# Metrization problem for linear connections and holonomy algebras

Archivum Mathematicum (2008)

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

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topVanžurová, 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

top- Anastasiei, M., Metrizable linear connections in vector bundles, Publ. Math. Debrecen 62 (3-4) (2003), 277–287. (2003) MR2008096
- Cheng, K. S., Ni, W. T., Necessary and sufficient conditions for the existence of metrics in two-dimensional affine manifolds, Chinese J. Phys. 16 (1978), 228–232. (1978)
- Eisenhart, L. P., Veblen, O., The Riemann geometry and its generalization, Proc. London Math. Soc. 8 (1922), 19–23. (1922)
- Gołab, S., Über die Metrisierbarkeit der affin-zusammenhängenden Räume, Tensor, N. S. 9 (1959), 132–137. (1959)
- Jakubowicz, A., Über die Metrisierbarkeit der affin-zusammenhängenden Räume, Tensor, N. S. 14 (1963), 132–137. (1963) Zbl0122.40501MR0161263
- Jakubowicz, A., Über die Metrisierbarkeit der affin-zusammenhängenden Räume, II Teil, Tensor, N.S. 17 (1966), 28–43. (1966) MR0195021
- Jakubowicz, A., Über die Metrisierbarkeit der vier-dimensionalen affin-zusammenhängenden Räume, Tensor, N.S. 18 (1967), 259–270. (1967) MR0215253
- Kobayashi, S., Nomizu, K., Foundations of Differential Geometry I, II, Wiley-Intersc. Publ., New York, Chichester, Brisbane, Toronto, Singapore, 1991. (1991)
- Kowalski, O., 10.1007/BF01110924, Math. Z. 125 (1972), 129–138. (1972) Zbl0234.53024MR0295250DOI10.1007/BF01110924
- Kowalski, O., Metrizability of affine connections on analytic manifolds, Note Mat. 8 (1) (1988), 1–11. (1988) Zbl0699.53038MR1050506
- Levine, J., Invariant characterization of two-dimensional affine and metric spaces, Duke Math. J. 14 (1948), 69–77. (1948) MR0025236
- Schmidt, B. G., 10.1007/BF01661152, Comm. Math. Phys. 29 (1973), 55–59. (1973) MR0322726DOI10.1007/BF01661152
- Thompson, G., Local and global existence of metrics in two-dimensional affine manifolds, Chinese J. Phys. 19 (6) (1991), 529–532. (1991)
- Vanžurová, A., Linear connections on two-manifolds and SODE’s, Proc. Conf. Aplimat 2007 (Bratislava, Slov. Rep.), Part II, 2007, pp. 325–332. (2007)
- Vilimová, Z., The problem of metrizability of linear connections, Master's thesis, Opava, 2004, supervisor: O. Krupková. (2004)

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