Cayley Hypersurfaces

Quasi-umbilical affine hypersurfaces congruent to their centre map

In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show how to construct them from (n-1)-dimensional affine hyperspheres.

On hypersurfaces satisfying conditions determined by the Opozda-Verstraelen affine curvature tensor

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.

Affine hypersurfaces admitting a pointwise SO(n-1) symmetry

In this article we obtain a classification of strictly locally convex affine hypersurfaces in A^{n+1} for which the geometrical structure is pointwise invariant under the group SO(n-1) represented by rotations around a fixed axis in the tangent space.

On $4$-dimensional Lorentzian affine hypersurfaces with an almost symplectic form

In this paper we study $4$-dimensional affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure $ω$. We prove that the rank of the shape operator is at most one if $R^k\cdot ω=0$ or $\nabla^kω=0$ for some positive integer $k$. This result is the final step in a classification of Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms.

一类新的完备仿射极大类型超曲面

A class of new complete affine maximal type hypersurfaces

本文分类了Calabi仿射几何中具有负常截面曲率的特殊Calabi超曲面，并构造了满足仿射极大类型方程和负常标量曲率Abreu方程的新完备仿射超曲面。

$\widetilde{J}$-tangent affine hypersurfaces with an induced almost paracontact structure

We study real affine hypersurfaces $f\colon M\rightarrow \mathbb{R}^{2n+2}$ with an almost paracontact structure $(φ,ξ,η)$ induced by a $\widetilde{J}$-tangent transversal vector filed, where $\widetilde{J}$ is the canonical paracomplex structure on $\mathbb{R}^{2n+2}$. We give a classification of hypersurfaces for which an induced almost paracontact structure is metric relative to the second fundamental form. Some other properties of such hypersurfaces are also studied.

3-dimensional affine hypersurfaces admitting a pointwise SO(2)- or Z_3-symmetry

In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every point there exists a linear transformation preserving the affine metric, the affine shape operator and the difference tensor K. The study of submanifolds which admit pointwise isometries was initiated by Bryant (math.DG/0007128). In this paper, we consider the 3-dimensional positive definite hypersurfaces for which at each point the group of symmetries is isom…

Affine hypersurfaces admitting a pointwise symmetry

An affine hypersurface is said to admit a pointwise symmetry, if there exists a subgroup of the automorphism group of the tangent space, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. In this paper, we deal with positive definite affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K,S) under the action of SO(3) on an Euclidean vectorspace (V,h) and find a representative (canonical form of K and S) of each (SO(3)/G)-orbit. Then, we classify hypersurfaces a…

Affine translation hypersurfaces in Euclidean and isotropic spaces

In this paper, we extend the notion of affine translation surfaces introduced by Liu and Yu (Proc. Japan Acad. Ser. A Math. Sci. 89, 111--113, 2013) in a Euclidean space R^{3} to higher dimensional ambient spaces. We provide that an affine translation hypersurface of constant Gauss-Kronocker curvature K_{0} in R^{n+1} is a cylinder, i.e. K_{0}=0. As further applications we describe such hypersurfaces in the isotropic spaces satisfying certain conditions on the isotropic curvatures and the Laplacian.

中心仿射几何中产生的一种永恒超曲面流

An eternal hypersurface flow arising in centro-affine geometry

本文研究 $R^{n+1}$ 中一类中心仿射不变超曲面流，证明其存在唯一性、长时间存在性及渐近收敛到椭球面，并分类永恒解。

仿射超曲面与超可积系统

Affine hypersurfaces and superintegrable systems

研究二阶共形超可积系统与仿射超曲面几何的对应关系，证明丰富流形对应于R^{n+1}中的非退化相对仿射超曲面正规化。

Centro-affine hypersurface immersions with parallel cubic form

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, and Köchers conic omega-domains, which are the maximal connected sets consisting of invertible elements in a real semi-simple Jordan algebra. Every level surface of the omega function in an omega-domain is an affine complete, Euclidean complete proper affine hypersphere with para…

Parallel Almost Paracontact Structures on Affine Hypersurfaces

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector field. Such vector field induces in a natural way an almost paracontact structure ${(φ,ξ,η)}$ on $M$ as well as the affine connection $\nabla$. In this paper we give the classification of hypersurfaces with the property that $φ$ or $η$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nablaφ=0$ (respectively $\nablaη…

Isoparametric hypersurfaces in Damek-Ricci spaces

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.

椭球中的等中心仿射极值超曲面

Equi-centro-affine extremal hypersurfaces in ellipsoid

本文研究椭球中的等中心仿射极值超曲面，推导不变面积变分公式，分类多维球面上的极值超曲面与曲线，并建立等中心仿射等周不等式。

Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations

An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a non-degenerate co-oriented hypersurface in flat affine space by its second fundamental form together with either the projective structure induced by the affine normal o…

R^4中Hessian秩为2的仿射齐次超曲面S^3的分类

Classification of Affinely Homogeneous Hessian Rank 2 Hypersurfaces S^3 in R^4

本文确定了R^4中Hessian具有常数秩2的所有仿射齐次超曲面S^3，发现34个不等价终端分支并利用幂级数等价方法通过显式Lie代数描述了这些齐次模型。

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…

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 …

Affine-geometric Wirtinger inequality

A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.

Geodesic Webs of Hypersurfaces

In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic we…

A classification theorem of nondegenerate equiaffine symmetric hypersurfaces

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing …

Equiaffine isoparametric functions and their regular level hypersurfaces

In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$, we first introduce the concept of equiaffine parallel hypersurfaces in $A^{n+1}$, obtaining some fundamental identities with the basic equiaffine geometric invariants, and then we define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces of constant affine mean curvature in $A^{n+1}$. Finally, we introdu…

Curvature-adapted submanifolds of symmetric spaces

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. Various classification results in the Cayley projective and hyperbolic planes and in complex two-plane Grassmannians are obtained under some additional assumptions.

On affine hypersurfaces with everywhere nondegenerate Second Quadratic Form

Consider a closed connected hypersurface in $\mathbb{R}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides $\mathbb{R}^n$ into two pieces. We prove that one of them contains a k-dimensional subspace, and another contains a l-dimensional subspace, thus proving an affine version of Arnold hypothesis. We construct an example of a surface of negative curvature in $\mathbb{R}^3$ with slightly different asymptotical behavior for which the previous claim is wrong.

超曲面上的射影与仿射不变 PDE

Projectively and affinely invariant PDEs on hypersurfaces

本文将构造齐性空间超曲面上 $G$-不变 PDE 的方法应用于射影与仿射空间，证明了此类 PDE 与无迹三次型空间的不变超曲面一一对应。

Weighted interpolation from certain singular affine hypersurfaces

We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.

Completeness in affine and statistical geometry

We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.

通过Calabi仿射几何构造的新的非二次欧氏完备仿射极大型超曲面

New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry

本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。

Singular points of lightlike hypersurfaces of the de Sitter space

The authors study singular points of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 and the geometry of hypersurfaces and use them for construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces.

Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces II

This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.

Lorentzian Geodesic Flows and Interpolation between Hypersurfaces in Euclidean Spaces

We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces viewed as affine hyperplanes. For naturality, we want the geodesic flow to be invariant under rigid transformations and homotheties. Consequently, we do not use the dual projective space, as the geodesic flow in this space is not preserved under translations. Instead we give an alternate approach using a Lorentzian space, which is semi-Riemannian with a metric of…

Bernstein Problem of Affine Maximal Type Hypersurfaces on Dimension N>=3

Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniofrmly convex C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and θ=3/4, and later extended by Jia-Li to N=2, θ\in(3/4,1] (see also [Zhou]). On the past twenty years, much efforts were done toward higher dimensional …

Möbius-flat hypersurfaces in projective space

I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied class of surfaces (n = 3). I extend an example of Akivis-Konnov, and use polynomial conserved quantities to characterise hypersurfaces with flat centro-affine metric among Moebius-flat hypersurfaces. The theory has an obvious counterpart in Lie sphere geometry.

A note on the index of closed minimal hypersurfaces of flat tori

Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients).

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

Quaternionic contact hypersurfaces in hyper-Kähler manifolds

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show that an embedded qc-hypersurface in a hyper-Kähler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kähler metric is degenerate along the hypersurface.

仿射空间中具有Li-规范化的常Gauss-Kronecker曲率超曲面

Hypersurfaces of constant Gauss-Kronecker curvature with Li-normalization in affine space

本文通过研究带梯度爆破边界条件的Monge-Ampère方程，证明了仿射空间中正则域被相对于Li-规范化的常Gauss-Kronecker曲率完备凸超曲面叶状化，并推广了$α=1$情形。

Affine focal sets of codimension $2$ submanifolds contained in hyper surfaces

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singularities. For a given Darboux vector field $ξ$ of the immersion $N\subset M$, one can define the affine metric $g$ and the affine normal plane bundle $\mathcal{A}$. We prove that the $g$-Laplacian of the position vector belongs to $\mathcal{A}$ if and only if $ξ$ is parallel. For umbil…

Curvature and Gauss-Bonnet defect of global affine hypersurfaces

The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.

The geometry of lightlike hypersurfaces of the de Sitter space

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 is directly connected with the geometry of hypersurfaces of the conformal space C^n. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.

On skew loops, skew branes and quadratic hypersurfaces

A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3-space. The paper extends recent work by M. Ghomi and B. Solomon.

On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Gorenstein Algebras

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $π$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_π\subset{\mathfrak m}$. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for ${\Bbb F}={\Bbb C}$ leads to interesting consequences in singularity theory. Also, for ${\Bbb F}={\Bbb R…

Every centroaffine Tchebychev hyperovaloid is ellipsoid

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding pr…

Functions dividing their Hessian determinants and affine spheres

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper affine spheres if some power of it equals a nonzero multiple of its Hessian determinant. The relative invariants of real forms of regular irreducible prehomogeneous vector spaces yield many such polynomials which are moreover irreducible. For example, the nonzero level sets of t…

A floating body approach to Fefferman's hypersurface measure

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

Algebraic theory of affine curvature tensors

We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory.

Ribaucour partial tubes and hypersurfaces of Enneper type

In this article we introduce the notion of a Ribaucour partial tube and use it to derive several applications. These are based on a characterization of Ribaucour partial tubes as the immersions of a product of two manifolds into a space form such that the distributions given by the tangent spaces of the factors are orthogonal to each other with respect to the induced metric, are invariant under all shape operators, and one of them is spherical. Our first application is a classification of all hypersurfaces with dimension at least three of a space form that carry a spherical foliation of codime…

On the affine Gauss maps of submanifolds of Euclidean space

It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface S the set of oriented lines that cross S orthogonally is a Lagrangian submanifold. Conversely, if \bar{S} an n-dimensional family of oriented lines is Lagrangian, there exists, locally, a 1-parameter family of immersed, oriented, parallel hypersurfaces S_t whose tangent spaces cross orthogonally the lines of \bar{S}. The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold S of R^m of dimen…

Ancient Solutions of the Affine Normal Flow

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Ampere equation modeled on those of Ben Andrews and Gutierrez-Huang, a decay estimate for the cubic form under the affine normal flow due to B…

Inexistence of Non-Product Hessian Rank 1 Affinely Homogeneous Hypersurfaces $H^n$ in $\mathbb{R}^{n+1}$ in Dimension $n \geqslant 5$

Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. Then what about higher dimensions? We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian matrix $(F_{x_i x_j})$, a relative affine inv…

Towards a Classification of Homogeneous Tube Domains in C^4

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.

Quadproj: a Python package for projecting onto quadratic hypersurfaces

Quadratic hypersurfaces are a natural generalization of affine subspaces, and projections are elementary blocks of algorithms in optimization and machine learning. It is therefore intriguing that no proper studies and tools have been developed to tackle this nonconvex optimization problem. The quadproj package is a user-friendly and documented software that is dedicated to project a point onto a non-cylindrical central quadratic hypersurface.

针对 $θ\in(0,(N-1)/N]$ 的非二次欧几里得完备仿射极大型超曲面

Non-quadratic Euclidean complete affine maximal type hypersurfaces for $θ\in(0,(N-1)/N]$

本文针对仿射极大型方程的Bernstein问题，显式构造出在改进范围 $N\geq2, θ\in(0,(N-1)/N]$ 内的非椭圆抛物面的欧几里得完备仿射极大型超曲面，为该领域提供了新的反例。