Exterior differentials of higher order and their covariant generalization

Abstract

We investigate a particular realization of generalized q-differential calculus of exterior forms on a smooth manifold based on the assumption that d N ϭ0 while d k 0 for kϽN. It implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order. Special attention is paid to the cases Nϭ3 and Nϭ4. A covariant basis of the algebra of such q-grade forms is introduced, and the analogs of torsion and curvature of higher order are considered. We also study a Z N -graded exterior calculus on a generalized Clifford algebra.

Generalized geodesic deviations: a Lagrangean approach

“…The same result can be obtained (30) from reduction of the geodesic equation and geodesic deviation equation in five-dimensional space-time, as particles with different q/m ratio in four dimensions correspond to particles with different momentum in fivedimensional space-time, as shown in [23].…”

The Two-Parameter Higher-Order Differential Calculus and Curvature on a Quantum Plane

“…In fact, in this limit one has q = j, due to (9), and all the results reduce to their counterparts obtained in [15]; especially, the two-parameter quantum plane becomes a reduced quantum plane.…”

n=3 differential calculus and gauge theory on a reduced quantum plane

“…In the literature, there exist different points of view [1][2][3][4][5][6] to generalize the ordinary differential calculus (d 2 ϭ0); see, for example, Refs. 7-8, to higher orders.…”