Download Foliations: Dynamics, Geometry and Topology by Masayuki Asaoka, Aziz El Kacimi Alaoui, Steven Hurder, Ken PDF
By Masayuki Asaoka, Aziz El Kacimi Alaoui, Steven Hurder, Ken Richardson, Jesús Álvarez López, Marcel Nicolau
This booklet is an advent to numerous lively examine subject matters in Foliation thought and its connections with different components. It includes expository lectures displaying the variety of principles and techniques converging within the learn of foliations. The lectures via Aziz El Kacimi Alaoui supply an advent to Foliation conception with emphasis on examples and transverse buildings. Steven Hurder's lectures follow principles from gentle dynamical structures to enhance helpful strategies within the examine of foliations: restrict units and cycles for leaves, leafwise geodesic stream, transverse exponents, Pesin conception and hyperbolic, parabolic and elliptic sorts of foliations. The lectures via Masayuki Asaoka compute the leafwise cohomology of foliations given by way of activities of Lie teams, and use it on describe deformation of these activities. In his lectures, Ken Richardson experiences the houses of transverse Dirac operators for Riemannian foliations and compact Lie team activities, and explains a lately proved index formulation. in addition to scholars and researchers of Foliation idea, this publication may be fascinating for mathematicians attracted to the functions to foliations of topics like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or worldwide Analysis.
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Fix a basis ξ1 , . . , ξl of g. , l [ξi , ξj ] = k=1 ckij ξk . Take a non-singular 1-form ω ∈ Ω1 (F) ⊗ g. Let Xi be a nowhere-vanishing vector field in X(F) given by Xi (x) = ωx−1 (ξi ). Then, (dF ω + [ω, ω])(Xi , Xj ) = Xi (ω(Xj )) − Xj (ω(Xi )) − ω([Xi , Xj ]) + [ω(Xi ), ω(Xj )] = −ω Xi , Xj ckij ξk + k = −ω Xi , Xj ckij ω(Xk ) + k ckij Xk =ω − X i , Xj . k Since ω is non-singular, dF ω + [ω, ω] = 0 if and only if [Xi , Xj ] = k ckij Xk for all Xi being a i and j. The latter condition is equivalent to the linear map ξi → homomorphism between Lie algebras.
For ω ∈ Ω1 (F; T F ⊥ ), we define a p-plane field Eω on M by Eω (x) = v + ω(v) | v ∈ Tx F . It gives a one-to-one correspondence between T F ⊥ -valued leafwise 1-forms and p-plane fields transverse to T F ⊥ . By a direct computation in a local coordinate system adapted to the pair (F, F ⊥ ), we obtain the following criterion for the integrability of Eω . 1. The p-plane field Eω generates a foliation if and only if ω satisfies the equation dF ω + ω, ω = 0. Fix β ∈ X(F ⊥ ) = Ω0 (F; T F ⊥ ). Let {ht }t∈R be a one-parameter family of diffeomorphisms such that h0 is the identity map and ht preserves each orbit of F ⊥ for all t.