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, gJ) on the 7-dimensional unit sphere S7, other than standard. The construction involved a Hopf bundle n: S7^CP3. 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 CP3. 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 gJ 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) -1, where R is a symmetric endomorphism R: TCP3 ^ TCP3 anticommuting with the standard structure J0, J ~{U 0J° 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 CP3. One of these matrices, ' 0 н!л 1 R R 0 (1+1 w |)4 f-1 2 3 n n Л www 0 0 r\ 1-2 3 n 0 www 0 0 0 w1w2w 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 CP3. It has been verified that the constructed structure J is not integrable.