&= {\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}_{\pi^{-1}(\gamma(s))}, & s&\in J,   \label{3.2}
\\
L^\gamma_{s\to t}(\lambda u + \mu v)                 \label{3.linear}
  &= \lambda L^\gamma_{s\to t}u + \mu L^\gamma_{s\to t}v,
    & \lambda,\mu &\in \mathbb{C},\quad u,v\in{\pi^{-1}(\gamma(s))},
    \end{aligned}$$ where $\circ$ denotes composition of maps and ${\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}_X$ is the identity map of a set $X$.

Equations [[(\[3.1\])]{}]{} and [[(\[3.2\])]{}]{} mean that $L$ is a *transport along paths* in the bundle $(E,\pi,B)$ [@bp-TP-general definition 2.1], while [[(\[3.linear\])]{}]{} specifies that it is *linear* [@bp-TP-general equation (2.8)]. In the present paper only linear transports will be used.

This definition generalizes the concept of a parallel transport in the theory of (linear) connections (see [@bp-TP-general; @bp-TP-parallelT] and the references therein for details and comparison).

A few comments on definition \[Defn3.1\] are now in order. According to equation [[(\[3.fibre\])]{}]{}, a linear transport along paths may be considered as a path-depending connection: it establishes a fibre (isomorphic - see below) correspondence between the fibres over the path along which it acts. By virtue of equation [[(\[3.linear\])]{}]{} this correspondence is linear. Such a condition is a natural one when vector bundles are involved, it simply represents a compatibility condition with the vectorial structure of the bundle (see [@bp-TP-general sect. 2.3] for details). Equation [[(\[3.2\])]{}]{} is a formal realization 