A generalization of Lelieuvre’s formula to equiaffine immersions of general codimension

A non-degenerate equiaffine immersion of codimension one into an equiaffine space is locally expressed in terms of its conormal map and its affine fundamental form. The expression is called the Lelieuvre’s formula. We recently defined the notions of an equiaffine immersion of general codimension and its transversal volume element map. In this paper, we locally express a non-degenerate equiaffine immersion of general codimension into an equiaffine space in terms of its transversal volume element map and its affine fundamental form.     

The total absolute curvature of an equiaffine immersion

For an equiaffine immersion of general codimension, we recently defined the Lipschitz-Killing curvature, where we fix a transversal bundle and a transversal volume element. The main purpose of this paper is to show that the integration of the absolute value of the Lipschitz-Killing curvature is independent of the choices of a transversal bundle and a transversal volume element.     

The Fundamental Theorems for Affine Immersionsinto Hyperquadrics and its Applications

 We prove the fundamental theorems for affine immersions into hyperquadrics (including affine spaces) with arbitrary codimension, which are generalizations of those for isometric immersions into space forms. As applications, the fundamental theorems for equiaffine immersions into hyperquadrics with arbitrary codimension are obtained. (Received 10 February 2000)     

The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type

In this paper, we define an equiaffine immersion of general codimension and the Lipschitz-Killing curvature for the immersion. Furthermore, we prove theorems of Gauss-Bonnet type and Chern-Lashof type for the immersion.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

The Center Map of an Affine Immersion

We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows. Received: May 24, 2006. Revised: July 26, 2006. Accepted: July 28, 2006.     

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.     

保持高斯映射的仿射形变

给定Ｒｉｅｍａｎｎ流形到欧氏空间的仿射浸入ｆ：　Ｍｎ→ＲＮ　　，我们建立存在另一个与ｆ有相同高斯映射的仿射浸入ｆ：Ｍｎ→ＲＮ　　的条件，进一步利用这个条件，解答了仿射浸入的高斯映射将其确定到何种程度的问题．     

Flat Affine Surfaces in \mathbb{R}^4 with Flat Normal Connection

In this paper we study nondegenerate affine surfaces in the 4-dimensional affine space . We assume that both the connection and the normal connection induced by the canonical equiaffine transversal bundle are flat. Surfaces with constant equiaffine transversal bundle are trivial examples of such surfaces. Here, we obtain a complete classification of all such surfaces which do not have constant equiaffine normal bundle.     

The fundamental theorem forC∞ immersed affine hypersurfaces with type changing Blaschke metric

We investigate the conormal geometry of relative affine hypersurfaces whose relative metric (Blaschke metric) degenerates on a codimension 1 submanifold. Such hypersurfaces arise in the investigation of compact hypersurfaces which are not diffeomorphic to the sphere. We give a fundamental theorem in terms of the conormal structure. Finally, we present a new, affinely invariant tensor which is defined at the set where the relative metric is degenerate.     

Gauge Theory, Isotropy, and Surfaces in Affine 4-Space

The surface theory in the equiaffine space R⁴ is developed on the basis of H. Weyl’s gauge theory. Rescaling of the Weyl geometry leads to a 1-parameter family of invariant transversal plane bundles containig former special constructions. A transversal bundle metric is gained via the notion of isotropy. The paper then proceeds with a general tensorial theory, including theorema egregium and Radon-type results and a discussion of cubic fundamental forms. Finally there is given an application to homogeneous surfaces.     

On the fundamental theorem for non-degenerate complex affine hypersurface immersions

Two geometric versions of the fundamental theorem for non-degenerate complex affine hypersurface immersions are proved. We consider non-degenerate complex affine hypersurface immersions with complex transversal connection form (or equivalently, with holomorphic normalization) and prove that the conormal map is a holomorphic map. These considerations inspired the definitions of complex semi-compatible and complex semi-conjugate connections. This allows us to formulate the integrability conditions of the fundamental theorem, on one hand in terms of the induced connection, which has to be complex semi-compatible and dualH-projective flat, and on the other hand, in terms of its semi-conjugate connection, which has to beH-projective flat. Using this results, we formulate the conditions of the fundamental theorem in terms of anyH-projective flat complex affine connection.Research partially supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridcki.     

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).     

Inducing, slopes, and conjugacy classes

We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.     

Immersion ofn-dimensional cylindrical metrics in the form of cylindrical surfaces into Lobachevsky spaces

A definition of l -dimensional k-cylindrical metrics is introduced and it is proved that k-cylindrical metrics admit an immersion in the form of k-cylindrical surfaces into Lobachevsky space L^l+p, where codimension p of the immersion coincides with the codimension of the immersion of the base of the cylindrical metric into a Euclidean or Lobachevsky space.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 66–69, 1991.     

Uniformisation de l’espace des feuilles de certains feuilletages de codimension un

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a “rigidly” embedded invariant hypersurface.     

A moduli space of minimal affine Lagrangian submanifolds

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Fréchet manifold (if it is not Ø).     

Translationsflächen in der äquiaffinen Differentialgeometrie

We discuss with equiaffine methods the surfaces of translation with plane generating curves in the three-dimensional affine space. Using (pseudo-) isothermic parameters we determine in this class all the affine minimal surfaces (which include the affine spheres, the quadrics and the ruled surfaces), all the surfaces with vanishing affine Gauss curvature (which include the surfaces with constant non vanishing affine mean curvature), and all the surfaces with only one family of affine lines of curvature.     

An extrinsic decomposition theorem and a slant tube theorem for a curvature netted hypersurface

In this paper, we treat hypersurfaces in a Euclidean space the number of whose distinct principal curvatures is constant almost everywhere. We call such a hypersurface satisfying certain additional condition a curvature netted hypersurface. First we shall define the notions of a twisted (or warped) sum immersion, a slant focal map and a slant tube. We shall investigate, in what case, a complete curvature netted hypersurface is immersed by a warped sum immersion or becomes a slant tube of the image of a slant focal map.