Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

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.     

Holomorphic affine vector fields on weil bundles

We obtain necessary and sufficient conditions for a holomorphic vector field to be affine for a holomorphic linear connection defined on aWeil bundle. We also prove that the Lie algebra over R of holomorphic affine vector fields for the natural prolongation of a linear connection from the base to theWeil bundle is isomorphic to the tensor product of theWeil algebra by the Lie algebra of affine vector fields on the base.     

Projectively flat affine surfaces

We study non-degenerate affine surfaces in A³ with a projectively flat induced connection. The curvature of the affine metric , the affine mean curvature H, and the Pick invariant J are related by . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

We study Lorentzian affine hypersurfaces in Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R⁴ is given.     

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.     

A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions

We study a decomposition of a holomorphic vector bundle with connection which need not be endowed with any metrics, which is a generalization of an orthogonal decomposition of a Hermitian holomorphic vector bundle. We first derive several results on the induced connections, the second fundamental forms of subbundles and curvature forms of the connections. We next apply these results to a complex affine immersion. Especially, we give elementary self-contained proofs of the fundamental theorems for a complex affine immersion to a complex affine space.     

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.     

Thomsen surfaces in affine 4-space

Surfaces which are both affine and Euclidean minimal are called Thomsen surfaces. In ³, these surfaces have been completely classified byBarthel, Volkmer andHaubitz. A similar problem, in the Lorentzian 3-space was solved byMagid. In the present paper, we study Thomsen surfaces in ⁴ and show that these surfaces are affine equivalent to the complex paraboloid.The author is a Senior Research Assistant of the National Fund, for Scientific Research (Belgium). This work was done while the author visited Brown University (Providence, USA) in April 93. He would like to thank Professors T. Cecil, M. Magid, and K. Nomizu for their hospitality.     

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).     

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.     

Holomorphic transforms with application to affine processes

In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.     

On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

In this paper, we study locally strongly convex affine hypersurfaces of Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric; it is known that they are affine spheres. In dimension n?7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres.     

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].     

A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft     

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.

     

The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.     

A class of affine maximal hypersurfaces

We describe a class of affine maximal surfaces lying in the threedimensional affine space. Every bounded domain in the plane may covered by an unbounded domain which is the range of definition of an affine maximal surface.     

The affine Cauchy problem

The aim of this paper is to solve the Cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional. These surfaces are called affine maximal surfaces and here, we give a new complex representation which let us describe the solution to the corresponding Cauchy problem. As applications, we obtain a generalized symmetry principle, characterize when a curve in R³ can be a geodesic or pre-geodesic of a such surface and study the helicoidal affine maximal surfaces. Finally, we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set.