This paper presents a new theoretical approach for the description of multidimensional objects for which 3-D and 4-D are particular cases. The approach is based on a curved space which is associated to each object. This curved space is characterised by Riemannian tensors from which invariant quantities are defined. A descriptor or index is constructed from those invariants for which a statistical and an abstract graph representation are associated. The obtained representations are invariant under general coordinate transformations. The statistical representation allows a compact description of the object while the abstract graph allows describing the relations in between the parts as well as the structure. | 