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Home , Ehresmann connection

Russian Mathematics Iz VUZ Izvestiya VUZ Matematika

Vol No pp UDC

EHRESMANN OF 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 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 M 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 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

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