No abstract

We introduce a generalization to the second order of the notion of the G1-structure, the so called generalized almost tangent structure. For this purpose, the concepts of the second order frame bundle H2 (V_), its structual group L,2n and its associated tangent bundle of second order T2 (V~) of a differentiable manifold V , are described from the point of view that is used . Then, a C1 -structure of second order -called Gi-structure-is constructed on V , by an endomorphism J acting on T2 (V ,), satisfying the relation J2 = 0 and some hypotheses on its rank . Its connection and characteristic cohomology class are defined.Some of the G-structures of the first order are those defined by nilpotent operators of degree r + 1 (r >_ 1) that is, the G,-structures, defined by J. ) and studied by H.A . Eliopoulos ([11]) .The G1-structure of the first order, briefly G1-structure, is defined ([15]) on an m-dimensional differentiable manifold V .,, of class C°°by means of an 1-form J, of constant rank p, with values in the tangent bundle, such that at each point x E V ,,, Jx = 0. dim Im Jx = p > 1, dime ker Jy = q, m = p + q and q independent of the point x of V, ,.The G1-structure is also studied by [1] ; it is called generalized almost tangent structure .Our objective in the present paper is to find a prolongation of this structure, that is, there is defined a G-structure of second order on V .,, called the G1-structure of order 2, briefly a G2-structure, by means of

The present work is based on a type of structures on a differential manifold V , called G-structures of the second kind, defined by endomorphism J on the second order tangent bundle T 2 (V). Our objective is to give conditions for a differential manifold to admit a real almost product and a generalised almost tangent structure of second order. The concepts of the second order frame bundle H 2 (V), its structural group L 2 and its associated tangent bundle of second order T 2 (V) of a differentiable manifold V , are used from the point of view that is described in papers [5] and [6]. Also, the almost tangent structure of order two is mentioned and its generalisation, the second order almost transverse structure, is defined.