Deformation spaces and normal forms around transversals

TL;DR

This paper introduces a framework using deformation spaces to analyze local geometric structures around transversals in manifolds, providing normal form theorems for complex structures like Lie algebroids and Courant algebroids.

Contribution

It develops a novel approach employing deformation spaces to derive normal form theorems for various geometric structures near transversals, including new examples like L_-algebroids and Courant algebroids.

Findings

Normal form theorems around transversals for complex structures

Introduction of deformation space techniques in geometric analysis

Application to new structures like L_-algebroids and Lie bialgebroids

Abstract

Given a manifold M with a submanifold N, the deformation space D(M,N) is a manifold with a submersion to R whose zero fiber is the normal bundle, and all other fibers are equal to M. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with N a submanifold transverse to the foliation. New examples include L_\infty-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around N, in terms of a model structure over the normal bundle.