The Gauss map of minimal surfaces in $${\mathbb {S}}^2 \times {\mathbb {R}}$$

In this work, we consider the model of $${{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}$$ isometric to $${\mathbb {R}}^3{\setminus } \{0\}$$ , endowed with a metric conformally equivalent Euclidean {R}}^3$$ and define Gauss map for surfaces in likewise 3-space. We show as main result that any two minimal conformal immersions same non-constant differ by only types ambient isometries: either $$f=({{\,\mathrm{\mathrm {Id}}\,}},T)$$ where T is translation on {R}}$$ or $$f=({\mathcal {A}},T)$$ $${\mathcal {A}}$$ denotes antipodal {S}}^2$$ . This means immersion determined its structure map, up those isometries.