Exterior differentials of higher order and their covariant generalization

We investigate a particular realization of generalized q-differential calcu lus of exterior forms on a smooth manifold based on the assumption that d(N )=0 while d(k)not equal 0 for k < N. It implies the existence of cyclic com mutation relations for the differentials of first order and their generaliz ation for the differentials of higher order. Special attention is paid to t he cases N=3 and N=4. A covariant basis of the algebra of such q-grade form s is introduced, and the analogs of torsion and curvature of higher order a re considered. We also study a Z(N)-graded exterior calculus on a generaliz ed Clifford algebra. (C) 2000 American Institute of Physics. [S0022- 2488(0 0)03008-5].