Fermi[-]{}Walker and Frenet[-]{}Serret transports , etc. Consequently definition \[Defn3.1\] is general enough to cover a list of important transports used in theoretical physics and mathematics. Thus studying the properties of the linear transports along paths we can make corresponding conclusions for any one of the transports mentioned.[^9]

From [[(\[3.1\])]{}]{} and [[(\[3.2\])]{}]{} we get that $L^\gamma_{s\to t}$ are invertible and $$\label{3.3}
\left(L^\gamma_{s\to t}\right)^{-1} = L^\gamma_{t\to s},
    \qquad s,t\in J.$$ Hence the linear transports along paths are in fact linear isomorphisms of the fibres over the path along which they act.

The following two propositions establish the general structure of linear transports along paths.

\[Prop3.1\] A map [[(\[3.fibre\])]{}]{} is a linear transport along $\gamma$ from $s$ to $t$ for every $s,t\in J$ if and only if there exist an isomorphic with $\pi^{-1}(x),\ x\in B$ vector space $V$ and family of linear isomorphisms $\{F(s;\gamma)\colon \pi^{-1}(\gamma(s))\to V,\ s\in J\}$ such that $$\label{3.4}
L_{s\to t}^{\gamma} =
    F^{-1}(t;\gamma) \circ  F(s;\gamma),\qquad s,t\in J.$$

If [[(\[3.fibre\])]{}]{} is a linear transport along $\gamma$ from $s$ to $t$, then fixing some $s_0\in J$ and using [[(\[3.2\])]{}]{} and [[(\[3.3\])]{}]{}, we get $
L_{s\to t}^{\gamma} = L_{s_0\to t}^{\gamma} \circ L_{s\to s_0}^{\gamma}
    = \bigl(L_{t\to s_0}^{\gamma}\bigr)^{-1} \circ L_{s\to s_0}^{\gamma}.
$ So [[(\[3.4\])]{}]{} holds for 