  We build a general formalism to describe thin viscous jets as one-dimensional
objects with an internal structure. We present in full generality the steps
needed to describe the viscous jets around their central line, and we argue
that the Taylor expansion of all fields around that line is conveniently
expressed in terms of symmetric trace-free tensors living in the two dimensions
of the fiber sections. We recover the standard results of axisymmetric jets and
we report the first and second corrections to the lowest order description,
also allowing for a rotational component around the axis of symmetry. When
applied to generally curved fibers, the lowest order description corresponds to
a viscous string model with circular sections. However, when including the
first corrections we find that curved jets generically develop elliptic
sections. Several subtle effects imply that the first corrections cannot be
described by a rod model, since it amounts to selectively discard some
corrections. For completeness, we also recover the constitutive relations for
forces and torques in rod models and exhibit a missing term in the lowest order
expression of viscous torque. Given that our method is based on tensors, the
complexity of all computations has been beaten down by using an appropriate
tensor algebra package such as {\it xAct}, allowing us to obtain a
one-dimensional description of curved viscous jets with all the first order
corrections consistently included. Finally, we find a description for straight
fibers with elliptic sections as a special case of these results, and recover
that ellipticity is dynamically damped by surface tension. An application to
toroidal viscous fibers is presented in the companion paper [Pitrou, Phys. Rev.
E 97, 043116 (2018)].
