Frank D. (Tony) Smith, Jr.

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

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 my e-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, I should have said

"parallelizable with a pseudo-Riemannian metric, invariant under the flat connection naturally associated with the parallelization, whose geodesics are the same as those of that connection".

Please read all my material about parallelizability accordingly.

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

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.research
Date: 2003-10-24 17:37:20 PST 

A Letter About Parallelizable Manifolds
(to appear in the AMS Notices)

Alan Weinstein and Joseph Wolf
Department of Mathematics, 
University of California, Berkeley, 
CA 94720 USA

It 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 

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, namely
that, among the pseudo-riemannian metrics which are invariant under
the flat connection naturally associated to a parallelization,
there is at least one whose geodesics are the same as those of the
connection.  Without this hypothesis, the Poincare conjecture would
be an easy corollary.

It is not hard to find counterexamples when the geometric hypothesis
is 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.


[1] Cartan, E., and Schouten, J.A., On riemannian geometries
admitting 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 of
absolute parallelisms. I,II, J. Diff. Geom. 6 (1971/72),
317-342, 7 (1972), 19-44. ...".


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