Homogeneous Geodesics in Homogeneous Affine Manifolds

Abstract

For studying homogeneous geodesics in Riemannian and pseudo-Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which involves the reductive decomposition \(\mathfrak{g} = \mathfrak{h} + \mathfrak{m}\) of the Lie algebra \(\mathfrak{g}\) of the isometry group G and the scalar product on \(\mathfrak{m}\) induced by the metric. In the affine differential geometry, there is not such a universal formula. In the present paper, we propose a simple method of investigation of affine homogeneous geodesics. As an application, we describe homogeneous geodesics for homogeneous affine connections in dimension 2 and we find families of affine g.o. spaces in dimension 2. We also solved the problem of the canonical re-parametrization of affine homogeneous geodesics.