Spinorial representation of submanifolds in a product of space forms

We present a method giving a spinorial characterization of an immersion in a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory in that spaces. We also study special cases: we recover previously known results concerning immersions in $\mathbb{S}^2\times\mathbb{R}$ and we obtain new spinorial characterizations of immersions in $\mathbb{S}^2\times\mathbb{R}^2$ and in $\mathbb{H}^2\times\mathbb{R}.$ We then study the theory of $H=1/2$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of $H=1/2$ surfaces in $\mathbb{R}^{1,2}.$