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.