On LP-Sasakian Manifolds - Mathematical Sciences

24A.A. Shaikh and Sudipta BiswasNow, from ( ∇ W ) ( U,V ) Z A(X ) W ( U,V ) Z , we getX =Hence from (4.2) we get( ∇ U ∇VW) ( X , Y ) Z = { UA(V ) + A(U ) A(V )} W ( X , Y ) Z .( R ( X , Y ) ⋅ W ) ( U,V ) Z = [ 2dA(X , Y )] W ( U,V ) Z = 0.Therefore, for a quasi-conformally recurrent manifold, we haveThus we can state the following:R ( X , Y ) ⋅ W = 0 for all X ,Y .Theorem 4.1. A quasi-conformally recurrent LP-Sasakian manifold ( , g)( n > 3)is an Einstein manifold provided that a − b ≠ 0 and a + ( n − 2) b ≠ 0.Corollary 4.1. A quasi-conformally recurrent LP-Sasakian manifold ( , g)( n > 3)is of constant scalar curvature for a = b and is a space of constant curvature ifa + ( n − 2) b = 0.Since for a quasi-conformally symmetric LP-Sasakian manifold ( , g)( n > 3) ,we have ( ∇U W ) ( X , Y ) Z = 0 which implies R ( X , Y ) ⋅ W = 0,we can state thefollowing:Corollary 4.2. A quasi-conformally symmetric LP-Sasakian manifold ( , g)( n > 3)is an Einstein manifold provided that a − b ≠ 0 and a + ( n − 2) b ≠ 0.M nM nM nM n5. LP-Sasakian manifolds satisfying the condition S ( X,ξ)⋅ R = 0We now consider an LP-Sasakian manifold ( , g)( n > 3)M nsatisfying the condition( S(X , ξ ) ⋅ R)( U,V ) Z = 0.(5.1)By definition we have( S(X , ξ ) ⋅ R) ( U,V ) Z = (( X ∧ S ξ ) ⋅ R) ( U,V ) Z= ( X ∧ S ξ ) R(U,V ) Z + R((X ∧ S ξ ) U,V ) Z+ R( U,( X ∧ ξ ) V ) Z + R(U,V ) ( X ∧ ξ ) Z , (5.2)SS