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We prove a Bonnet theorem for isometric immersions of submanifolds into theproducts of an arbitrary number of simply connected real space forms. Then, weprove the existence of associated families of minimal surfaces in suchproducts. Finally, in the case of $\mathbb{S}^2\times\mathbb{S}^2$, we give acomplex version of the main theorem in terms of the two canonical complexstructures of $\mathbb{S}^2\times\mathbb{S}^2$.

A Fundamental Theorem for submanifolds of multiproducts of real space forms