Martin A. Magid

Flat affine spheres in R 3

Nondegenerate affine surfaces in R3 which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x2+y2)z=1.

SHAPE OPERATORS OF EINSTEIN HYPERSURFACES IN INDEFINITE SPACE FORMS

The possible shape operators for an Einstein hypersurface in an indefi- nite space form are classified algebraically. If the shape operator A is not diagonal- izable then either ^2-0or^2 = -?>2Id. Introduction. In (F) A. Fialkow classifies Einstein hypersurfaces in indefinite space forms, if the shape operator is diagonalizable at each point. He calls such an immersion proper (p. 764). This paper investigates what happens if the immersion is improper, i.e., if the shape operator is not diagonalizable at a point. It is possible for such a shape operator to have complex eigenvalues or eigenvectors with zero length. The main tool is Petrovs classification of symmetric operators in an indefinite inner product space (P).

Affine 3-spheres with constant affine curvature

We classify the affine hyperspheres in R 4 which have constant curvature in the affine metric h and whose Pick invariant is nonzero. In particular, the metric h must be flat

Affine translation surfaces with constant sectional curvature

In this paper we characterize affine translation surfaces with constant Gaussian curvature. We show that such surfaces must be flat and that one of the defining curves must be planar.

AFFINE UMBILICAL SURFACES IN R(4)

SummaryA surface in ℝ4 is called affine umbilical if for each vector belonging to the affine normal plane the corresponding shape operator is a multiple of the identity. We will classify affine umbilical definite surfaces which either have constant curvature or which satisfy ∇⊥g⊥. Furthermore, it will be shown that for an affine umbilical definite surface, the affine mean curvature vector can not have constant non-zero length.

Indefinite Khler submanifolds with positive index of relative nullity

We deal with complex submanifolds in indefinite space forms. In particular, submanifolds with large index of relative nullity are emphasized. In that context, we prove cylinder theorems in terms of indefinite metrics. We also give a systematic way of constructing a family of new complete and closed indefinite complex submanifolds in the projective setting.In the appendix, we show that the method used for complex cases can be applied to real indefinite geometry. We prove real cylinder theorems including B-scrolls in the general signature. We also show two decomposition lemmas which clarify the relationships between the Hartman-Nirenberg cylinder theorem and slanted cylinder theorems in indefinite geometry.

Affine surfaces in R5 with zero cubic form

Abstract Affine surfaces in R 5 with zero cubic form are classified. The result depends on the rank of the Ricci tensor. For example, if the rank is two, then the surface is part of a Veronese surface

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

AbstractIn this paper we study nondegenerate affine surfaces in the 4-dimensional affine space

Björling problem for timelike surfaces in the Lorentz–Minkowski space

\mathbb{R}^4