Σ 附近的差分变形组成的 D(F) 子群。文中还举例说明了此类叶形的群 D(F) 的同调类型计算。

引用次数: 0

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