Topology

## I Juhasz's Cardinal functions in topology, ten years later PDF

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.