Associated contact metric structures on the $7$-dimensional unit sphere~$S^7$

Ya. V. Slavolyubova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2016), pp. 44-57

Résumé

In this paper, we construct new examples of associated contact metric structures $(\eta, \xi, \varphi, g^J)$ on the $7$-dimensional unit sphere $S^7$, other than standard. The construction involved a Hopf bundle $\pi: S^7\to\mathbf{CP^3}$. This projection maps affinor $\varphi$ into an almost complex structure $J$. Therefore, it became necessary to build new examples of associated almost complex structures $J$ in the $3$-dimensional complex projective space $\mathbf{CP^3}$. Let $\Phi$ be a nondegenerate $2$-form (a Fubini–Study form). An almost complex structure $J$ is called positively associated with the form $\Phi$ if the following conditions are satisfied for any vector fields $X$, $Y$: $$ \Phi(JX, JY)=\Phi(X, Y)\text{ and }\Phi(X, JX)>0, \text{ if } X\ne0. $$ Each positively associated almost complex structure $J$ defines a Riemannian metric $g^J$ by the equality $g(X, Y)=\Phi(X, JY)$; the metric is also called associated. The associated metric has the following properties: $$ g(JX, JY)=g(X, JY),\ g(JX, Y)=\Phi(X, Y). $$ The positively associated almost complex structure can be obtained as follows: $$ J=J_0(1+R)(1-R)^{-1}, $$ where $R$ is a symmetric endomorphism $R: TCP^3\to TCP^3$ anticommuting with the standard structure $J_0$, $$ J_0= \begin{pmatrix} iI0\\ 0-iI \end{pmatrix}. $$ In this paper, we have found a series of matrices $R$ satisfying these conditions. Each matrix of this kind defines an associated almost complex structure in the space $\mathbf{CP^3}$. One of these matrices, $$ R=\frac{1}{(1+|w|)^4} \begin{pmatrix} 0 \overline{R_{\alpha}^{\overline{\beta}}}\\ R_{\alpha}^{\overline{\beta}}0 \end{pmatrix}, $$ where the block $R_{\alpha}^{\overline{\beta}}=\begin{pmatrix} \overline{w}^1w^2w^3 0 0\\ 0 w^1\overline{w}^2w^3 0\\ 0 0 w^1w^2\overline{w}^3 \end{pmatrix}$, has been considered in more detail. For this endomorphism, the relevant almost complex structure $J$ and a Hermite metric $g^J$ have been found in the space $\mathbf{CP}^3$. It has been verified that the constructed structure $J$ is not integrable.

Détail

Mots-clés : contact structures
Keywords: associated contact metric structures, $7$-dimensional sphere.