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I'm currently reading William Burke's book Applied Differential Geometry and he uses a lot in the development of Lagrangian Mechanics the notion of a Contact Bundle. He does explain intuitively what the contact bundle should be, but I really didn't get yet the intuition, nor I got the rigorous construction of it.

He says: "Locally an unparametrized curve is just a one-dimensional submanifold. The local approximation to this is called a line element... If you wish to be precise, a line element is an equivalence class of curves, and is also, therefore, an equivalence class of tangent vectors, say, $[v]$, under $v\sim kv$, where $k$ can be positive or negative, but nonzero. The line-element contact bundle that I call $CM$ consists of pairs, a point in the manifold $M$, and a line element at that point".

So, my understanding is that we construct it this way: at each point $p\in M$ we consider $T_p M$ the tangent space at $p$ and introduce an equivalence relation $\sim$ by $v\sim kv$. Then we call the quotient set of this relation $C_p M$ and define $CM = \bigcup_{p\in M} \{p\}\times C_pM$ like we do for the tangent bundle. Now, what is the topology on $CM$?

Apart from that, I do not understood yet the intuition on this bundle. Is there some material out there more detailed on that matter?

Thanks very much in advance.

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  • $\begingroup$ See here en.wikipedia.org/wiki/Contact_geometry $\endgroup$ – user40276 Feb 11 '14 at 0:58
  • $\begingroup$ Highly recommended: Arnold and Givental article on Symplectic geometry in the Encyclopedia of Mathematical Sciences, vol 4 (Dynamical Systems IV). amazon.com/Dynamical-Systems-IV-Symplectic-Applications/dp/… $\endgroup$ – Gil Bor Feb 11 '14 at 3:28
  • $\begingroup$ Another way to express your construction (giving a natural topology): $CM$ is the projectivization of $TM$, i.e. the quotient of $TM$, with the zero section removed, by the action of multiplication by non-zero scalars. But beware: the standard definition of the manifold of contact elements is the set of hyperplanes (codimension 1 subpsaces) in $T_x M$ for all $x\in M$ (the projectivized cotangent bundle). If $M$ is Riemennian you can identify the two definitions, but the advantage of the second definition is that it gives $CM$ a natural, metric independent, contact structure. $\endgroup$ – Gil Bor Feb 12 '14 at 14:49

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