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

In this paper, we construct new examples of associated contact metric structures (n, Ф, g) on the 7-dimensional unit sphere S, other than standard. The construction involved a Hopf bundle n: S^CP. 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. 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 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) -, where R is a symmetric endomorphism R: TCP ^ TCP anticommuting with the standard structure J₀, J ~{° 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. One of these matrices, ' 0 н! 1 R R 0 (1+1 w |) f-1 2 3 _n n Л www 0 0 r\ 1-2 3 _n 0 www 0 0 0 www 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. It has been verified that the constructed structure J is not integrable.

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