Deformation spaces and normal forms around transversals

Abstract

Given a manifold$M$with a submanifold$N$, the deformation space${\mathcal{D}}(M,N)$is a manifold with a submersion to$\mathbb{R}$whose zero fiber is the normal bundle$\unicode[STIX]{x1D708}(M,N)$, 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$\unicode[STIX]{x1D708}(M,N)$.