Locally homogeneous affine connections on compact surfaces

Abstract

Abstract. 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 .The best known locally homogeneous affine connections are flat ones. J. Milnor, [3], proved that there are no flat affine connections on surfaces of genus different from 1. We prove an analogous result for nonflat locally homogeneous affine connnections. We prove, in fact, that a nonflat torsion-curvature homogeneous of order 1 connection on a compact surface of genus different from 1 must be torsionfree metric and locally symmetric; that is, it must be a Levi-Civita connection of constant curvature.In contrast with this situation, there do exist nonlocally symmetric nonmetrizable locally homogeneous connections on the torus of genus 1. We give a family of such connections.Note that, unlike in the case of Riemannian structures, the class of locally homogeneous affine connections on 2-dimensional manifolds is much richer than the class of locally symmetric ones. Some results concerning local homogeneity are proved in [8], [9] and [10]. A local classification of locally homogeneous connections on 2-dimensional manifolds is given in [5] and [12].An important question in the theory of affine connections is whether a manifold equipped with a connection ∇ can be realized as a hypersurface (immersed or imbedded) in a standard affine space in such a way that there exists a transversal bundle inducing ∇. We study the realization problem for the torus equipped with nonflat locally homogeneous connections. Using the notion of extrinsic completeness, we prove that there is no immersion of the torus into R 3 for which an induced connection is nonflat and locally homogeneous.

Locally homogeneous rigid geometric structures on surfaces

“…The previous result combined with proposition 3.4 imply the main result in [32]: Theorem 3.6. (Opozda) A compact surface M bearing a locally homogeneous affine connection of non Riemannian type is a torus.…”

“…If φ is an affine connection this was first proved by B. Opozda [32] for the case torsion-free (see also the group-theoretical approach in [22]), and then by T. Arias-Marco and O. Kowalski in the case of arbitrary torsion [3]. Theorem 1.1 stands, in particular, for projective connections.…”

Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

“…The connections on the 2-torus, discussed below, are among those described by Opozda [17,Example 2.10]. Our description is different, for reasons dictated by our applications.…”

“…where the unknown is a twice-covariant symmetric tensor field τ on a surface Σ carrying an equiaffine projectively flat torsionfree connection D, while ±α is a D-parallel area element (Section 7), and L is given by (15). The value of ε is of no consequence for solvability of ( 17), since L is linear; in other words, solving (17) means, up to a factor, finding τ such that Lτ is parallel and nonzero. We have the following result.…”

The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor

“…The theory of connections with torsion plays an important role in string theory [1,13,15,17], they are important in almost contact geometry [14,18,25,28], they play a role in non-integrable geometries [1,2,3,7], they are important in spin geometries [19], they are useful in considering almost hypercomplex geometries [24], they appear in the study of compact solvmanifolds [12], and they have been used to study the non-commutative residue for manifolds with boundary [29]. The following result was first proved in the torsion free setting by Opozda [26] and subsequently extended to surfaces with torsion by Arias-Marco and Kowalski [4], see also [4,11,16,21,22,27] for related work.…”