Tensor calculus and deformation theory on on a local Lie group

TL;DR

This paper explores how tensor calculus can be developed from Lie's transformation group concept, leading to a natural framework for deformation theory and the Kodaira-Spencer map.

Contribution

It introduces a Lie group-based foundation for tensor calculus, connecting it with deformation theory and the Kodaira-Spencer map.

Findings

Tensor calculus can be founded on Lie's transformation groups.

The approach naturally leads to deformation of transformation groups.

Connections to the Kodaira-Spencer map are established.

Abstract

Historically tensor calculus emerged in an attempt to formalize Rie- mann's ideas. We show that tensor calculus can be based also on Lie's idea of a transformation group and this approach leads quite naturally to the concept of deformation of a transformation group and the Kodaira- Spencer map.