Locally symmetric connections on surfaces

Opozda, Barbara

doi:10.1007/bf03323207

Cited by 22 publications

(17 citation statements)

References 6 publications

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“…Then the Ricci tensor of ∇ has rank 1, the connection is locally symmetric nonmetrizable, and, as we shall prove in the next section, none of the connections can be globally realized on a surface in R 3 . Recall that all locally symmetric connections on 2-dimensional manifolds can be locally realized on various surfaces; see [7].…”

Section: Lemma 23 If the Torsion Tensor Of ∇ Is Pointwise Homogeneomentioning

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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Section: Lemma 23 If the Torsion Tensor Of ∇ Is Pointwise Homogeneomentioning

confidence: 99%

Locally homogeneous affine connections on compact surfaces

Opozda

2004

Proc. Amer. Math. Soc.

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“…From the equality dim im R p + dim ker Ric p = 2 [5], where R is the curvature tensor of ∇, Ric its Ricci tensor, im…”

Section: Locally Symmetric Connections On Two-dimensional Manifoldsmentioning

confidence: 99%

On some flat connection associated with locally symmetric surface

Robaszewska

2014

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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Abstract. For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P (M, G). Associated with -satisfying some particular conditions -local basis of T M local connection form of such a connection is an R(G)-valued 1-form Ω build from the dual basis ω 1 , ω 2 and from the local connection form ω of ∇. The structural equations of (M, ∇) are equivalent to the condition dΩ − Ω ∧ Ω = 0.This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.

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“…Here we consider the case of a non-degenerate immersion, dim C M = 2 and we shall study connections of ranks 1 and 2. By the rank of locally symmetric connection we mean, following [8], the (complex) dimension of the subspace…”

Section: H(x Sy ) − H(sx Y ) = 2 Dτ (X Y ) (110)mentioning

confidence: 99%