Diffeological Levi-Civita connections

TL;DR

This paper explores the concept of Levi-Civita connections within diffeological spaces, proposing a framework based on the dual of the cotangent bundle, especially when it is finite-dimensional.

Contribution

It introduces a novel approach to defining Levi-Civita connections in diffeological spaces using the dual of the cotangent bundle, bypassing the lack of a standard tangent bundle theory.

Findings

A diffeological Levi-Civita connection can be defined via the dual of the cotangent bundle.

When the cotangent bundle is finite-dimensional, an equivalent Levi-Civita connection exists.

The approach extends compatibility and symmetry notions to the diffeological setting.

Abstract

A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual of the just-mentioned counterpart of the cotangent bundle in place of the tangent bundle (without making any claim about its geometrical meaning). To it, the notions of compatibility with a pseudo-metric and symmetricity can be easily extended, and therefore the notion of a Levi-Civita connection makes sense as well. In the case when $\Lambda^1(X)$, the counterpart of the cotangent bundle, is finite-dimensional, there is an equivalent Levi-Civita connection on it as well.