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We classify hypersurfaces of rank two of Euclidean space $${\mathbb{R}^{n+1}}$$ that admit genuine isometric deformations in $${\mathbb{R}^{n+2}}$$. That an isometric immersion $${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$$ is a genuine isometric deformation of a hypersurface $${f\colon M^n\to\mathbb{R}^{n+1}}$$ means that $${\hat f}$$ is nowhere a composition $${\hat f=\hat F\circ f}$$, where $${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$$ is an isometric immersion of an open subset V containing the hypersurface. [ABSTRACT FROM AUTHOR]

In this paper we give a conformal classification of all euclidean submanifolds carrying a parallel principal curvature normal under the intrinsic additional assumption that the associated conformal conullity is involutive and the leaves are extrinsic spheres in the submanifold in the sense of Nomizu. We also provide several applications of this result. [ABSTRACT FROM AUTHOR]

We first extend the classical Sbrana-Cartan theory of isometrically deformable euclidean hypersurfaces to the sphere and hyperbolic space. Then we construct and characterize a large family of hypersurfaces which admit a unique deformation. This is used to show, by means of explicit examples, that different types of hypersurfaces in the Sbrana-Cartan classification can be smoothly attached. Finally, among other applications, we discuss the existence of complete deformable hypersurfaces in hyperbolic space. [ABSTRACT FROM AUTHOR]