Combinatorial differential geometry and ideal Bianchi-Ricci identities II - the torsion case

TL;DR

This paper extends previous work on differential geometry by characterizing natural vector-field operators for connections with torsion, describing their structure, and establishing an ideal basis satisfying generalized Bianchi-Ricci identities.

Contribution

It introduces a comprehensive characterization of generators for vector-field operators in the torsion case and constructs an ideal basis satisfying generalized Bianchi-Ricci identities.

Findings

Characterization of all generators for vector-field operators with torsion

Determination of the dimension of the space of such operators

Existence of an ideal basis satisfying generalized Bianchi-Ricci identities

Abstract

This paper is a continuation of arXiv:0809.1158, dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without corrections.