Consider the usual formulas for the Torsion and Curvature of an affine connection:
$$T(X,Y)=\nabla_X Y -\nabla_Y X-[X,Y]$$ $$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]} $$
These formulas are clearly formally alike; can one still obtain tensors (i.e. geometric quantities) by wisely applying "higher-order" derivatives?
