Metrization problem for linear connections and holonomy algebras. (English) Zbl 1212.53021

Vanžurová, Alena

Arch. Math., Brno 44, No. 5, 511-521 (2008).

Summary: We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. The compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of the 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 systems of ODEs, which define the geodesics of a linear connection, components of a metric compatible with the connection play the role of variational multipliers).

MSC:

53B05	Linear and affine connections
53B20	Local Riemannian geometry

Keywords:

manifold; linear connection; pseudo-Riemannian metric; holonomy group; holonomy algebra