On metrizability of locally homogeneous affine 2-dimensional manifolds

Alena Vanžurová

Archivum Mathematicum (2013)

  • Volume: 049, Issue: 5, page 347-357
  • ISSN: 0044-8753

Abstract

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In [19] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated by examples. Note that in higher dimension, the problem is not easy to solve. Here we try to apply this apparatus to the two main types (A and B from [9], [1]) of torsion-less locally homogeneous connections defined in open domains of 2-manifolds. We prove that in dimension two a symmetric linear connection with constant Christoffels is metrizable if and only if it is locally flat. On the other hand, in the class of connections of type B there are even non-flat metrizable connections.

How to cite

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Vanžurová, Alena. "On metrizability of locally homogeneous affine 2-dimensional manifolds." Archivum Mathematicum 049.5 (2013): 347-357. <http://eudml.org/doc/260800>.

@article{Vanžurová2013,
abstract = {In [19] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated by examples. Note that in higher dimension, the problem is not easy to solve. Here we try to apply this apparatus to the two main types (A and B from [9], [1]) of torsion-less locally homogeneous connections defined in open domains of 2-manifolds. We prove that in dimension two a symmetric linear connection with constant Christoffels is metrizable if and only if it is locally flat. On the other hand, in the class of connections of type B there are even non-flat metrizable connections.},
author = {Vanžurová, Alena},
journal = {Archivum Mathematicum},
keywords = {manifold; connection; metric; manifold; connection; metric},
language = {eng},
number = {5},
pages = {347-357},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On metrizability of locally homogeneous affine 2-dimensional manifolds},
url = {http://eudml.org/doc/260800},
volume = {049},
year = {2013},
}

TY - JOUR
AU - Vanžurová, Alena
TI - On metrizability of locally homogeneous affine 2-dimensional manifolds
JO - Archivum Mathematicum
PY - 2013
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 049
IS - 5
SP - 347
EP - 357
AB - In [19] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated by examples. Note that in higher dimension, the problem is not easy to solve. Here we try to apply this apparatus to the two main types (A and B from [9], [1]) of torsion-less locally homogeneous connections defined in open domains of 2-manifolds. We prove that in dimension two a symmetric linear connection with constant Christoffels is metrizable if and only if it is locally flat. On the other hand, in the class of connections of type B there are even non-flat metrizable connections.
LA - eng
KW - manifold; connection; metric; manifold; connection; metric
UR - http://eudml.org/doc/260800
ER -

References

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