The Complex Structures on S²ⁿ

2006/08/15 by Jianwei Zhou, Zhou, Jianwei
Mathematics · #53B21 #53B35 #53C28 #57R20 #Advanced Algebra and Geometry #Advanced Mathematical Physics Problems #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0608368

openalex publication_date 2006/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \widetilde\cal J(S²ⁿ) be the set of orthogonal complex structures on TS²ⁿ. We show that the twistor space \widetilde\cal J(S²ⁿ) is a Kaehler manifold. Then we show that an orthogonal almost complex structure J_f on S²ⁿ is integrable if and only if the corresponding section f\colon S²ⁿ→ \widetilde\cal J(S²ⁿ) is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere S²ⁿ for n>1. We also show that there is no complex structure in a neighborhood of the space \widetilde\cal J(S²ⁿ). The method is to study the first Chern class of T^(1,0)S²ⁿ.

Citations

• The Geometry and Topology on Grassmann Manifolds

Related

• Almost Complex Structure on S²ⁿ
• Grassmann Manifold G(2,8) and Complex Structure on S⁶
• Integrable Complex Structures on Twistor Spaces
• On cotangent manifolds, complex structures and generalized geometry
• On Twistor Almost Complex Structures