On a generic defining function of projective motion in the rigid 6-dimensional h-spaces

圆堆积的射影刚性

Projective Rigidity of Circle Packings

本文研究亏格至少为二的曲面上与给定三角剖分相容的圆堆积，证明其在复射影结构下具有射影刚性，并表明圆堆积空间是复射影结构空间的子流形。

锥结构的刚性定理

Rigidity theorems for cone structures

该研究针对从常微分方程经典范构造得到的锥结构，证明以n维射影空间一般曲线或3维射影空间直纹曲面为模型的平凡锥结构是平坦的，并探讨了其在相关领域的应用。

Convex real projective structures and Weil's local rigidity Theorem

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using real projective structures.

平面上点线构型的射影刚性

Projective rigidity of point-line configurations in the plane

本文提出“射影刚性矩阵”框架研究平面上点线构型的保关联运动，并引入对称自适应矩阵分析对称构型及其保对称运动。

On projective group properties of the $6D$ pseudo-Riemannian space

We study the six-dimensional pseudo-Riemannian spaces with two time-like coordinates that admit non-homothetic infinitesimal projective transformations. The metrics are manifestly obtained and the projective group properties are determined. We also find a generic defining of projective motion in the 6-dimensional rigid h-space.

Projective and Finsler metrizability: parameterization-rigidity of the geodesics

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

On projective and affine equivalence of sub-Riemannian metrics

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and a…

Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with $C^1$ boundary are the projective balls.

Rigid 6-dimensional h-spaces of constant curvature

In this paper, we continue studying the 6-dimensional pseudo-Riemannian space V^6(g_{ij}) with signature [++--], which admits projective motions, i. e. continuous transformation groups preserving geodesics. In particular, we determine a necessary and sufficient condition that the 6-dimensional rigid h-spaces have constant curvature.

调和带理论：从2-环面到复射影空间的非零度调和映射的刚性

Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space

研究非零度调和映射的刚性，通过全纯嵌入与完备线性系统关联的holomorphic嵌入构造，证明其刚性，并扩展到无特殊超振荡点的全纯嵌入，附加对Fubini-Study辛形式拉回的假设。

Invariant rigid geometric structures and smooth projective factors

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra i…

Riemannian流形的面积宽度的刚性定理

Rigidity theorems for the area widths of Riemannian manifolds

本文证明了紧致Riemannian流形体积谱的刚性定理，包括二维射影平面度量由体积谱决定、三维球面Zoll度量由面积宽度刻画，并建立了三维射影空间的锐利面积systolic不等式。

Spectral rigidity of complex projective spaces, revisited

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact Kähler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this Kähler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one i…

Rigid Flat Webs on the Projective Plane

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations which g…

Rigidity of convex divisible domains in flag manifolds

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.

Projective duality and K-energy asymptotics

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry.

Differential geometry of submanifolds of projective space

These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.