Contact metric structures on odd-dimensional unit spheres

In this work, the contact structure on the 3-dimensional unit sphere S c R = C which arises in Hopfs map S ^ S ^ S is considered. The group S acts on the sphere S c R = C by the rule (z ₁, z ₂)e = (z ₁e , z ₂e ). The field of speeds of this action defines a characteristic vector field 4 and 2-dimensional subspaces E _x orthogonal to the vector field 4 form a contact structure. The contact form п is defined by the equality п(V) = (4, X). These constructions are generalized in the case of considering the 7-dimensional unit sphere S . On the 3-dimensional unit sphere S , expressions of the contact metric structure in local coordinates of a stereographic projection are received, the corresponding characteristics are determined: contact form п, external differential of the contact form d^ characteristic vector field 4, contact distribution E, and affinor ф. A contact metric structure on the 7-dimensional unit sphere is constructed. For the sphere, main characteristics are determined: contact form п, external differential of the contact form d^ characteristic vector field 4, contact distribution E, and affinor ф are determined. The relation between the contact structure on the 7-dimensional unit sphere S and almost complex structure J established by means of a projection n: S ^ CP on the 3-dimensional projective.

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