2017/01/18 by Ekaterina Pervova, Pervova, Ekaterina
Mathematics · #53C15 (primary) #57R45 (secondary) #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1701.04988
openalex publication_date 2017/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Λ1(X), the counterpart of the cotangent bundle, is finite-dimensional, there is an equivalent Levi-Civita connection on it as well.