Related papers: Euclidean Hypersurfaces with Genuine Conformal Def…

We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$…

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds is a classical subject in…

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature…

Let $ M^{n+1} $ ($ n \ge 2 $) be a simply-connected space form of sectional curvature $ -\kappa^2 $ for some $ \kappa \geq 0 $, and $ I $ an interval not containing $ [-\kappa,\kappa] $ in its interior. It is known that the domain of a…

We study generic conformally flat (local-)hypersurfaces in the Euclidean 4-space $\mathbb{R}^4$. Such a hypersurface $f$ has the dual (hypersurface) $f^*$ in $\mathbb{R}^4$, which is also generic and conformally flat. By repeating the…

We prove that a minimal hypersurfaces $f\colon M^{3} \to \mathbb{Q}^4(c)$ with nonzero three distinct principal curvature cannot be isometrically immersed in $\mathbb{Q}^4(\tilde{c}), \ \tilde{c}\neq c$. In the other cases, we present a…

Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. If $2p\leq 2n-1$, we show that generic rank conditions on the second…

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of…

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the…

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine…

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures.…

We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers,…

In this paper, we first study isometric immersions $f: M^n\rightarrow M^{n+k}(c), n\geq 3,$ into space forms with flat normal bundle and constant scalar curvature $R.$ Under a suitable multiplicity condition on the second fundamental form…

We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic space forms $\mathbb HP^m$ and $\mathbb HH^m$ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where…

We prove that strong finite total curvature complete hypersurfaces of (n+1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of…

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension…

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do…

We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled,…