By I. Moerdijk, J. Mrcun
In line with a graduate path taught at Utrecht collage, this booklet offers a quick advent to the speculation of Foliations and Lie Groupoids to scholars who've already taken a primary direction in differential geometry. Ieke Moerdijk and Janez Mrcun contain special references to let scholars to discover the considered necessary history fabric within the study literature. The textual content positive factors many workouts and labored examples.
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Extra resources for Introduction to Foliations and Lie Groupoids
Ge = (dg)x ◦ e for any e ∈ Fx (Ui ). 11. In particular it follows that F (Ui )/Gi is a manifold equipped with a right action of GL(n, R). Note that the map φi induces a map pi : F (Ui )/Gi → Q in the obvious way and that the action of GL(n, R) is along the ﬁbres of this map. Now take any embedding λ: (Ui , Gi , φi ) → (Uj , Gj , φj ) between orbifold charts. The composition with the derivative dλ induces an embed˜ F (Ui ) → F (Uj ). This embedding factors as ding λ: λ∗: F (Ui )/Gi −→ F (Uj )/Gj .
5, we will present a ‘global’ stability theorem, which applies to foliations of codimension 1, and states that under certain conditions, the holonomy group has to be trivial and the foliation has to be simple (in the technical sense of being given by the ﬁbres of a submersion). 6 we also give Thurston’s more reﬁned version of these stability theorems. The notion of holonomy is closely related to that of a ‘transverse’ Riemannian structure on the foliation. Any manifold can be equipped with a Riemannian metric, but it is a special property for a foliated manifold to be equipped with a Riemannian metric for which the length of 19 20 Holonomy and stability tangent vectors or curves which are transversal to the leaves is invariant under the holonomy group.
Proof Choose a Riemannian metric on M and let X be a normalized normal ﬁeld of F (this exists since F is transversely orientable). Let γ be an integral curve of X with γ(0) ∈ L0 . 27 all the leaves hit by γ are diﬀeomorphic to L0 . If γ is periodic with period p, then we take σ to be γ|[0,p] with the natural reparametrization. Otherwise γ is an injective immersion which is not closed. Take a point p in the boundary of γ(R), and choose a foliation chart ϕ: U → Rn−1 × R with p ∈ U . We may also assume that X is tangent to the ﬁbres of pr1 ◦ ϕ.