The Complex Structures on $S^{2n}$

Details The Complex Structures on $S^{2n}$ The Complex Structures on $S^{2n}$

2006-08-15

arxiv.org/abs/math/0608368v4

Mathematics

Differential Geometry

17 pages

Scientific paper

In this paper, we show that the twistor space ${\cal J}(R^{2n+2})$ on Euclidean space $R^{2n+2}$ is a Kaehler manifold and the orthogonal twistor space $\widetilde{\cal J}(S^{2n})$ of the sphere $S^{2n}$ is a Kaehler submanifold of ${\cal J}(R^{2n+2})$. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\colon S^{2n}\to \widetilde{\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$.

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