Frank D. (Tony) Smith, Jr.

I may change my e-mail address from time to time to avoidspam.

A valid address (as of around October 2007) is:

f75m17h at ignore-this bellsouth dot ignore-this net
It is possible that if spam gets too bad, I may not check mye-mail regularly, or even stop using e-mail altogether.


Some post - 6 October 2003 Notes about Corrections,Updates, etc:

 


25 October 2003 note -

Whenever I used the term "parallelizable" for a manifold, Ishould have said

"parallelizable with a pseudo-Riemannian metric, invariantunder the flat connection naturally associated with theparallelization, whose geodesics are the same as those of thatconnection".

Please read all my material about parallelizabilityaccordingly.

This matter came to my attention on 25 October 2003 when I read apost to sci.physics.research on the subject parallelizable manifoldsby Alan Weinstein, which post is at

http://groups.google.com/groups?dq=&hl=en&lr=&ie=UTF-8&group=sci.physics.research&selm=3f99c56b%241%40news.sentex.net

and the text of which says (the links are not his- I added them):

"... From: Alan Weinstein (alanw@RemoveThis.Math.AndThis.Berkeley.EDU)Subject: parallelizable manifolds This is the only article in this thread View: Original FormatNewsgroups: sci.physics.researchDate: 2003-10-24 17:37:20 PST A Letter About Parallelizable Manifolds(to appear in the AMS Notices)Alan Weinstein and Joseph WolfDepartment of Mathematics, University of California, Berkeley, CA 94720 USAIt has recently come to the attention of one of us (AW) that an old result due to Cartan and Schouten [1] and the other of us [3] is frequently misquoted in the mathematics and physics literature (on the sci.physics.research newsgroup, as well as in published books and papers).  We hope that this letter will help to prevent further misquotations.The "theorem" is frequently stated in a form like:"Every compact, simply-connected, parallelizable manifold is(diffeomorphic to) a product of 7-spheres and Lie groups."  In fact, the theorem requires a strong geometric hypothesis, namelythat, among the pseudo-riemannian metrics which are invariant underthe flat connection naturally associated to a parallelization,there is at least one whose geodesics are the same as those of theconnection.  Without this hypothesis, the Poincare conjecture wouldbe an easy corollary.It is not hard to find counterexamples when the geometric hypothesisis dropped.  For instance, Kervaire [2] proved that a product of spheres is parallelizable as long as at least one of them has odd dimension; most such products are not diffeomorphic to products of Lie groups, since a compact, simply connected Lie group has nontrivial third cohomology.  We would like to thank Robert Bryant, Rob Kirby, and Jack Lee for some interesting discussion of this matter.Bibliography[1] Cartan, E., and Schouten, J.A., On riemannian geometriesadmitting an absolute parallelism, Nederl. Akad. Wetensch. Proc.Ser. A 29 (1926), 933-946.[2] Kervaire, M., Courbure integrale generalise et homotopie,Math. Ann. 131} (1956), 219-252.[3] Wolf, J.A., On the geometry and classification ofabsolute parallelisms. I,II, J. Diff. Geom. 6 (1971/72),317-342, 7 (1972), 19-44. ...".

 


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