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 [question on this site](https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere) 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.



Hsiung on the Complex Structure of $S^6$
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:


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'):


**Theorem 6.1.** 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 $i\_1,i\_2,i\_3,k$.


Now, this result cannot possibly be correct, as you can see from the following observations. 


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 *any* 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 *any* local pair $J$ and $g$.
(Moreover, since this doesn't involve any derivatives of $J$, the hypothesis that $J$ be integrable is irrelevant.) 


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.)


