Shyamal Kumar Hui and Richard Santiago Lemence*
Department of Mathematics, The University of Burdwan, Burdwan –713104, West Bengal, India, e-mail: shyamal_hui@yahoo.co.in, Institute of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City 1101, Philippines, e-mail: rslemence@math.upd.edu.ph
Received: March 25, 2015; Accepted: March 21, 2018
A Kenmotsu manifold Mn(φ, ξ, η, g), (n = 2m+1 > 3) is called a generalized φ-recurrent if its curvature tensor R satisfies φ2((∇WR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z
for all X, Y, Z, W ∈ χ(M), where ∇ denotes the operator of covariant differentiation with respect to the metric g, i.e. ∇ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y )Z = g(Y, Z)X −g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a φ-symmetric. Now, a Kenmotsu manifold Mn(φ, ξ, η, g), (n = 2m + 1 > 3) is said to be generalized φ-Ricci recurrent if it satisfies φ2((∇WQ)(Y))=A(X)QY+B(X)Y
for any vector field X, Y ∈ χ(M), where Q is the Ricci operator, i.e., g(QX, Y ) = S(X, Y ) for all X, Y. In this paper, we study generalized φ-recurrent and generalized φ-Ricci recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized φ-recurrent Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.
Keywords: generalized φ-recurrent, generalized φ Ricci-recurrent, Kenmotsu manifold, η-Einstein manifold, quarter-symmetric metric connection
