Symmetric Cartan calculus, the Patterson-Walker metric and Killing vector fields

TL;DR

This paper introduces symmetric Cartan calculus, linking it to the Patterson-Walker metric and classical Cartan calculus, highlighting the role of Killing vector fields in this new framework.

Contribution

It develops a symmetric analogue of Cartan calculus, explores its geometric structures, and establishes a novel connection with the Patterson-Walker metric.

Findings

Symmetric derivative choices are parametrized by torsion-free affine connections.

Symmetric Cartan calculus is shown to be a complete analogue of classical Cartan calculus.

Killing vector fields are central in the symmetric Cartan calculus framework.

Abstract

We develop symmetric Cartan calculus, an analogue of classical Cartan calculus for symmetric differential forms. We first show that the analogue of the exterior derivative, the symmetric derivative, is not unique and its different choices are parametrized by torsion-free affine connections. We use a choice of symmetric derivative to generate the symmetric Lie derivative and the symmetric bracket, and give geometric interpretations of all of them. By proving the structural identities and describing the role of affine morphisms, we reveal an unexpected link of symmetric Cartan calculus with the Patterson-Walker metric, which we recast as a direct analogue of the canonical symplectic form on the cotangent bundle. We show that, in the light of the Patterson-Walker metric, symmetric Cartan calculus becomes a complete analogue of classical Cartan calculus. In this analogy, its Killing vector fields play a central role.