Hsiung on the Complex Structure of \$S^6\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:59:50Z http://mathoverflow.net/feeds/question/101888 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101888/hsiung-on-the-complex-structure-of-s6 Hsiung on the Complex Structure of \$S^6\$ HeWhoHungers 2012-07-10T20:30:30Z 2012-07-12T00:51:19Z <p>In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his proof. Yet answers to the 2009 <a href="http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere" rel="nofollow">question on this site</a> all agree that the existence of complex structures on \$S^6\$ is still an open problem. Some recent preprints answering the question with opposite answers are also cited there. I would like to know if there are any known mistakes in Hsiung's approach and if so I would appreciate some reference to a paper that points them out.</p> http://mathoverflow.net/questions/101888/hsiung-on-the-complex-structure-of-s6/101910#101910 Answer by YangMills for Hsiung on the Complex Structure of \$S^6\$ YangMills 2012-07-11T02:49:40Z 2012-07-11T02:49:40Z <p>I just found <a href="http://streaming.ictp.trieste.it/preprints/P/90/398.pdf" rel="nofollow">this paper</a> by B. Datta (later published in J. Indian Math. Soc. 60 (1994), no. 1-4, 171–190) that explains in details why one key equation in Hsiung's paper is wrong. See the whole discussion in section 4.</p> <p>Also, in a 2001 book containing Hsiung's selected papers he added <a href="http://books.google.com/books?id=s85vcMYZDJgC&amp;pg=PA673" rel="nofollow">a short paper</a> titled "Nonexistence of a Complex Structure on the Six-Sphere II" which looks like an erratum to his other one. He still claims the same result about \$S^6\$. This paper was never published.</p> http://mathoverflow.net/questions/101888/hsiung-on-the-complex-structure-of-s6/102008#102008 Answer by Robert Bryant for Hsiung on the Complex Structure of \$S^6\$ Robert Bryant 2012-07-12T00:51:19Z 2012-07-12T00:51:19Z <p>While it's good to have a source, such as Datta's paper that points out the error, I find that his explanation of why the key equation is wrong is not as clear as it could be. In fact, with a little thought (requiring essentially no computation), it's clear why this equation must be wrong and what is wrong with the approach. Since it's relatively short, I thought I'd put it in:</p> <p>On page 263 of Hsuing's monograph "Almost Complex and Complex Structures", he claims the following result, from which, if it were correct, the non-existence of a complex structure on the \$6\$-sphere would follow immediately (and, in fact, Hsiung 'applies' this result to get exactly this 'conclusion'):</p> <p><strong>Theorem 6.1.</strong> Let \$J\$ be an almost complex structure on a Riemannian \$2n\$-manifold \$M^{2n}\$ (\$n\ge2\$) with a Riemannian metric \$g_{ij}\$ but without a flat metric or a nonzero constant sectional curvature or both, and let \$J_i^j\$ and \$R_{hijk}\$ be respectively the components of the tensor of \$J\$ and the Riemann curvature tensor of \$M^{2n}\$ with respect to \$g_{ij}\$, where all indices take the values \$1,2,\ldots,2n\$. If \$J\$ is complex structure on \$M^{2n}\$, then \$\$ J_{i_1}^iJ_{i_2}^jR_{iji_3k}+J_{i_2}^iJ_{i_3}^jR_{iji_1k}+J_{i_3}^iJ_{i_1}^jR_{iji_2k}=0 \$\$ for all \$i_1,i_2,i_3,k\$.</p> <p>Now, this result cannot possibly be correct, as you can see from the following observations. </p> <p>First, note that no relation between \$g\$ and \$J\$ is supposed. If it weren't for the peculiar assumptions about \$M\$ not admitting a flat or constant curvature metric (which might have nothing to do with \$g\$), this would be a purely local statement, but, no matter, let's let \$M\$ be \$\mathbb{CP}^n\$ and note that, since \$n\ge2\$, \$M\$ cannot carry either kind of metric. Let \$J\$ be the standard complex structure on \$M\$. Then the above 'Theorem' would imply that, for <em>any</em> metric \$g\$ on \$M\$, its Riemann curvature tensor \$R\$ would satisfy the above equation. Since any metric in dimension \$2n\$ can be locally transplanted onto \$\mathbb{CP}^n\$ and since all complex structures are locally equivalent, it follows easily that the above 'Theorem' implies that the above relation (which is a purely pointwise statement) must hold identically as an algebraic relation for <em>any</em> local pair \$J\$ and \$g\$. (Moreover, since this doesn't involve any derivatives of \$J\$, the hypothesis that \$J\$ be integrable is irrelevant.) </p> <p>Second, it's easy to check that this 'identity' does not hold: Just choose a metric \$g\$ of nonzero constant sectional curvature and any local \$J\$ that is \$g\$-orthogonal, and you'll see that this says that the \$2\$-form \$\Omega\$ associated to \$J\$ by \$g\$ must satisfy \$\Omega^2 = 0\$, contradicting the fact that \$\Omega^n\$ cannot vanish because \$\Omega\$ must be nondegenerate. (This is, in fact, Hsuing's argument as to why \$S^6\$ can't carry an integrable complex structure, because it has a metric of constant sectional curvature.)</p>