Submanifolds with homothetic Gauss map in codimension two

Abstract

Let \(f{:}\;M^n\rightarrow \mathbb {R}^{n+p}\) be an isometric immersion of an n-dimensional Riemannian manifold \(M^n\) into the (\(n+p\))-dimensional Euclidean space. Its Gauss map \(\phi {:}\;M^n\rightarrow G_n(\mathbb {R}^{n+p})\) into the Grassmannian \(G_n(\mathbb {R}^{n+p})\) is defined by assigning to every point of \(M^n\) its tangent space, considered as a vector subspace of \(\mathbb {R}^{n+p}\). The third fundamental form \(\text{ III }\) of f is the pullback of the canonical Riemannian metric on \(G_p(\mathbb {R}^{n+p})\) via \(\phi \). In this article we derive a complete classification of all those f with codimension two for which the Gauss map \(\phi \) is homothetic; i.e., \(\text{ III }\) is a constant multiple of the Riemannian metric on \(M^n\). We furthermore study and classify codimension two submanifolds with homothetic Gauss map in real space forms of nonzero curvature. To conclude, based on a strong connection established between homothetic Gauss map and minimal Einstein submanifolds, we pose a conjecture suggesting a possible complete classification of the submanifolds with the former property in arbitrary codimension.