Isometric immersions of $${{\mathbb R}^2}$$ into $${{\mathbb R}^4}$$ and perturbation of Hopf tori

Abstract

We produce a new general family of flat tori in \({{\mathbb R}^4}\) , the first one since Bianchi’s classical works in the nineteenth century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in \({{\mathbb S}^3}\) , we first present a global description of all isometric immersions of \({{\mathbb R}^2}\) into \({{\mathbb R}^4}\) with flat normal bundle.