A coordinate-free description for dynamic systems described by explicit (nonlinear) difference equations in one independent variable via a differential geometric framework is presented. Based on this covariant approach suitable geometric objects for discrete-time dynamics are introduced. Especially, the observability along a trajectory is discussed and transformations to normal forms are derived. In addition, the obtained (local) observability criteria can be checked by computer algebra algorithms. Some examples illustrate the proposed approach. (C) 2008 Elsevier Ltd. All rights reserved.