From: "Robert L. Bryant" <bryant@math.duke.edu>
Newsgroups: sci.math.research
Subject: Re: 6-sphere
Date: 9 Feb 1996 20:38:20 GMT

pmt5el@sun.leeds.ac.uk (E Loubea) wrote:

>Hello,
>
>
>I've just come across a reference for a paper by C.C. Hsiung (Bull. Acad. Math.
>Sinica 14 (1986) n.3 pp. 231-?, there is a recent book as well) where he shows 
>that there is no integrable complex structure on $S^{6}$. Now I don't have direct 
>access to this paper and I have only read the review in Math. Reviews. 
>So I looked through recent papers on the subject expecting them to at least 
>mention this theorem which solved a long-standing conjecture but none of them did.
>This surprises me a fair bit. 
>Can anyone shed light on the matter?
>
>eric

	This 'proof' is not correct, hence the problem is still regarded as open.
That this paper is in error is known to many people, but for some reason
no one seems to address the mistake in print.  I have a reprint of the 
original as well as the erratum that is supposed to fix the matter. The
mistake is not a particularly subtle one and one can see, without much
difficulty that something is wrong just by looking at the Main Theorem,
which asserts that, for any integrable almost complex structure J on a 
Riemannian manifold M with metric g, the identity

			R(JX, JY) Z  +  R(JY, JZ) X  +  R(JZ, JX) Y = 0 

holds for all vector fields X, Y, and Z on M, where R is the Riemann 
curvature tensor of g.  Since no relation between J and g has been assumed, 
this is clearly erroneous.  The erratum tries to save this theorem
by adding the hypothesis that J `is not the natural complex structure
from  a flat metric', which turns out to mean that J does not have 
constant coefficients in any (local) coordinate system.  However, this 
is exactly what rules out an almost complex structure being integrable
in the first place, so the modified main theorem, though true, is vacuous.
	In any event, the application to the 6-sphere fails.

	Yours,

	Robert Bryant