Contact metric structures on odd-dimensional unit spheres

Y. V. Slavolyubova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2014), pp. 46-54

Résumé

In this work, the contact structure on the $3$-dimensional unit sphere $S^3\subset\mathrm{R}^4=\mathrm{C}^2$ which arises in Hopf's map $S^1\to S^3\to S^2$ is considered. The group $S^1$ acts on the sphere $S^3\subset\mathrm{R}^4=\mathrm{C}^2$ by the rule $(z_1,z_2)e^{i\varphi}=(z_1e^{i\varphi}, z_2e^{i\varphi})$. The field of speeds of this action defines a characteristic vector field $\xi$ and 2-dimensional subspaces $E_x$ orthogonal to the vector field $\xi$ form a contact structure. The contact form $\eta$ is defined by the equality $\eta(X)=(\xi,X)$. These constructions are generalized in the case of considering the $7$-dimensional unit sphere $S^7$. On the $3$-dimensional unit sphere $S^3$, expressions of the contact metric structure in local coordinates of a stereographic projection are received, the corresponding characteristics are determined: contact form $\eta$, external differential of the contact form $d\eta$, characteristic vector field $\xi$, contact distribution $\mathrm{E}$, and affinor $\varphi$. A contact metric structure on the $7$-dimensional unit sphere is constructed. For the sphere, main characteristics are determined: contact form $\eta$, external differential of the contact form $d\eta$, characteristic vector field $\xi$, contact distribution $\mathrm{E}$, and affinor $\varphi$ are determined. The relation between the contact structure on the $7$-dimensional unit sphere $S^7$ and almost complex structure $\mathrm{J}$ established by means of a projection $\pi$: $S^7\to\mathbf{CP}^3$ on the $3$-dimensional projective.

Détail

Mots-clés : contact structures
Keywords: contact metric structures, $3$-dimensional sphere, $7$-dimensional sphere, Riemannian metrics.