Franki Dillen

Conjugate Connections and Radon's Theorem in Affine Differential Geometry.

For a given nondegenerate hypersurfaceMn in affine space ℝn+1 there exist an affine connection ∇, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=∇h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ∇ and a nondegenerate metrich on a differentiable manifoldMn such that ∇h is totally symmetric and satisfies the apolarity condition relative toh, canMn be locally immersed in ℝn+1 in such a way that (∇,h) is realized as the induced structure?In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ∇. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (∇,h) is that the conjugate connection of ∇ relative toh is projectively flat.

Calabi-type composition of affine spheres

Abstract We give several ways of constructing new nondegenerate affine hypersurfaces in R nn+1 starting from some given proper or improper affine spheres. The resulting hypersurfaces can be forced to have nice properties, if the original affine spheres are chosen appropriately. In this way we obtain new examples of nondegenerate locally homogeneous affine hypersurfaces and of proper and improper affine spheres. We also discuss the properties of the affine metric of the hypersurface, such as the curvature or completeness. Finally we discuss the relation with projective transformations.

Constant angle surfaces in ℍ2 × ℝ

We classify all surfaces in ℍ2 × ℝ for which the unit normal makes a constant angle with the ℝ-direction. Here ℍ2 is the hyperbolic plane.

Totally real submanifolds in ⁶(1) satisfying Chen’s equality

In this paper, we study 3-dimensional totally real submanifolds of S6(1). If this submanifold is contained in some 5-dimensional totally geodesic S5(1), then we classify such submanifolds in terms of complex curves in CP 2(4) lifted via the Hopf fibration S5(1)→ CP 2(4). We also show that such submanifolds always satisfy Chen’s equality, i.e. δM = 2, where δM (p) = τ(p)−inf K(p) for every p ∈ M . Then we consider 3-dimensional totally real submanifolds which are linearly full in S6(1) and which satisfy Chen’s equality. We classify such submanifolds as tubes of radius π/2 in the direction of the second normal space over an almost complex curve in S6(1).

Lagrangian isometric immersions of a real-space-form M^n(c) into a complex-space form M^n(4C)

It is well known that totally geodesic Lagrangian submanifolds of a complexspace-form f M n (4c) of constant holomorphic sectional curvature 4c are real-spaceforms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form f M n (4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1 I1 fk Ik 1 N n k (c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simplyconnected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-spaceform f M n (4c). Conversely, if L: M n (c) ! f M n (4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-formM n (c) of constant sectional curvaturec into a complex-space-form f M n (4c), thenM n (c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

Einstein, conformally flat and semi-symmetric submanifolds satisfying chen’s equality

AbstractIn a recent paper, B. Y. Chen proved a basic inequality between the intrinsic scalar invariants infK andτ ofMn, and the extrinsic scalar invariant |H|, being the length of the mean curvature vector fieldH ofMn in

Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms

\mathbb{E}^m

Constant Angle Surfaces in

. In the present paper we classify the submanifoldsMn of