Contact Riemannian manifolds satisfying C(^)S=0 AND 4e(k, n)-nullity distribution

Abstract

Let ,cp,Ç,n,g) be a contact Riemannian manifold of dimension 2n+l>3. Tanno [6] proved that (M^’,cp,§,r|,g) is an Einstein manifold and Ç belongs to the k-nullity distribution, then M is a Sasakian manifold and Perrone [4] proved that if M is a contact Riemannian manifold with R(X,Ç)S=0 and Ç belongs to the k-nulhty distribution, where ke R, then M is either an Einstein-Sasakian manifold or the produet E"^’(0)xS"(4). Papantoniou [1] generahzing this result proved that if M is a contact Riemannian manifold with R(X,Ç)S=0 and § belongs to the (k,ıx)-nullity distribution, where (k,)j.)e R^, then M is local isometric to E“^'(0)xS"(4) or an Einstein-Sasakian manifold or, an r|-Einstein manifold.The purpose of this paper is to classify the contact- manifolds satisfying C(X,Ç)S=0 under the condition that characteristic vector field belongs to the (k,p,)-nullity distribution.