I'm not so good on geometry, so I fear this is a relatively basic question.


For any $N \in \mathbb{N}$, let us identify the tangent bundle of $\mathbb{R}^N$ with $\mathbb{R}^{2N}$ in the obvious manner. Let $M$ be a smooth manifold. For any chart $\,\varphi:U \, \to \, \varphi(U) \! \subset \! \mathbb{R}^n\,$ on $M$, we can regard the derivative $\,\mathrm{d}\varphi:TU \, \to \, T(\varphi(U)) \! \subset \! \mathbb{R}^{2n}\,$ as a chart on $TM$; and hence we can also regard the second derivative $\,\mathrm{d}^2\varphi:TTU \, \to \, TT(\varphi(U)) \! \subset \! \mathbb{R}^{4n}\,$ as a chart on $TTM$.


Now suppose we have two charts $\,\varphi:U \to \varphi(U)\,$ and $\,\psi:U \to \psi(U)\,$ on $M$ defined over the same open set $U \subset M$. Letting $\Phi:=\psi \, \circ \, \varphi^{-1}$, we have that


$\hspace{5mm} \mathrm{d}^2\psi \circ (\mathrm{d}^2\varphi)^{-1}(\mathbf{x},\mathbf{u},\mathbf{v},\mathbf{w}) \ = \ (\,\Phi(\mathbf{x})\,,\,\Phi'(\mathbf{x})\mathbf{u}\,,\,\Phi'(\mathbf{x})\mathbf{v}\,,\, \textrm{more complicated term symmetric in } \mathbf{u} \textrm{ and } \mathbf{v}\,)$.


Observe in particular that if we swap $\mathbf{u}$ and $\mathbf{v}$ in the LHS, this has the effect of swapping the 2nd and 3rd terms (out of the four terms) in the RHS.



> 
> So then, we can define a function $\theta:TTM \to TTM$ by
> 
> 
> $\hspace{5mm} \mathrm{d}^2\varphi \circ \theta \circ (\mathrm{d}^2\varphi)^{-1}(\mathbf{x},\mathbf{u},\mathbf{v},\mathbf{w}) \ = \ (\mathbf{x},\mathbf{v},\mathbf{u},\mathbf{w}) \hspace{2mm}$ for any chart $\varphi$ on $M$.
> 
> 
> My question: Is there a standard name or notation for this function $\theta$?
> 
> 
> 



Is there a name for the function on $TTM$ swapping the 2nd and 3rd coordinates?
It is called the **canonical flip**, (see for example Michor, *Topics in differential geometry*, 2008, § 8.13). It can be characterised as the unique map such that 
$$
 \frac{\partial}{\partial s} \frac{\partial}{\partial t} \sigma (t, s)
= \theta \frac{\partial}{\partial t} \frac{\partial}{\partial s} \sigma (t, s).
$$


