Level surfaces of non-degenerate functions inR n+1

We study level surfaces of non-degenerate functions inR ⁿ⁺¹. Such level surfaces are non-degenerate in the sense of affine differential geometry. In affine differential geometry, the affine normal plays an important role for the study of a non-degenerate hypersurface. In this note, being motivated by Koszul's work we take a canonical vector field for level surfaces of a non-degenerate function and give certain characterizations of when is transversal, by the shape operatorS, the transversal connection , and consider the difference between and the affine normal.     

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given： The affine immersions with pointwise planar normal sections （with respect to the equiaffine transversal bundle） are parallel. This result is verified for surfaces in R4 and for immersions with the maximal codimension.     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

Willmore surfaces in the four-sphere

Willmore immersions of an orientable surface X in the n-dimensionalsphere appear as the extremal points of a conformally invariant variational problem in the space of all immersions f: X S ⁿ.In this paper we will study Willmore immersions of the differentiable two-sphere in S ⁴, using the method of moving frames and Cartan's conformal structures.The work on this paper was partially supported by a Fellowship of the Consiglio Nazionale delle Ricerche.     

Linear spaces withn + n + 2lines

It is shown that every non-degenerate linear space withn² + n + 2lines,n ≥ 6, hasv n² + 1 − epoints, whereeis the unique positive real number withn =½e(e + 1). For values ofnfor whicheis an integer, it is shown that the linear spaces withn² + 1 − epoints andn² + n + 2lines are related to symmetric divisible designs.     

Integral Geometry on Discrete Grassmannians in Z n

We study the Radon transform R on the discrete Grassmannian of rank-d affine sublattices of Z ⁿ for 0 <  d <  n, extending and building on previous work of the first- and third-named authors in codimension 1. By analogy with the integral geometry on Grassmannians in R ⁿ , various natural questions are treated, such as definition and properties of R and its dual transform R ^*, function space setting, support theorems and inversion formulas.     

Nonholonomic (n + 1)-webs

We consider a nonholonomic (n + 1)-web NW on an n-dimensional manifold M, i.e., n + 1 codimension 1 distributions on M. We prove that a web NW on M is equivalent to a G-structure with structure group λE, the group of scalar matrices. We find the structure equations of a web NW and the integrability conditions of the distributions of a web NW. It is shown that on a manifold with nonholonomic (n + 1)-web an affine connection Γ arises naturally for which the distributions of the web are totally geodesic. We consider the case when the connection Γ has zero curvature and, in particular, when a web NW is defined by invariant distributions on a Lie group. In the case when all distributions of a web NW on a Lie group are integrable, we find the equations of this group in terms of local coordinates.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations

We present an algorithm for linear programming which requires O(((m+n)n ²+(m+n)^1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of .     

Bijections of #x211D; n onto itself

We characterize affine and continuous maps within the class of bijections of #x211D; ⁿ onto itself by the preservation of various geometric or topological figures. A characterization of similarity maps of Hilbert space is given.Supported in part by a grant from the National Science Foundation.     

Parallel surfaces in affine 4- space

We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in R⁴ which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the beginning of this century by Blaschke and his collaborators.     

Euler Tours of Maximum Girth in K2 n +1 and K2 n ,2 n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g^E(G)= maxg(T) where the maximum is taken over all Euler tours of G.We prove that g^E(K_2n,2n)=4n–4 and 2n–3g^E(K_2n+1)2n–1 for any n2. We also show that g^E(K₇)=4. We use these results to prove the following:1)The graph K_2n,2n can be decomposed into edge disjoint paths of length k if and only if k4n–1 and the number of edges in K_2n,2n is divisible by k.2)The graph K_2n+1 can be decomposed into edge disjoint paths of length k if and only if k2n and the number edges in K_2n+1 is divisible by k.     

Graph immersions with parallel cubic form

We consider non-degenerate graph immersions into affine space ${\mathbb{A}}^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,\gamma )$, where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.     

The cardinality of FM(1+1+n)

The aim of this note is to develop a counting formula for the modular lattice FM(1+1+n) freely generated by two single elements and an n-element chain. This answers Problem 44 in Birkhoff [1] which asks one to determine FM(1+1+n). The proof of our recursive formula is based on the scaffolding method developed by R. Wille.     

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

The Graded Cobordism Group of Codimension-One Immersions

The cobordism group N(Mⁿ) of codimension-one immersions in the n-manifold Mⁿ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group gr*N(Mⁿ). We introduce some new invariants for immersions enlightening the Atiyah–Hirzebruch spectral sequence associated to N(M), which are of combinatorial-geometric nature. Explicit computations are developed for n 7, and the group structure is also investigated for orientable 4-manifolds.     

3·2 r+1 -Cycle Systems of K n −C n

 In this paper we investigate the necessary and sufficient conditions for the existence of 3 · 2 ^{r +1}-cycle system of K _n – C _n , where C _n is any Hamilton cycle of K _n and r ≥ 0. Received: December 17, 1999 Final version received: July 25, 2000 Present address: 15 Guang Ming 10 St. Sec. 1, Chu-Bei, Hsin-Chu, 302, Taiwan, ROC     

Skewing (n+1) tensor indices where the index range is n

It is known that any totally skew quantity with (n?+?1) indices, each of which ranges over n values, vanishes identically. The aim of this short note is to show that this is equivalent to the simple fact that any (n?+?1) vectors in an n-dimensional vector space are linearly dependent.