On certain operators associated with tensor fields

The theory of differential concomitants has been developed by Frolicher, Nijenhuis [1], [2], [6], Schouten [9] and others. 1 ' Schouten [9] introduced a concomitant [P, Q] formed with tensor fields P and Q of type (p+l, 0) and (q+1, 0) respectively, which is a tensor field of type (p+q+1, 0).Nijenhuis [6] introduced a concomitant [S, T] formed with a vector s-form S and a vector /-form T which is a vector (s+ί)-form.Frolicher and Nijenhuis [1], [2], also introduced a concomitant [S, ω] formed with a vector s-form S and a scalar /-form ω which is a scalar (s-K)-form.On the other hand it was found that, in the study of a differentiable manifold M with an almost complex structure F, the tensor = [FX, FY]-F[X, FY]-F[FX, Y]+F*[X, Y]introduced by Nijenhuis [5] plays an important part.It is now well known [4] that it is necessary and sufficient for an almost complex manifold to be complex that the Nijenhuis tensor N formed with F vanishes identically.Nijenhuis [6] proved also thatand it seemed that there is no concomitant formed only with F and its partial derivatives and being essentially distinct from N.However Walker [13] found a tensor field of type (1,4) involving the second partial derivatives of an almost complex structure F.Willmore [14] introduced a tensor field W of type (1,4)