Associated contact metric structures on the 7-dimensional unit sphere S<sup>7</sup>
In this paper, we construct new examples of associated contact metric structures (n, Ф, g<sup>J</sup>) on the 7-dimensional unit sphere S<sup>7</sup>, other than standard. The construction involved a Hopf bundle n: S<sup>7</sup>^CP<sup>3</sup>. This projection maps affinor ф 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 CP<sup>3</sup>. Let Ф be a nondegenerate 2-form (a Fubini-Study form). An almost complex structure J is called positively associated with the form Ф if the following conditions are satisfied for any vector fields X, Y: Ф(JX, JY) = Ф^,Y) and Ф^, JX) > 0, if X * 0. Each positively associated almost complex structure J defines a Riemannian metric g<sup>J</sup> by the equality g(X,Y) = Ф^, JY); the metric is also called associated. The associated metric has the following properties: g(JX, JY) = g(X, JY), g(JX,Y) = Ф(X,Y). The positively associated almost complex structure can be obtained as follows: J = J 0(1 + R)(1 - R) -<sup>1</sup>, where R is a symmetric endomorphism R: TCP<sup>3</sup> ^ TCP<sup>3</sup> anticommuting with the standard structure J<sub>0</sub>, J ~{<sup>U 0</sup><sup>J</sup>° 4 0 -iI 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 CP<sup>3</sup>. One of these matrices, ' 0 н!<sup>л</sup> 1 R R 0 (1+1 w |)<sup>4</sup> f-1 2 3 <sub>n</sub> n Л www 0 0 r\ 1-2 3 <sub>n</sub> 0 www 0 0 0 w<sup>1</sup>w<sup>2</sup>w where the block Ra =, has been considered in more detail. For this endomorphism, the relevant almost complex structure J and a Hermite metric g have been found in the space CP<sup>3</sup>. It has been verified that the constructed structure J is not integrable.