By I Juhasz

ISBN-10: 9061961963

ISBN-13: 9789061961963

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**Extra info for Cardinal functions in topology, ten years later **

**Example text**

In particular, this corank is always even. This proposition has immediate consequences for the computation of the homology of the orbit 0 in the case where we have a positive-definite invariant scalar product on r and the splitting and exponential map are relative to this scalar product. For example, if we choose y to be a regular point not on the orbit 0, then the distance function from y to 0 is a Morse function whose critical points are the points E 0 at which the line from y to ,B is perpendicular to 0.

2), is just the canonical one-form ab of Fb = T * Xb. Hence i **,tw restricts to the canonical two-form of Fb. Let us now show that the orthogonal complement of T (Fb) with respect to i ytw is the fiber at b of the horizontal bundle of the connection on M. For this we recall our definition of the connection. Let v be a vector field on B. The connection on X then defines a horizontal vector field v# on X. Let 01 be the local oneparameter group generated by v#. Then 4t carries fibers into fibers, and hence defines a one-parameter group of transformations *t of M: If Ot (Xa) = Xb then F.

Then 4t carries fibers into fibers, and hence defines a one-parameter group of transformations *t of M: If Ot (Xa) = Xb then F. = T*(XQ) and Fb = T*(Xb) and Vit: Fa - * Fb is given by *t = (07)-1. If iQ denotes the embedding of FQ into M as the fiber over a, then it follows that it (ibar) = iQar. 3) The infinitesimal generator of ft is the horizontal vector field ii associated to v. What we must show is that [t(v)i,*ytw](w) = 0 for any vertical vector field w. Now t(U)i,*ytw = t(v')dar = L;ar, - dt(v')ar.

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