Centroaffine First Order Invariants of Surfaces in IR4

An investigation of the centroaffine geometry of surfaces in IR⁴ leads to the centroaffine first order invariants: the vector bundle valued second fundamental form, the affine semiconformal structure, the h³-semiconformal structure and the centroaffine metric. A classification of surfaces by their semiconformal structures according to signature and rank is given. This involves the study of the orbits of two pencils of symmetric bilinear forms on IR² under a change of basis. Combined with previous results ([Nomizu-Sasaki 93]) a complete classification of the zero-degenerate surfaces is obtained and examples of the other surface types are constructed.     

Centroaffine space curves with constant curvatures and homogeneous surfaces

We study the geometric characters of a centroaffine space curve with vanishing centroaffine curvatures, and classify the centroaffine space curves with constant centroaffine curvatures, which are centroaffine homogeneous curves in ${\mathbb{R}^3}$ . Moreover, we can find a centroaffine homogeneous surface on which such a space curve lies.     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

Configurations of 2 n - 2 Quadrics in R n with 3 · 2 n-1 Common Tangent Lines

   Abstract. We construct 2n-2 smooth quadrics in R ⁿ whose equations have the same degree 2 homogeneous parts such that these quadrics have 3⋅ 2 ^n-1 isolated common real tangent lines. Special cases of the construction give examples of 2n-2 spheres with affinely dependent centres such that all but one of the radii are equal, and of 2n-2 quadrics which are translated images of each other.     

Classification of flat indefinite Equicentroaffinely Homogeneous Surfaces in R4

A nondegenerate equi-centroaffine surface in R⁴ is called homogeneous if for any two points p and q on the surface there exists an equi-centroaffine transformation in R⁴ which takes the surface to itself and takes p to q. In this paper we classify the equi-centroaffinely homogeneous surfaces with flat indefinite metric in R⁴ up to centroaffine transformations.     

The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group

LetG be any compact non-commutative simple Lie group not locally isomorphic to SO(3). We present a generalization of a theorem of Lubotzky, Phillips and Sarnak on distributing points on the sphere S² (or S³) to any homogeneous space ofG, in particular, to all higher dimensional spheres. Our results can also be viewed as a quantitative solution to the generalized Ruziewicz problem for any homogeneous space ofG. Partially supported by DMS-0070544 and DMS-0333397.     

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H²×R or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.     

Centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants

We classify centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants locally, which also gives classification of centroaffine minimal surfaces whose centroaffine curvature and generalized Pick function are constants locally.     

Helly-type theorems for spheres

We prove that (i) a familyF of at leastn+3 spheres inE ⁿ has nonempty intersection if eachn+1 spheres ofF have nonempty intersection, and (ii) if a familyF of spheres inE ⁿ has nonempty intersection, then there existn+1 or fewer spheres inF whose intersection coincides with the intersection of all spheres ofF.Dedicated to Professor Itiro Tamura on his 60th birthday     

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n ² are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n ²)-dimensional submanifolds in for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

Maximum density space packing with parallel strings of spheres

A string of spheres is a sequence of nonoverlapping unit spheres inR ³ whose centers are collinear and such that each sphere is tangent to exactly two other spheres. We prove that if a packing with spheres inR ³ consists of parallel translates of a string of spheres, then the density of the packing is smaller than or equal to . This density is attained in the well-known densest lattice sphere packing. A long-standing conjecture is that this density is maximum among all sphere packings in space, to which our proof can be considered a partial result. The work of A. Bezdek and E. Makai was partially supported by the Hungarian National Foundation for Scientific Research under Grant Number 1238.     

Isoparametric functions on exotic spheres

This paper extends widely the work in [11]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells–Kuiper projective planes are established. In particular, every homotopy n  -sphere (n>4$n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234–235 of [3]).     

On a class of compact homogeneous spaces I

Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M = K/L of positive Euler characteristic are diffeomorphic if and only if the graded rings H ^*(M,Z) and H ^*(M,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Gaussian processes on compact symmetric spaces

Paul Lévy studied Gaussian processes (a) with the parameter a running over Euclidean d-space R ^d and he also studied the case when a runs over the d-sphere S ^d . His results were extended by Gangolli in a number of directions, one being the extension to the cases where the parameter a lies in the other two-point homogeneous Riemannian manifolds. In the compact cases Gangolli showed there was a distinction between spheres and projective spaces, in that the process discovered by Lévy which he called Brownian motion parametrized by spheres does not exist for projective spaces. However many interesting Gaussian process exist with parameters running through projective spaces as we show.Sponsored in part by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation Grant MPS75-06687AO2     

Planforms in three dimensions

The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are calledplanforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE inR ² (see [6, 7]), In this article we attempt to give such a classification for Euclidean equivariant systems of PDE inR ³. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found.We attempt to find the planforms on all lattices inR ³ that are forced to exist near a steady-state bifurcation from a trivial solution. The proof of our classification uses Liapunov-Schmidt reduction with symmetry (which can be used if we assume spatial periodicity of the solutions) and the Equivariant Branching Lemma. The analytical problem of finding planforms for systems of PDE is reduced to the algebraic problem of computing isotropy subgroups with one dimensional fixed point subspaces.The Navier-Stokes equations and reaction-diffusion equations (with constant diffusion coefficients) are examples of systems of PDE that satisfy the conditions of our classifications. In this article, we show that our classification applies to the Kuramoto-Sivashinsky equation.     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

Holomorphic centroaffine immersions and the Lelieuvre correspondence

The rigidity and intrinsic characterization of holomorphic centroaffine immersions are given. We also obtain a method to construct nondegenerate holomorphic affine hypersurfaces from centroaffine immersions and metrics satisfying some conditions. Mathematics subject classification: 53A15.