New PDF release: Introduction to the Geometry of Foliation, Part B:

By Dr. Gilbert Hector, Dr. Ulrich Hirsch (auth.)

ISBN-10: 3322856194

ISBN-13: 9783322856197

ISBN-10: 3528085681

ISBN-13: 9783528085681

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Extra info for Introduction to the Geometry of Foliation, Part B: Foliations of Codimension One

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SI. 2. ytic. d bundle. ptionai. mtrUmai. ~ u. Following Raymond's thesis [Ra] we construct an analytic foliation of codimension one with an exceptional minimal set. 1 the manifold will be B x SI and the foliation is obtained by suspending a Fuchsian group In part developed for the construction of a COO C this example will be further foliation on S3 (or any odd- dimensional sphere) with an arbitrary finite number of exceptional mini-mal sets. We remark that Raymond's example is closely related to Sacksteder's example of a COO foliation on B x SI with an exceptional minimal set.

0) to ... -e: s Js we see that 0 ... 0 g~e:I(o) JI Supposing that the lift of fk goes from = dk · Thus 0 dk to equals w. - !! the representation H: nIB ~ Oiff:(SI) factors through Or (S I) then i;H has Euler class ~. 2, v». For P~oo6: we can find a representative I. Pk(gl""'~) = H(y. 3 representing be taken-to be the trivial one and consequently e(i;H) e(i;H) o. = may [J This corollary has two interesting applications the first one of which is obvious (cf. 2, iv». The second one is due to Wood; see [Woo; theorem 2].

Consists of finite orbits. Taking the G G (cf. I; 3. I. I), the above argument for of shows that an orbit in Z(P G). e. Z(P G) is closed also in this case. d) Finally, the proposition holds trivially when that is Z(P G) = F. 0 We observe that for G c Homeo+(I) PG is minimal, F = I the only minimal sets of are stationary points. From the preceding proof we deduce two corollaries. rwise would not be closed. Furthermore, if then thus showing 0 is another closed orbit o then ~ or x E Z(P G). G(x) mal set for i l) x E F.

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Introduction to the Geometry of Foliation, Part B: Foliations of Codimension One by Dr. Gilbert Hector, Dr. Ulrich Hirsch (auth.)


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