On a Structure Defined by a Tensor Field f of Type (1,1) Satisfying Pi(k)(j-1) [F-2+ a(j) F+ lambda(2)(j) I]= 0

The differentiable manifold with f - structure were studied by many authors, for example: K. Yano [7], Ishihara [8], Das [4] among others but thus far we do not know the geometry of manifolds which are endowed with special polynomial F-a(j)x( j) - structure satisfying Pi(k)(j-1) [F-2+ a(j) F+ lambda(2)(j) I]= 0 However, special quadratic structure manifold have been defined and studied by Sinha and Sharma [8]. The purpose of this paper is to study the geometry of differentiable manifolds equipped with such structures and define special polynomial structures for all values of j = 1, 2, ..., K is an element of N, and obtain integrability conditions of the distributions pi(j)(m) and (pi) over tilde (j)(m).