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The structure of a class of K-contact manifolds. (English) Zbl 0924.53024

Let (M,g,φ,ξ,η) be a contact metric manifold with a Killing ξ structure vector field (called a K-contact manifold) and C its Weyl conformal tensor. Then T p M, pM decomposes into φ(T p M)(ξ p ), where (ξ p ) is a 1-dimensional linear subspace of T p M generated by ξ p . It is natural to study the following particular cases:

(i) C:T p M×T p M×T p M(ξ p ) ,

(ii) C:T p M×T p M×T p Mφ(T p M),

(iii) C:φ(T p M)×φ(T p M)×φ(T p M)(ξ p ).

It was shown by the last and first author that in case (i) M is locally isometric to the unit sphere; in case (ii) M is an η-Einstein Sasakian manifold. This paper shows that in case (iii) if M is compact and φ 2 C(φX,φY)φZ=0 (i.e., M is φ-conformally flat), then M is a principal S 1 -bundle over an almost Kähler space of constant holomorphic sectional curvature.

53C15Differential geometric structures on manifolds
53C55Hermitian and Kählerian manifolds (global differential geometry)