Recently I learned about the following definition, but I'm not sure what's the precise meaning of this: Let $Y$ be an odd-dimensional manifold. An almost complex structure $J$ on $\mathbb{R} \times Y$ is $\mathbb{R}$-invariant if it is invariant under the translation $(s,y) \mapsto (s + c,y)$ for all $c \in \mathbb{R}$. I was confused because $J$ is a bundle map on $T(\mathbb{R} \times Y)$, whereas translation map is defined on the product manifold $\mathbb{R} \times Y$.
Nevertheless, I'm convinced myself that translation invariance here means $J$ commutes with the derivative of the translation map. Indeed, fix such an almost complex structure $J$ and consider the space of $J$-holomorphic curves \begin{equation*} \mathcal{M}_J = \{u: \Sigma \to \mathbb{R} \times Y \mid du \circ j = J \circ du\}/\sim \end{equation*} where $(\Sigma,j)$ is a Riemann surface with (almost) complex structure $j$, and $u \sim u'$ if they differ by a biholomorphic reparametrization of the domain. One can define an $\mathbb{R}$-action on $\mathcal{M}_J$ by translating the $\mathbb{R}$-coordinate in the image of $u$ (using the proposed definition of translation invariance). The resulting moduli space $\mathcal{M}_J/\mathbb{R}$ is studied by many authors (for example, in this paper). I just want to make sure if that's the correct definition. Thanks in advance!