Download Introduction to Foliations and Lie Groupoids by I. Moerdijk, J. Mrcun PDF

By I. Moerdijk, J. Mrcun

According to a graduate direction taught at Utrecht collage, this e-book presents a quick creation to the speculation of Foliations and Lie Groupoids to scholars who've already taken a primary path in differential geometry. Ieke Moerdijk and Janez Mrcun comprise distinctive references to allow scholars to discover the needful history fabric within the study literature. The textual content good points many workouts and labored examples.

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23 Let Q be an orbifold of dimension n equipped with a Riemannian metric. Then Q is isomorphic to the orbifold associated to the foliated action of the group U (n) on the unitary frame bundle UF (Q). 19. 5 Global Reeb stability in codimension 1 In this section M is a compact connected manifold, equipped with a transversely orientable foliation F of codimension 1. 24 (Global Reeb stability) If F is a codimension 1 transversely orientable foliation of a compact connected manifold M and admits a compact leaf L0 with finite fundamental group, then F is the foliation given by the fibres of a fibre bundle projection φ: M → S 1 .

23 Let Q be an orbifold of dimension n equipped with a Riemannian metric. Then Q is isomorphic to the orbifold associated to the foliated action of the group U (n) on the unitary frame bundle UF (Q). 19. 5 Global Reeb stability in codimension 1 In this section M is a compact connected manifold, equipped with a transversely orientable foliation F of codimension 1. 24 (Global Reeb stability) If F is a codimension 1 transversely orientable foliation of a compact connected manifold M and admits a compact leaf L0 with finite fundamental group, then F is the foliation given by the fibres of a fibre bundle projection φ: M → S 1 .

This is now a complex vector bundle of rank n over Ui , and we may again consider the ordered (complex) bases in the fibres Tx (Ui )⊗C. Such a basis is now represented by a complex linear isomorphism e: Cn → Tx (Ui ) ⊗ C. The collection of such bases yield the complex frame bundle CF (Ui ) which is now a principal GL(n, C)-bundle. Again there is a free Gi -action on CF (Ui ), now given by the composition with dg ⊗ idC , for g ∈ Gi . Similarly an embedding λ between orbifold charts induces an equivariant embedding ˜ by the composition with dλ ⊗ idC .

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