Metrization problem for linear connections and holonomy algebras

Alena Vanžurová

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 511-521
  • ISSN: 0044-8753

Abstract

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We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. Compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of holonomy algebra in comparison with certain more classical approaches, and propose a possible application in the Calculus of Variations (for a particular type of second order system of ODE’s, which define geodesics of a linear connection, components of a metric compatible with the connection play the role of variational multipliers).

How to cite

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Vanžurová, Alena. "Metrization problem for linear connections and holonomy algebras." Archivum Mathematicum 044.5 (2008): 511-521. <http://eudml.org/doc/250501>.

@article{Vanžurová2008,
abstract = {We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. Compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of holonomy algebra in comparison with certain more classical approaches, and propose a possible application in the Calculus of Variations (for a particular type of second order system of ODE’s, which define geodesics of a linear connection, components of a metric compatible with the connection play the role of variational multipliers).},
author = {Vanžurová, Alena},
journal = {Archivum Mathematicum},
keywords = {manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra; manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra},
language = {eng},
number = {5},
pages = {511-521},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Metrization problem for linear connections and holonomy algebras},
url = {http://eudml.org/doc/250501},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Vanžurová, Alena
TI - Metrization problem for linear connections and holonomy algebras
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 511
EP - 521
AB - We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. Compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of holonomy algebra in comparison with certain more classical approaches, and propose a possible application in the Calculus of Variations (for a particular type of second order system of ODE’s, which define geodesics of a linear connection, components of a metric compatible with the connection play the role of variational multipliers).
LA - eng
KW - manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra; manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra
UR - http://eudml.org/doc/250501
ER -

References

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  12. Schmidt, B. G., 10.1007/BF01661152, Comm. Math. Phys. 29 (1973), 55–59. (1973) MR0322726DOI10.1007/BF01661152
  13. Thompson, G., Local and global existence of metrics in two-dimensional affine manifolds, Chinese J. Phys. 19 (6) (1991), 529–532. (1991) 
  14. Vanžurová, A., Linear connections on two-manifolds and SODE’s, Proc. Conf. Aplimat 2007 (Bratislava, Slov. Rep.), Part II, 2007, pp. 325–332. (2007) 
  15. Vilimová, Z., The problem of metrizability of linear connections, Master's thesis, Opava, 2004, supervisor: O. Krupková. (2004) 

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