A Classification of Locally Homogeneous Affine Connections with Skew-Symmetric Ricci Tensor on 2-Dimensional Manifolds

Kowalski, Oldřich; Opozda, Barbara; Vlášek, Zdeněk

doi:10.1007/s006050070041

Cited by 26 publications

(21 citation statements)

References 6 publications

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“…Thus the only possibly non-zero Christoffel symbols are Γ 11 2 and Γ 22 1 . Since it is not possible that (a − 2) = 0 and (−1 + 2a) = 0 simultaneously, we also have Γ 11 2 Γ 22 2 = 0. This implies ρ = 0 so this case is ruled out.…”

Section: Casementioning

confidence: 99%

Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons

Brozos‐Vázquez¹,

Garcı́a-Rı́o²,

Gilkey³

2018

J. Math. Soc. Japan

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The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.2010 Mathematics Subject Classification. 53C21.

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Section: Casementioning

confidence: 99%

Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons

Brozos‐Vázquez¹,

Garcı́a-Rı́o²,

Gilkey³

2018

J. Math. Soc. Japan

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“…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].…”

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confidence: 99%

Locally homogeneous affine connections on compact surfaces

Opozda

2004

Proc. Amer. Math. Soc.

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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.

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“…Subsequently, Arias-Marco and Kowalski [1] extended this classification to the more general setting; a different proof of this result has been given recently by Brozos-Vázquez et al [2]. Previous studies of locally homogeneous surfaces in the torsion free setting include [13,14]. For a different approach in higher dimensions we refer to [4].…”

Section: Symmetric Affine Surfaces With Vanishing Torsionmentioning

confidence: 96%

Symmetric affine surfaces with torsion

D'Ascanio

Gilkey

Pisani

2020

Journal of Geometry and Physics

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We study symmetric affine surfaces which have non-vanishing torsion tensor. We give a complete classification of the local geometries possible if the torsion is assumed parallel. This generalizes a previous result of Opozda in the torsion free setting; these geometries are all locally homogeneous. If the torsion is not parallel, we assume the underlying surface is locally homogeneous and provide a complete classification in this setting as well.2010 Mathematics Subject Classification. 53C21.

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A Classification of Locally Homogeneous Affine Connections with Skew-Symmetric Ricci Tensor on 2-Dimensional Manifolds

Cited by 26 publications

References 6 publications

Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons

Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons

Locally homogeneous affine connections on compact surfaces

Symmetric affine surfaces with torsion

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