Equivalence theorems in affine differential geometry

In this paper we establish an affine equivalence theorem for affine submanifolds of the real affine space with arbitrary codimension. Next, this theorem is used to prove the classical congruence theorem for submanifolds of the Euclidean space, and to prove some results on affine hypersurfaces of the real affine space.Research Assistant of the National Fund for Scientific Research (Belgium).     

Generic equi-centro-affine differential geometry of plane curves

We study the equi-centro-affine invariants of plane curves from the view point of the singularity theory of smooth functions. We define the notion of the equi-centro-affine pre-evolute and pre-curve and establish the relationship between singularities of these objects and geometric invariants of plane curves.     

Oversampling and reconstruction functions with compact support

Assume that a sequence of samples of a filtered version of a function in a shift-invariant space is given. This paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. This is done in the light of the generalized sampling theory by using the oversampling technique. A necessary and sufficient condition is given in terms of the Smith canonical form of a polynomial matrix. Finally, we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm.     

Sub-Finsler geometry in dimension three

We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.     

A sextic holomorphic form of affine surfaces with constant affine mean curvature

We extend an original idea of Calabi for affine maximal surfaces and define a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. We get some rigidity results for affine complete surfaces by using this sextic holomorphic form. Received: 17 May 2003     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, V _D ^m (r) the volume of a small geodesic ball for the connection D and C C _D ¹ the first non-trivial term of the Taylor expansion of V _D ^m (r). NK-manifolds are characterized in terms of C C _D ¹ and a family of Hermitian manifolds for which _M C C _D ¹ dvol is a spectral invariant is given and one proves that C C _D ¹ and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.     

Strictly locally order affine complete lattices

We call a latticeL strictly locally order-affine complete if, given a finite subsemilatticeS ofL ⁿ, every functionf: S L which preserves congruences and order, is a polynomial function. The main results are the following: (1) all relatively complemented lattices are strictly locally order-affine complete; (2) a finite modular lattice is strictly locally order-affine complete if and only if it is relatively complemented. These results extend and generalize the earlier results of D. Dorninger [2] and R. Wille [9, 10].     

Lagrangian submanifolds in affine symplectic geometry

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.     

Some classes of sprays in projective spray geometry

This paper studies some properties of projective changes in spray and Finsler geometry. Firstly, it obtains a comparison theorem on Ricci curvature for projectively related Finsler metrics. Secondly, it studies the properties of a class of projectively flat sprays, which particularly shows that there exist many isotropic sprays that cannot be induced by any (even singular) Finsler metrics.     

CR-structures and Lorentzian geometry

As a recent excellent example of mutual interplay between the Cauchy-Riemann structure and physical spacetime geometry, we present, in this paper, a few fresh ideas on this fruitful relationship with respect to the conformal geometry and the groups of motions of Lorentzian manifolds.     

Structure of Hiai-Petz parametrized geometry for positive definite matrices

Recently Hiai-Petz (2009) [10] discussed a parametrized geometry for positive definite matrices with a pull-back metric for a diffeomorphism to the Euclidean space. Though they also showed that the geodesic is a path of operator means, their interest lies mainly in metrics of the geometry. In this paper, we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a Finsler one. Also we discuss another type of geometry in Hiai and Petz (2009) [10] which is a generalization of Corach-Porta-Recht’s one [3].     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

Singularities of the Minkowski set and affine equidistants for a curve and a surface

Generic singularities of envelopes of families of chords and bifurcations of affine equidistants defined by a pair of a curve and a surface in R³ are classified. The chords join pairs of points of the curve and the surface such that the tangent line to the curve is parallel to the tangent plane to the surface. The classification contains singularities of stable Lagrange and Legendre projections, boundary singularities and some less known classes appearing at the points of the surface and the curve themselves.     

The Ribaucour transformation in Lie sphere geometry

We discuss the Ribaucour transformation of Legendre (contact) maps in its natural context: Lie sphere geometry. We give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer-Tojeiro as well as a higher dimensional version with the combinatorics of a cube. We also show how these theorems descend to the corresponding results for submanifolds in space forms.     

On some properties of the curvature and Ricci tensors in complex affine geometry

We present a new approach — which is more general than the previous ones — to the affine differential geometry of complex hypersurfaces inC ⁿ⁺¹. Using this general approach we study some curvature conditions for induced connections.The research supported by Alexander von Humboldt Stiftung and KBN grant no. 2 P30103004.     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

Harmonicity of gradient mappings of level surfaces in a real affine space

In our previous paper [4] we have investigated level surfaces of a non-degenerate function in a real affine space A ⁿ⁺¹ by using the gradient vector field . We gave characterizations of by means of the shape operatorS, the transversal connection , and studied the difference between and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and =0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].