∇N-EINSTEIN ALMOST CONTACT METRIC MANIFOLDS

Abstract

On an almost contact metric manifold M, an N-connection ∇N defined by the pair (∇,N), where ∇ is the interior metric connection and N: TМ → TM is an endomorphism of the tangent bundle of the manifold M such that Nξ = 0, 􀁇 􀁇 N (D) ⊂ D , is considered. Special attention is paid to the case of a skew-symmetric N-connection ∇N, which means that the torsion of an N-connection considered as a trivalent covariant tensor is skew-symmetric. Such a connection is uniquely defined and corresponds to the endomorphism N = 2ψ, where the endomorphism ψ is defined by the equality ω( X ,Y ) = g (ψX ,Y ) and is called in this work the second structure endomorphism of an almost contact metric manifold. The notion of a ∇N-Einstein almost contact metric manifold is introduced. For the case N = 2ψ, conditions under which almost contact manifolds are ∇N-Einstein manifolds are found.