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Register a new userClassification of codimension two deformations of rank two Riemannian manifolds
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The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also determine which deformations are genuine and honest, allowing us to find the first known examples of honestly locally deformable rank two submanifolds in codimension two. In addition, we study which of these submanifolds admit isometric immersions as Euclidean hypersurfaces, a property that gives rise to several applications to the Sbrana-Cartan theory of deformable Euclidean hypersurfaces.
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In this paper we classify Euclidean hypersurfaces $fcolon M^n rightarrow mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $tilde{f}colon M^n rightarrow mathbb{R}^{n+2}$. That $tilde{f}colon M^n rightarrow mathbb{R}^{n+2}$ is a genuine conformal deformation of $f$ means that it is a conformal immersion for which there exists no open subset $U subset M^n$ such that the restriction $tilde{f}|_U$ is a composition $tilde f|_U=hcirc f|_U$ of $f|_U$ with a conformal immersion $hcolon Vto mathbb{R}^{n+2}$ of an open subset $Vsubset mathbb{R}^{n+1}$ containing $f(U)$.
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We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.
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