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Deformation spaces and normal forms around transversals

Part of: Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  17 February 2020

Francis Bischoff ,
Henrique Bursztyn ,
Hudson Lima  and
Eckhard Meinrenken
Francis Bischoff
Affiliation:
Mathematical Institute and Exeter College, University of Oxford, Oxford, OX2 6GG, UK email Francis.Bischoff@maths.ox.ac.uk
Henrique Bursztyn
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil email henrique@impa.br
Hudson Lima
Affiliation:
Departamento de Matemática – UFPR, Centro Politécnico, Curitiba, 81531-980, Brazil email hudsonlima@ufpr.br
Eckhard Meinrenken
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, CanadaM5S 2E4 email mein@math.toronto.edu
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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 $\unicode[STIX]{x1D708}(M,N)$.

Keywords

deformation spacesplitting theoremsL∞-algebroidsCourant algebroids

MSC classification

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 53D17: Poisson manifolds; Poisson groupoids and algebroids 53D18: Generalized geometries (à la Hitchin)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 4 , April 2020 , pp. 697 - 732
Copyright
© The Authors 2020

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