In this paper we shall obtain certain results on the structure defined by F ( K , − ( − ) K + 1 ) and satisfying F K − ( − ) K + 1 F = 0 , where F is a non null tensor field of the type ( 1 , 1 ) Such a structure on an n -dimensional differentiable manifold M n has been called F ( K , − ( − ) K + 1 ) structure of rank “ r ”, where the rank of F is constant on M n and is equal to “ r ” In this case M n is called an F ( K , − ( − ) K + 1 ) manifold. The case when K is odd has been considered in this paper