Barbara Opozda

Affine Versions of Singer's Theorem on Locally Homogeneous Spaces

A theorem of Singer says that an infinitesimaly homogeneous Riemannian manifold is locally homogeneous. We propose two result on affine connections similar to the theorem of Singer. As an application we prove a theorem giving a sufficient condition for local homogeneity in case of affine connections on 2-dimensional manifolds.

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

On Locally Nonhomogeneous Pseudo–Riemannian Manifolds with Locally Homogeneous Levi–Civita Connections

In this paper we make the first steps to a classification of (pseudo-) Riemannian manifolds which are not locally homogeneous but their Levi–Civita connections are homogeneous. The full classification is given for dimension n = 2; in higher dimensions we prove some substantial partial results. In more generality, we are also interested in the difference between the dimension of the algebra of affine Killing vector fields and that of the algebra of metric Killing vector fields (without any homogeneity properties).

Locally homogeneous affine connections on compact surfaces

Global properties of locally homogeneous and curvature homogeneous affine connections are studied. It is proved that the only locally homogeneous connections on surfaces of genus different from 1 are metric connections of constant curvature. There exist nonmetrizable nonlocally symmetric locally homogeneous affine connections on the torus of genus 1. It is proved that there is no global affine immersion of the torus endowed with a nonflat locally homogeneous connection into R 3 .

Some relations between riemannian and affine geometry

The aim of the paper is to characterize metric normals in terms of affine geometry and derive from that some consequences for affine geometry. Also, an affine affine version of the theorema egregium is proved.

A characterization of affine cylinders

A class of non-metrizable connections is studied. It contains the only non-flat locally symmetric connections existing on affine hypersurfaces of type number 1.

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 inCn+1. Using this general approach we study some curvature conditions for induced connections.

Some equivalence theorems in affine hypersurface theory

Equivalence problems in affine geometry of hypersurfaces with type number greater than one are considered.

On Locally Homogeneous G-structures

The notion of infinitesimal homogeneity is extended to arbitrary connections on G-structures. Two theorems of Singer type are proved for the extended notion. The results are applied to conformal and Weyl structures.

MINIMALITY OF AFFINE LAGRANGIAN SUBMANIFOLDS IN COMPLEX EQUIAFFINE SPACES

The notion of complex equiaffine manifolds is an affine generalization of Calabi–Yau manifolds. Similarly as in the Riemannian case the minimality of affine Lagrangian submanifolds in complex equiaffine spaces can be studied via calibrations, phase functions and variational formulas.