Some Remarks on Existence of a Complex Structure on the Compact Six Sphere

Abstract

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on S6 was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on S6. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem.

Keywords

• almost complex structure
• integrable
• complex structure
• six sphere
• nil-manifold
• Riemann connection