Differential Geometry of Weightings

TL;DR

This paper introduces the concept of weightings along submanifolds and explores their geometric implications, including weighted normal bundles, deformation spaces, and blow-ups, with applications to Lie algebroids and groupoids.

Contribution

It provides a comprehensive framework for understanding weightings in differential geometry and extends these notions to Lie algebroids and groupoids, offering new tools for geometric analysis.

Findings

Defined weighted normal bundles and deformation spaces.

Connected weightings to subbundles of higher tangent bundles.

Extended the concept to multiplicative weightings in Lie algebroids and groupoids.

Abstract

We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted blow-ups. We give a description of weightings in terms of subbundles of higher tangent bundles, which leads us to notions of multiplicative weightings for Lie algebroids and Lie groupoids.