ABSTRACT. We present two curvature identities and study the corresponding classes R1 and R2 of normal manifolds of Killing type.KEY WORDS: almost contact metric manifold, normal manifold of Killing type, almost Hermitian manifold, Riemannian connection, fundamental form, curvature, CNK-structure.The geometry of almost contact metric manifolds with additional structures is a very interesting object for geometric investigations. Normal manlfolds of Killing type (below we call them CNK-ma=ifolds) form an important class of such metric manlfolds. They generalize Sasakian, quasi-Sasakian, cosymplectic, and weakly cosymplectic manifolds, which are the topic of active research of geometers from abroad ([1-3], etc.). Manifolds of this class become more interesting when they are locally conformal almost contact metric manifolds with closed fundamental 2-form (manifolds of this type are said to be Cc-maaifolds). In this case we get a contact analog of locally conformal K/ihler manifolds, which are important for modem mathematical physics, in particular, for constructing models of supergravity in the Kaluza-Klein theory [4].As A. Gray indicates, curvature identities that define certain classes of almost Hermitian marrifolds are the key to the geometry of Hermitian structures [5]. This approach led to the study of para-K~der manifolds, RE-manifolds, and other important classes of almost Hermitian manifolds. However, almost contact manifolds admit even more curvature identities that define additional symmetries of the manifold. In the present paper we introduce two curvature identities and study the corresponding classes Ri, R2 of almost contact metric manifolds. We obtain analytical tests for a CNK-mardfold to belong to these classes. We study the geometric structure of CNK-manifolds satisfying the curvature identities that we consider. We introduce an operator B (defined by the identity B(X) = Vx~, where ~ is the structural vector) and prove that the distribution Ira B on a Cc-manifold M of class R1 is involutive if and only if M is locally equivalent to the product of a quasi-Sasakian manifold and a K/ihler manifold. Finally, we obtain some tests for the structure of a CNK-manifold on M to be integrable.Let M 2"+1 be a smooth manifold with an ~]most contact metric structure {&, ~, q, g = (-9 )} such thatLet X(M) be the Lie algebra of smooth vector fields on M and d be the exterior differentiation. By x7 denote the Riemaamian connection on M and by f~(X, Y) = (X, OY) the hmdamental form of the structure. Finally, let C~176 be the algebra of smooth ffmctions on M.Recall (see [6]) that am almost contact metric structure is said to be normal if 2N + ~ @ &l = 0, whereis the Nijenhuis tensor of the structure operator ~. A CNK-str~cfurc is a normal almost contact structure with structural covector of Killing type. The condition that the structural covector is a Killing type covector means that
Vx(~7)Y + VYO?)X = O, X, Y E X(M).
