Metrizability of connections on two-manifolds

Alena Vanžurová; Petra Žáčková

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2009)

  • Volume: 48, Issue: 1, page 157-170
  • ISSN: 0231-9721

Abstract

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We contribute to the reverse of the Fundamental Theorem of Riemannian geometry: if a symmetric linear connection on a manifold is given, find non-degenerate metrics compatible with the connection (locally or globally) if there are any. The problem is not easy in general. For nowhere flat 2 -manifolds, we formulate necessary and sufficient metrizability conditions. In the favourable case, we describe all compatible metrics in terms of the Ricci tensor. We propose an application in the calculus of variations.

How to cite

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Vanžurová, Alena, and Žáčková, Petra. "Metrizability of connections on two-manifolds." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 157-170. <http://eudml.org/doc/35190>.

@article{Vanžurová2009,
abstract = {We contribute to the reverse of the Fundamental Theorem of Riemannian geometry: if a symmetric linear connection on a manifold is given, find non-degenerate metrics compatible with the connection (locally or globally) if there are any. The problem is not easy in general. For nowhere flat $2$-manifolds, we formulate necessary and sufficient metrizability conditions. In the favourable case, we describe all compatible metrics in terms of the Ricci tensor. We propose an application in the calculus of variations.},
author = {Vanžurová, Alena, Žáčková, Petra},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Manifold; linear connection; metric connection; pseudo-Riemannian geometry; linear connection; metric connection; pseudo-Riemannian geometry},
language = {eng},
number = {1},
pages = {157-170},
publisher = {Palacký University Olomouc},
title = {Metrizability of connections on two-manifolds},
url = {http://eudml.org/doc/35190},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Vanžurová, Alena
AU - Žáčková, Petra
TI - Metrizability of connections on two-manifolds
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 157
EP - 170
AB - We contribute to the reverse of the Fundamental Theorem of Riemannian geometry: if a symmetric linear connection on a manifold is given, find non-degenerate metrics compatible with the connection (locally or globally) if there are any. The problem is not easy in general. For nowhere flat $2$-manifolds, we formulate necessary and sufficient metrizability conditions. In the favourable case, we describe all compatible metrics in terms of the Ricci tensor. We propose an application in the calculus of variations.
LA - eng
KW - Manifold; linear connection; metric connection; pseudo-Riemannian geometry; linear connection; metric connection; pseudo-Riemannian geometry
UR - http://eudml.org/doc/35190
ER -

References

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