John Baez
Aug13-04, 05:56 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIn article <e58d56ae.0406151159.26a2b6a3@posting.google.com>, \nAlfred Einstead <whopkins@csd.uwm.edu> wrote:\n\n>The only teleparallel manifolds are Lie groups and S_7.\n\nThis isn\'t true: for example, one can take products of\nthese guys, and open sets of those products.\n\nIt\'s also worth being very clear what one means by\n"teleparallel" here. Alan Weinstein and Joseph Wolf\nwrote a letter to the Notices of the American Mathematical\nSociety clarifying this issue and refuting an urban\nlegend that was going around. They also posted their\nletter to sci.physics.research. Here it is again:\n\n........................................ ..........................\n\nA Letter About Parallelizable Manifolds\n(to appear in the AMS Notices)\n\nAlan Weinstein and Joseph Wolf\nDepartment of Mathematics,\nUniversity of California, Berkeley,\nCA 94720 USA\n\n\nIt has recently come to the attention of one of us (AW) that an\nold result due to Cartan and Schouten [1] and the other of us\n[3] is frequently misquoted in the mathematics and physics literature\n(on the sci.physics.research newsgroup, as well as in published books\nand papers). We hope that this letter will help to prevent further\nmisquotations.\n\nThe "theorem" is frequently stated in a form like:\n"Every compact, simply-connected, parallelizable manifold is\n(diffeomorphic to) a product of 7-spheres and Lie groups."\n\nIn fact, the theorem requires a strong geometric hypothesis, namely\nthat, among the pseudo-riemannian metrics which are invariant under\nthe flat connection naturally associated to a parallelization,\nthere is at least one whose geodesics are the same as those of the\nconnection. Without this hypothesis, the Poincare conjecture would\nbe an easy corollary.\n\nIt is not hard to find counterexamples when the geometric hypothesis\nis dropped. For instance, Kervaire [2] proved that a product of\nspheres is parallelizable as long as at least one of them has odd\ndimension; most such products are not diffeomorphic to products of\nLie groups, since a compact, simply connected Lie group has\nnontrivial third cohomology. We would like to thank Robert\nBryant, Rob Kirby, and Jack Lee for some interesting discussion\nof this matter.\n\n\nBibliography\n\n[1] Cartan, E., and Schouten, J.A., On riemannian geometries\nadmitting an absolute parallelism, Nederl. Akad. Wetensch. Proc.\nSer. A 29 (1926), 933-946.\n\n[2] Kervaire, M., Courbure integrale generalise et homotopie,\nMath. Ann. 131} (1956), 219-252.\n\n[3] Wolf, J.A., On the geometry and classification of\nabsolute parallelisms. I,II, J. Diff. Geom. 6 (1971/72),\n317-342, 7 (1972), 19-44. ...".\n\n......................................... .............................\n\nAlso available at\nhttp://www.ams.org/notices/200401/commentary.pdf\n\n\n\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <e58d56ae.0406151159.26a2b6a3@posting.google.com>,
Alfred Einstead <whopkins@csd.uwm.edu> wrote:
>The only teleparallel manifolds are Lie groups and S_7.
This isn't true: for example, one can take products of
these guys, and open sets of those products.
It's also worth being very clear what one means by
"teleparallel" here. Alan Weinstein and Joseph Wolf
wrote a letter to the Notices of the American Mathematical
Society clarifying this issue and refuting an urban
legend that was going around. They also posted their
letter to sci.physics.research. Here it is again:
.................................................. ................
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
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, 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.
Bibliography
[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. ...".
.................................................. ....................
Also available at
http://www.ams.org/notices/200401/commentary.pdf
Alfred Einstead <whopkins@csd.uwm.edu> wrote:
>The only teleparallel manifolds are Lie groups and S_7.
This isn't true: for example, one can take products of
these guys, and open sets of those products.
It's also worth being very clear what one means by
"teleparallel" here. Alan Weinstein and Joseph Wolf
wrote a letter to the Notices of the American Mathematical
Society clarifying this issue and refuting an urban
legend that was going around. They also posted their
letter to sci.physics.research. Here it is again:
.................................................. ................
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
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, 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.
Bibliography
[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. ...".
.................................................. ....................
Also available at
http://www.ams.org/notices/200401/commentary.pdf