Formal frames and deformations of affine connections

TL;DR

This paper introduces formal frames as a generalization of ordinary frames, explores their properties, defines torsions, and studies deformations of affine connections with applications to foliations.

Contribution

It develops the theory of formal frames, their canonical forms, and torsions, and analyzes deformations of affine connections and foliations.

Findings

Vanishing torsions imply formal frames are realizable as ordinary frames.

Formal frames generalize classical frames with new properties.

Deformations of linear connections relate to foliation structures.

Abstract

We will introduce formal frames of manifolds, which are a generalization of ordinary frames. Their fundamental properties are discussed. In particular, canonical forms are introduced, and torsions are defined in terms of them as a generalization of the structural equations. It will be shown that the vanishing of torsions are equivalent to the realizability of given formal frames as ordinary frames. We will also discuss deformations of linear connections on tangent bundles. An application to deformations of foliations are then given.