Associated left-invariant contact metric structures on the 7- dimensional Heisenberg group H⁷

In this paper, we construct new nonstandard associated left-invariant contact metric structures (η,ξ,ϕ,gλ) on the 7-dimensional Heisenberg group H⁷. The associated left-invariant contact metric structures for the contact structure η on the contact Lie group(H7,η) were given by the affinor φ and the (pseudo-)Riemannian metric gλ such that ϕ ker η=J , ϕ(ξ) =0, gλ(X,Y)=dη(ϕX,Y)+ λη(X)η(Y), (1) where J is an almost complex structure compatible with the restriction of gλ on ker η , gλ kerη . The parameter λ provided deformation of the associated metric gλ along the Reeb field ξ. The affinor 0 0 0 0 0 ϕ =⎛⎜J ⎞⎟ ⎝ ⎠ and the metric 0 0 0 I g =⎛⎜⎝ λ⎞⎟⎠ are fixed. The new affinors ( )( ) 1 0 Id P Id P- ϕ = ϕ + - are given by an operator P:L(H7)→L(H7)such that P(ξ) =0 and |ker A B D P B C F D F N η ⎛ ⎞ =⎜⎜ ⎟⎟ ⎝ ⎠ , where A u v v u =⎛⎜⎝ - ⎞⎟⎠ , B s t t s =⎝⎛⎜ - ⎠⎞⎟ , С k l l k =⎛⎜⎝ - ⎞⎟⎠ , D x y y x =⎛⎜⎝ - ⎞⎟⎠ , F q r r q =⎛⎜⎝ - ⎞⎟⎠ , and N w z z w =⎛⎜⎝ - ⎞⎟⎠ are symmetric matrices; u, v, s, t, k, l, x, y, q, r, w, and z are real parameters. Each new affinor φ defines a new associated metric gλ by formula (1). We have considered some particular classes of associated metrics corresponding to the affinors φ which were given by the operators P of the following types ker 0 0 | 0 0 0 0 0 B P B η ⎛ ⎞ =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ , ker 0 0 | 0 0 0 0 0 D P D η ⎛ ⎞ =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ , ker 0 0 0 | 0 0 0 0 P F F η ⎛ ⎞ =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ , ker 0 0 | 0 0 0 0 A P C N η ⎛ ⎞ =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ . The following theorem was received for any associated (pseudo-)Riemannian metric gλ(X,Y)=dη(ϕX,Y)+ λη(X)η(Y). Theorem 1. Any left-invariant contact metric structure (η,ξ,ϕ, gλ ) on the Heisenberg group H7 is a Sasaki, K-contact, and η-Einstein structure. The squares of the norms of a Riemann tensor R and Ricci tensor Ric(X ,Y) = gλ (ARicX ,Y) of associated left-invariant metric gλ have the following expressions: 2 2 69 4 R λ = , 2 2 15 4 Ric λ = . The Ricci operator has the following matrix: 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 3 2 ARic ⎛⎜- λ ⎞⎟ ⎜ λ ⎟ ⎜ - ⎟ ⎜⎜ - λ ⎟⎟ ⎜ ⎟ =⎜⎜ - λ ⎟⎟ ⎜ λ ⎟ ⎜ - ⎟ ⎜ ⎟ ⎜ - λ ⎟ ⎜ ⎟ ⎜ ⎟λ ⎜ ⎟ The sign of the scalar curvature of associated left-invariant metric gλ is not constant and 3 2 S λ = - . In addition, the following theorem has been proved for any (2n+1)-dimensional Heisenberg group H2n+1 with a given (pseudo-)Riemannian metric *2 *2 *2 0 1 2 2 1 ... n n g e e e+ = + + +λ . Theorem 2. A left-invariant contact metric structure (η,ξ,ϕ0,g0)on the Heisenberg group H2n+1 is η-Einstein, and 0 0 ( , ) ( , ) ( ) ( ) ( ) g 2 2 Ric X Y g X Y n X Y λ +λ λ = - + η η , X,Y∈L(H2n+1).

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