Cartan calculus for $C^\infty$-ringed spaces

TL;DR

This paper develops a Cartan calculus framework for local $C^0$-ringed spaces, defining tangent sheaves, contractions, Lie derivatives, and proving standard Cartan identities in this setting.

Contribution

It introduces a sheaf of vector fields and establishes Cartan calculus operations on differential forms for local $C^0$-ringed spaces, extending classical differential geometry.

Findings

Defined the tangent sheaf on local $C^0$-ringed spaces.

Proved Cartan's identities for vector fields and differential forms.

Extended differential geometric tools to the setting of $C^0$-ringed spaces.

Abstract

In an earlier paper (arXiv:2212.11163) I constructed a complex of differential forms on a local $C^\infty$-ringed space. In this paper I define a sheaf of vector fields (``the tangent sheaf'') on a local $C^\infty$-ringed space, define contractions of vector fields and forms, Lie derivatives of forms with respect to vector fields, and show that the standard equations of Cartan calculus hold for vector fields and differential forms on local $C^\infty$-ringed spaces.