6-球面上近复结构和G2结构的相关几何研究

6-球面上的近复结构和G2结构及其关联的几何一直是人们关注的研究课题，因为它联系着6-球面上是否存在复结构的著名Hopf猜想。本申请项目旨在研究6-球面上与近复结构、G2结构相关的几何问题，以深入了解和认识6-球面上近复结构及G2结构的性质；6-球面上与近复结构及G2结构相关的几何问题也联系着Harvey和Lawson的Calibrated Geometries，其中有着许多有待解决的问题。具体地讲，我们会利用6-球面上的近复结构研究到6-球面的Schrodinger流的几何性质，在了解欧氏7-空间（视为八元数的虚部）中沿G2-副法线运动曲线方程的基础上，试图建立欧氏7-空间中vortex filament理论，并通过这种研究反过来来了解6-球面上的近复结构与G2结构的关系和它们的几何性质。

As related to the famous Hopf conjecture (Does there exist a complex structure on 6-sphere?), the study of almost complex structures and the G2 structure on 6-sphere has been attracted since 1950's. The aim of this proposal is to give a geometric study related to almost complex structures and the G2 structure on 6-sphere and then to understand some deep properties of almost complex structures and the G2 structure on 6-sphere. On the other hand, such a study also relates to calibrated geometries proposed by Harvey and Lawson. More exactly, in this proposal, based on the understanding the motion of space curves in the Euclidean 7-space along the G2 binormal direction, we try to established the theory of vortex filament on the Euclidean 7-space and to grasph the properties of Schrodinger flows to 6-sphere, so that we shall find our further information about almost complex structures and the G2 structure on 6-sphere.