MATHEMATICS

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

Ya. V. Slavolyubova

Kemerovo Institute of Plekhanov Russian University of Economics, Kemerovo, Russian Federation

Abstract: 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 $J_0$,
$$ J_0= \begin{pmatrix} iI&0\\ 0&-iI \end{pmatrix}. $$

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.

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

UDC: 514.76

Received: 15.12.2015

DOI: 10.17223/19988621/42/5

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