Infinitesimal bendings of complete Euclidean hypersurfaces

Infinitesimal Bendings for Classes of Two Dimensional Surfaces

Infinitesimal bendings for classes of two-dimensional surfaces in $\mathbb{R}^3$ are investigated. The techniques used to construct the bending fields include reduction to solvability of Bers-Vekua type equations and systems of differential equations with periodic coefficients.

半黎曼流形中超曲面的无穷小性和局部凸性

Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold

给定一个黎曼流形M和M中的一个超曲面H，已知H中一点邻域上的无穷小凸蕴涵局部凸。本文证明了同样的结果在半黎曼流形中也成立。我们对只涉及类时测地线、零测地线或类空测地线的情形作了一些说明。几何凸性的概念也进行了审查，并给出了一些应用的洛伦兹流形的开子集的测地连通性。

平移曲面的等距形变

Isometric deformations of surfaces of translation

本文研究平移曲面的等距形变，给出了无穷小与有限弯曲的存在条件及闭式解，并证明所有平移曲面均允许纯扭转弯曲，特定情形下可获无扭转弯曲。

无穷小 Moebius 可弯超曲面

Infinitesimally Moebius bendable hypersurfaces

本文引入无穷小 Moebius 变分的概念，刻画了允许非平凡无穷小 Moebius 变分的等距浸入，并完成了对 $n\geq 5$ 维无脐点 Euclidean 超曲面在该变分下的分类。

伪欧几里得空间超曲面的同余

Congruence of hypersurfaces of a pseudo-Euclidean space

本文讨论了伪欧几里得空间的超曲面的弯曲问题。证明了相应的定理。

Variation of the Affine Connection in the kinematics of hypersurfaces

Affine deformations serve as basic examples in the continuum mechanics of deformable 3-dimensional bodies (referred as homogeneous deformations). They preserve parallelism and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, the parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time - dependent deformations (motions) after establishing formulas for the variation of the connection, but in the more general context of hypersurfac…

Conformal infinitesimal variations of submanifolds

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds is a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.

有限体积双曲流形中的分支弯曲

Branched Bending in Finite-Volume Hyperbolic Manifolds

本文研究了有限体积双曲流形中的分支弯曲变形，推导了支撑在分支弯曲复形上的变形空间维度下界，并推广了Bart和Scannell的结果。

A topological lower bound for the energy of a unit vector field on a closed hypersurface of the euclidean space. The 3-dimensional case

In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.

具有对称边界的超曲面的对称性

Symmetry of hypersurfaces with symmetric boundary

本文证明具有对称边界和接触角的最小及常平均曲率超曲面内部具有对称性，并推广至高阶平均曲率和Helfrich型超曲面。

Classification of Hessian Rank 1 Affinely Homogeneous Hypersurfaces H^n in R^{n+1} in Dimensions n = 2, 3, 4

In a previous memoir 2202.03030, we showed that in every dimension $n \geq 5$, there exists (unexpectedly) no affinely homogeneous hypersurface $H^n \subset \mathbb{R}^{n+1}$ having Hessian of constant rank 1 (and not being affinely equivalent to a product with $\mathbb{R}^{m \geqslant 1}$). The present work is devoted to determine all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimensions $n = 2, 3, 4$, the cases $n = 1, 2$ being known. With complete details in the case $n = 2$, we illustrate the main features of what can be termed …

一类带正曲率和平点的二维曲面的非刚性

Nonrigidity of a class of two dimensional surfaces with positive curvature and planar points

构造了一类二维曲面的非平凡无穷小弯曲。本文所考虑的曲面是可定向的、紧的、有边界的、除了在许多平点处外处处具有正曲率的、第一同调群为零的曲面，从而得到了这类曲面的非刚性结果.

曲面翻译和折纸镶嵌中无伸长弯曲问题的完整闭式解

Complete closed-form solutions to the problem of inextensional bending for surfaces of translation and origami tessellations

本文提出曲面翻译和折纸镶嵌中无伸长弯曲问题的完整闭式解，揭示三种无限小无伸长变形模式及其对应的伸展、弯曲和扭曲效应。

关于奇异切触结构的切触哈密顿函数

On contact Hamiltonian functions of singular contact structures

该研究针对奇异切触结构上无穷小切触变换研究不足的问题，证明了当切触结构具有带结构光滑Martinet超曲面的第一类奇点时，切触哈密顿函数对应映射为单射，并给出其逆映射的显式局部公式。

$\mathbb{C}^N$ 中 $2$-非退化 CR 超曲面的模型

Models of $2$-nondegenerate CR hypersurface in $\mathbb{C}^N$

本文研究一致 $2$-非退化 CR 超曲面的典范模型，刻画其结构与无限维模空间，给出有限维 Lie 群作用下唯一的定义方程范式，并推广修正符号、导出无穷小对称性的显式公式。

$\mathbb{C}^N$ 中具有任意幂零符号的 $2$-非退化实超曲面的新例子

New examples of $2$-nondegenerate real hypersurfaces in $\mathbb{C}^N$ with arbitrary nilpotent symbols

本文引入了一类在 $\mathbb{C}^N$（$N>3$）中具有秩 $1$ Levi 核的一致 $2$-非退化 CR 超曲面，构成了首个显式无限维族，解决了单 Jordan 块符号结构的等价问题与对称性分类，并证明多块符号结构可由单块结构构造。

7维模型CR超曲面的定义方程

Defining equations of $7$-dimensional model CR hypersurfaces

本文研究了$\mathbb{C}^4$中Levi退化且有限非退化的CR超曲面，获得了实解析一致$2$-非退化模型的完整正规形式，并推导了$9$个齐次$2$-非退化模型的定义方程及其无穷小对称性的显式公式。

三维严格拟凸CR流形的形变与嵌入

Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

本文研究三维严格拟凸CR流形的形变与嵌入问题，证明标准CR三维球面的可嵌入形变空间在原点附近是Frechet子流形，建立修正的Cheng-Lee切片定理并用球谐函数精确刻画可嵌入形变，还构造了从无穷小可嵌入形变出发的规范可嵌入形变族。