# Metrization problem for linear connections and holonomy algebras

## Alena Vanžurová

Address: Department of Algebra and Geometry Faculty of Science, Palacký University Tomkova 40, 779 00 Olomouc, Czech Republic

E-mail: alena.vanzurova@upol.cz

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).

AMSclassification: primary 53B05; secondary 53B20.

Keywords: manifold, linear connection, pseudo-Riemannian metric, holonomy group, holonomy algebra.