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Ehresmann Connection of Foliation with Singularities

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Ehresmann Connection of Foliation with Singularities Russian Mathematics Iz VUZ Izvestiya VUZ Matematika Vol No pp UDC EHRESMANN CONNECTION OF FOLIATION WITH SINGULARITIES AND GLOBAL STABILITY OF LEAVES NI Zhukova Intro duction The theory of connections in b er bundles plays central role in mo dern dierential geometry Unlike the base of b er bundle the top ological space of foliation leaves is in general bad For example the leaf space of linear irrational foliation on the torus T has trivial top ology In the notion of Ehresmann connection for foliations is intro duced as a natural generalization of the Ehresmann connection in b er bundle The construction of Ehresmann connection is based on the existence of parallel transp ort of horizontal curves along vertical curves by the verticalhorizontal homotopy Note that this homotopy has b een already used in the global dierential geometry see e g In the present pap er we dene the Ehresmann connection for the foliation M F with singu larities considered by HJ Sussman and P Stefan as a generalized distribution M on M whichmakes it p ossible to transp ort horizontal curves a curve is horizontal if its tangent vectors lie in M along vertical curves Unlike the regular case such a transp ort is a manyvalued map Therefore we dene the Mholonomy group H L a of an arbitrary leaf L of M F as the M transformation group of the quotient set of horizontal curves A foliation with singularities admitting an Ehresmann connection is called an Ehresmann foli ation We also intro duce generalized Ehresmann foliations such that the horizontal curves are the integral curves of M without singular p oints except for the endp oints In the present pap er under natural assumptions we prove theorems on global stability of compact leaf with nite fundamental group and nite holonomy group for generalized Ehresmann foliations and Ehresmann foliations with singularities These theorems can b e considered as analogs of the classical Reeb theorem on global stability of compact leaf with nite fundamental group r which holds true for a C foliation r of co dimension one on a compact manifold Also totally we prove that the transversally complete Riemannian foliations and vertically complete geo desic foliations with singularities admit natural Ehresmann connections Foliations with singularities r We assume that all manifolds under consideration are of class C r connected second countable and Hausdor unless otherwise stated All neighb orho o ds are op en Paths curves and maps are piecewise dierentiable Let M be an ndimensional manifold AmapT assigning to each p oint x M a pxdimensional subspace T in the tangent space T M is called a generalized distribution on M The distribution T x x is said to be dierentiable if for anyvector Y T there exists a dierentiable vector eld X in a x T for any y U neighborhood U of x suchthat X Y X y y x x x c by Allerton Press Inc Authorization to photo copy individual items for internal or p ersonal use or the internal or p ersonal use of sp ecic clients is granted by Allerton Press Inc for libraries and other users registered with the Copyright Clearance Center CCC Transactional Rep orting Service provided that the base fee of p er copy is paid directly to CCC Rosewo o d Drive Danvers MA .
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