Combinatorial differential geometry and ideal Bianchi-Ricci identities

TL;DR

This paper uses graph complex methods to analyze vector fields and symmetric connections, characterizing generators, describing the space of operators, and establishing an ideal basis satisfying generalized Bianchi-Ricci identities.

Contribution

It introduces a novel graph complex approach to characterize and construct vector-field valued operators with Bianchi-Ricci identities, including an ideal basis with specified leading terms.

Findings

Characterized all generators for vector-field valued operators.

Described the size of the operator space.

Proved existence of an ideal basis satisfying Bianchi-Ricci identities.

Abstract

We apply the graph complex method to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without the correction terms.