Euclidean Hypersurfaces Research Articles

Infinitesimally Moebius bendable hypersurfaces

AbstractLi, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$, the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $f\colon M^n\to \mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_t\colon M^n\to \mathbb{R}^m$, with $t\in (-\epsilon, \epsilon)$ and $f_0=f$, such that the Moebius metrics determined by ft coincide up to the first order. Then we characterize isometric immersions $f\colon M^n\to \mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension $n\geq 5$ that admit non-trivial infinitesimal Moebius variations.

Euclidean hypersurfaces isometric to spheres

<p>Given an immersed hypersurface $ M^{n} $ in the Euclidean space $ E^{n+1} $, the tangential component $\boldsymbol{\omega }$ of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function $ \sigma $ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface $ M^{n} $ in $ E^{n+1} $ of positive Ricci curvature with shape operator $ T $ invariant under $\boldsymbol{\omega }$ and the support function $ \sigma $ satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface $ M^{n} $ in $ E^{n+1} $ with the gradient of support function $ \sigma $, an eigenvector of the shape operator $ T $ with eigenvalue function the mean curvature $ H $, and the integral of the squared length of the gradient $ \nabla \sigma $ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $ M^{n} $ of positive Ricci curvature in $ E^{n+1} $ has an incompressible basic vector field $\boldsymbol{\omega }$, if and only if $ M^{n} $ is isometric to a sphere.</p>

A construction of Sbrana-Cartan hypersurfaces in the discrete class

The classical classifications of the locally isometrically deformable Euclidean hypersurfaces obtained by U. Sbrana in 1909 and E. Cartan in 1916 include four classes, among them the one formed by submanifolds that allow just a single deformation. The question of whether these Sbrana-Cartan hypersurfaces do, in fact, exist was not addressed by either of them. Positive answers to this question were given by Dajczer-Florit-Tojeiro in 1998 for the ones called of hyperbolic type and by Dajczer-Florit in 2004 when of elliptic type which is the other possibility. In both cases the examples constructed are rather special. The main result of this paper yields an abundance of examples of hypersurfaces of either type and seems to point in the direction of a classification although that goal remains elusive.

Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces

Abstract We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.

Translation Hypersurfaces in Euclidean 4-Spaces

In this article, the translation hypersurfaces in Euclidean 4- space are defined as the sum of three curves with distinct parameters with unit speed, and non-planar. These curves are called the generator curves of the hypersurface. Utilizing the hypersurface theory in Euclidean 4-space, unit normal vector field, shape (Weingarten) operator matrix, fundamental forms, Gaussian curvature and mean curvature have been expressed for the translation hypersurfaces. Finally, the computational example is stated to efficiency of the theoretical results.

Cyclic conformally flat hypersurfaces revisited

In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$, that is, the conformally flat hypersurfaces of $\mathbb{R}^4$ with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$ as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of $\mathbb{R}^4$ whose tangent component is an eigenvector field correspondent to one of its principal curvatures.

Conformal infinitesimal variations of Euclidean hypersurfaces

In the realm of conformal geometry, we give a parametric classification of the hypersurfaces in Euclidean space that admit nontrivial conformal infinitesimal variations. A parametric classification of the Euclidean hypersurfaces that allow a nontrivial conformal variation was obtained by E. Cartan in 1917. In particular, we show that the class of hypersurfaces studied here is much larger than the one characterized by Cartan.

2-Ruled hypersurfaces in Euclidean 4-space

In this paper, firstly we obtain the Gauss map (unit normal vector field) of a 2-ruled hypersurface in Euclidean 4-space with the aid of its general parametric equation. After, we obtain Gaussian and mean curvatures of the 2-ruled hypersurface and give some characterizations about its minimality. Finally, we deal with the first and second Laplace–Beltrami operators of 2-ruled hypersurfaces in E4 and give some results about them.

Conformal infinitesimal variations of submanifolds

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 are a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.

Curvature computations for the intersection curves of hypersurfaces in Euclidean n-space

The aim of this paper is to derive the curvatures and Frenet vectors of the intersection curve of (n−1) transversally intersecting hypersurfaces represented either in implicit or in parametric form in Euclidean n-space. Since the intersection of (n−1) implicit hypersurfaces defines an implicit curve in n-dimensions, the method derived for the implicit case can be considered as a partial answer to the open problem posed by Ron Goldman in 2005. Also, an application is provided for the implicit case where the results are presented by using computer codes currently produced in MATLAB.

Equiaffine Braneworld

Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the directions of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note, in this paper, that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry. In particular, we show that a geometric origin for dark energy can be proposed within this framework.

<i>h</i>-Almost Ricci solitons with concurrent potential fields

In this paper, we will focus our attention on the structure of h-almost Ricci solitons. A complete classification of h-almost Ricci solitons with concurrent potential vector fields is given. Also, we obtain conditions on a submanifold of a Riemannian h-almost Ricci soliton to be an h-almost Ricci soliton. Finally, we classify h-almost Ricci soliton on Euclidean hypersurface with λ = h.

A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere $\mathbb

Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam’s result is known to be equivalent to the Riemannian positive mass theorem. In this paper, we provide a supplement to Shi–Tam’s result by including the boundary effect of minimal hypersurfaces. More precisely, given a compact manifold $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_H$ and $\Sigma_O$, where $\Sigma_H$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_O$ is assumed to be isometric to a suitable 2-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of mass $m$, we establish an inequality relating $m$, the area of $\Sigma_H$, and two weighted total mean curvatures of $\Sigma_O$ and $\Sigma$. In $3$-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of $\Sigma_O$ being greater than or equal to the Hawking mass of $\Sigma_H$. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in an asymptotically flat $3$-manifold $M$ into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of $M$. It follows that our result on the compact manifold $\Omega$ is equivalent to the Riemannian Penrose inequality.

Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

In this paper we classify Euclidean hypersurfaces \(f:M^n\rightarrow {\mathbb {R}}^{n+1}\) with a principal curvature of multiplicity \(n-2\) that admit a genuine conformal deformation \({\tilde{f}}:M^n\rightarrow {\mathbb {R}}^{n+2}\). That \({\tilde{f}}:M^n\rightarrow {\mathbb {R}}^{n+2}\) 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 \({\tilde{f}}|_U=h\circ f|_U\) of \(f|_U\) with a conformal immersion \(h:V\rightarrow {\mathbb {R}}^{n+2}\) of an open subset \(V\subset {\mathbb {R}}^{n+1}\) containing f(U).

Hypersurfaces of space forms carrying a totally geodesic foliation

In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was already given in Dajczer et al. (Geometriae Dedicata 176:215–224, 2015) for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two. We also extend the classification under the global assumption of completeness of the leaves for hypersurfaces of the sphere and show that there exist plenty of examples in hyperbolic space.

A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition

We show that a curvature condition on the Gauss-Kronecker curvature and scalar curvature of a convex smooth hypersurface lying in the four dimensional Euclidean space yields a lower bound for the mean curvature. The curvature condition we investigate is suggested by the local geometry of cylinders in the four dimensional Euclidean space.

Yamabe and Quasi-Yamabe Solitons on Euclidean Submanifolds

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.

Lie Algebroid Invariants for Subgeometry

We investigate the infinitesimal invariants of an immersed submanifold $\Sigma $ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $\Sigma $. The invariants are derived from the logarithmic derivative of the immersion of $\Sigma $ into $M$, a complete invariant introduced in the companion article, 'A characterization of smooth maps into a homogeneous space'. Applications of the Lie algebroid approach to subgeometry include a new interpretation of Cartan's method of moving frames and a novel proof of the fundamental theorem of hypersurfaces in Euclidean, elliptic and hyperbolic geometry.

Convexity, Rigidity, and Reduction of Codimension of Isometric Immersions into Space Forms

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by do Carmo and Lima, as well as by do Carmo and Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.