Derivative strings: contravariant aspect

Abstract

A concept termed strings, which generalizes those of derivatives of scalars, tensors and affine connections, was introduced and studied in two previous papers by the authors. The results obtained were, essentially, of a ‘covariant nature’. In this paper a corresponding contravariant theory is developed and combined with the previous results. A fairly complete theoretical framework is thereby established. In particular, the previously defined operations of intertwining and of covariant differen­tiation for strings are generalized to arbitrary strings of mixed covariant-contravariant type and further properties of these operations are derived. To distinguish strings in the sense of this and the two foregoing papers from various other concepts that are also termed strings and that are now of great interest in physics and cosmology, we shall occasionally refer to the present concept as derivative strings, as has already been done in the title.