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Deformation spaces and normal forms around transversals

Published online by Cambridge University Press:  17 February 2020

Francis Bischoff
Affiliation:
Mathematical Institute and Exeter College, University of Oxford, Oxford, OX2 6GG, UK email [email protected]
Henrique Bursztyn
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil email [email protected]
Hudson Lima
Affiliation:
Departamento de Matemática – UFPR, Centro Politécnico, Curitiba, 81531-980, Brazil email [email protected]
Eckhard Meinrenken
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, CanadaM5S 2E4 email [email protected]

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 $\unicode[STIX]{x1D708}(M,N)$.

Type
Research Article
Copyright
© The Authors 2020

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