Geodesic
On metrizability of locally homogeneous affine 2-dimensional manifolds
Archivum mathematicum, Tome 49 (2013) no. 5, pp. 347-357 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.5817/AM2013-5-347
Classification : 53B05, 53B20
Keywords: manifold; connection; metric 
@article{10_5817_AM2013_5_347,
     author = {Van\v{z}urov\'a, Alena},
     title = {On metrizability of locally homogeneous affine 2-dimensional manifolds},
     journal = {Archivum mathematicum},
     pages = {347--357},
     year = {2013},
     volume = {49},
     number = {5},
     doi = {10.5817/AM2013-5-347},
     mrnumber = {3159333},
     zbl = {06383796},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2013-5-347/}
}
TY  - JOUR
AU  - Vanžurová, Alena
TI  - On metrizability of locally homogeneous affine 2-dimensional manifolds
JO  - Archivum mathematicum
PY  - 2013
SP  - 347
EP  - 357
VL  - 49
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2013-5-347/
DO  - 10.5817/AM2013-5-347
LA  - en
ID  - 10_5817_AM2013_5_347
ER  - 
%0 Journal Article
%A Vanžurová, Alena
%T On metrizability of locally homogeneous affine 2-dimensional manifolds
%J Archivum mathematicum
%D 2013
%P 347-357
%V 49
%N 5
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2013-5-347/
%R 10.5817/AM2013-5-347
%G en
%F 10_5817_AM2013_5_347
Vanžurová, Alena. On metrizability of locally homogeneous affine 2-dimensional manifolds. Archivum mathematicum, Tome 49 (2013) no. 5, pp. 347-357. doi: 10.5817/AM2013-5-347

Cité par Sources :