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-FX, FY-FFX, Y+F*X, Yintroduced 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)