Geodesic
Metrization problem for linear connections and holonomy algebras
Archivum mathematicum, Tome 44 (2008) no. 5, pp. 511-521

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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).
Classification : 53B05, 53B20
Keywords: manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra 
@article{ARM_2008__44_5_a11,
     author = {Van\v{z}urov\'a, Alena},
     title = {Metrization problem for linear connections and holonomy algebras},
     journal = {Archivum mathematicum},
     pages = {511--521},
     publisher = {mathdoc},
     volume = {44},
     number = {5},
     year = {2008},
     mrnumber = {2501581},
     zbl = {1212.53021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008__44_5_a11/}
}
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Vanžurová, Alena. Metrization problem for linear connections and holonomy algebras. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 511-521. http://geodesic.mathdoc.fr/item/ARM_2008__44_5_a11/