Canonical affine connections of the first and second orders

K. V. Polyakova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Tome 203 (2021), pp. 71-83

Résumé

We examine affine connections in the bundle of frames of the first and second orders over an $m$-dimensional smooth manifold using the canonical forms of these bundles. We construct the components of the objects of affine connections of the first and second orders determined by zero covariant derivatives of the fiber coordinates of a smooth manifold. These connections are canonical flat affine connections. We examine the objects of deformation from arbitrary affine connections of the first and second orders to the canonical connections of the corresponding orders. Tangent vectors to a smooth manifold are horizontal vectors of the first order for the canonical connections of the first and second orders; these vectors are called first-order canonical vectors. We construct horizontal operators that transform first-order canonical vectors into horizontal vectors of various orders for affine connections of the first and second order. Horizontal vectors are represented as the sums of horizontal vectors of the canonical connection and vertical vectors with the coefficients equal to the components of the deformation tensor from the canonical connection to the given connection.

Détail

Keywords: frame bundle, basic coordinates, layer coordinates, affine connection, covariant derivative, horizontal vector.