I am learning jet bundle theory for my research in physics. I have read 6 (of 7) chapters of "The Geometry of Jet Bundles" (D. J. Saunders), but I am confused about the relationships between jet bundles, projectivised vector bundles and contact manifolds. I summarize what I have learnt from Saunders' book and other sources as follows.
Let $M$ be an $m$-dimensional manifold, $R$ $1$-dimensional, and $E = M \times R$. The total space $J^1 E$ of the first jet bundle of the bundle $\pi: E \rightarrow M$ can be described locally by some coordinates $\{x^i, t, t_i\}$. $J^1 E$ is a contact manifold with a contact form $dt - t_i dx^i$ living on it, and it is diffeomorphic to $T^* M \times R$. The latter is diffeomorphic to a submanifold of $T^* E$.
The projectivised cotangent bundle $PT^{*} E$ can be described locally by some coordinates $\{ x^i, t, [p_i, p_t] \}$. $PT^* E$ is a contact manifold with a contact form locally expressible as $dt +\frac{p_i}{p_t} d x^i$ ($p_t \neq 0$).
There is a ( canonical?! ) map $PT^* E \rightarrow J^1 E$ given by sending ($x^i, t, [p_i, p_t])$ to $(x^i, t, t_i)$ with $p_t \neq 0$ and $p_i = - t_i p_t$.
My question: how are $PT^* E$ and $J^1 E$ related (as manifolds and as fiber bundles)? They have similar contact forms, but are they diffeomorphic manifolds and bundles (at least locally)?
I imagine their relationship is similar to the relationship between the 2-plane $z=1$ in $R^3$ and the projective plane $RP^2$.
What is their relationship in exact mathematical terms?
A similar question may be asked about the relationship between $PTE$ and $J^1 E$ of the bundle $\pi_2: E \rightarrow R$ (with no emphasis on contact structure).