Normalized Mean Curvature Vector Field Research Articles

Biconservative Riemannian Submanifolds in Minkowski 5-Space

In this study, we focus biconservative Riemannian submanifolds with parallel normalized mean curvature vector field (PNMCV) in $\mathbb{E}^5_1$. We obtain explicit classifications for these submanifolds with exactly two distinct principal curvatures of the shape operator along the mean curvature vector field. In particular, we investigate these submanifolds which have time-like and space-like mean curvature vector field in $\mathbb{E}^5_1$.

Timelike surfaces with parallel normalized mean curvature vector field in the Minkowski 4-space

Timelike surfaces with parallel normalized mean curvature vector field in the Minkowski 4-space

Basic classes of timelike general rotational surfaces in the four-dimensional Minkowski space

In the present paper, we consider timelike general rotational surfaces in the Minkowski 4- space which are analogous to the general rotational surfaces in the Euclidean 4-space introduced by C. Moore. We study two types of such surfaces (with timelike and spacelike meridian curve, respectively) and describe analytically some of their basic geometric classes: flat timelike general rotational surfaces, timelike general rotational surfaces with flat normal connection, and timelike general rotational surfaces with non-zero constant mean curvature. We give explicitly all minimal timelike general rotational surfaces and all timelike general rotational surfaces with parallel normalized mean curvature vector field.

Biconservative Submanifolds with Parallel Normalized Mean Curvature Vector Field in Euclidean Spaces

Biconservative Submanifolds with Parallel Normalized Mean Curvature Vector Field in Euclidean Spaces

On complete submanifolds with parallel normalized mean curvature in product spaces

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$ , where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$ , it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

Classification of f-biharmonic submanifolds in Lorentz space forms

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

A Simons type integral inequality for closed submanifolds in the product space [formula omitted

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces Sn×R, where Sn is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in Sn×R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S1(1−r2)×Sn−1(r) with 0<r<1.

A sharp integral inequality for closed spacelike submanifolds immersed in the de Sitter space

In this paper, we establish a sharp integral inequality for n-dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space $$\mathbb S_p^{n+p}$$ of index p, and we use it to characterize totally umbilical round spheres $$\mathbb S^n(r)$$ , with $$r>1$$ , of $$\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$$ . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.

Complete linear Weingarten spacelike submanifolds immersed in the anti-de Sitter space

We deal with $n$-dimensional complete linear Weingarten spacelike submanifolds having nonnegative sectional curvature and immersed in the anti-de Sitter space $\mathbb H_p^{n+p}$ of index $p$ with parallel normalized mean curvature vector field. We recall

On the umbilicity of linear Weingarten spacelike submanifolds immersed in the de Sitter space

We investigate the umbilicity of n-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field in the de Sitter space [Formula: see text] of index [Formula: see text]. We recall that a spacelike submanifold is said to be linear Weingarten when its mean curvature function H and its normalized scalar curvature [Formula: see text] satisfy a linear relation of the type [Formula: see text], for some constants [Formula: see text]. Under suitable constraints on the values of [Formula: see text] and [Formula: see text], we apply a generalized maximum principle for a modified Cheng–Yau operator [Formula: see text] in order to show that such a spacelike submanifold must be either totally umbilical or isometric to a product [Formula: see text], where the factors [Formula: see text] are totally umbilical submanifolds of [Formula: see text] which are mutually perpendicular along their intersections. Moreover, we also study the case in which these spacelike submanifolds are [Formula: see text]-parabolic.

On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Let $$M^{n}$$ be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $$L_{p}^{n+p}$$ of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold $$M^{n}$$ be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold $$L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}$$, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.

LW-surfaces with higher codimension and Liebmann’s Theorem in the hyperbolic space

We deal with complete surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in the hyperbolic space $${\mathbb {H}}^{2+p}$$ . Supposing that such a surface $$M^{2}$$ satisfies a linear Weingarten condition of the type $$K=aH+b$$ for some appropriate real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively, we show that $$M^{2}$$ must be either totally umbilical or isometric to one of the following flat surfaces: $${\mathbb {S}}^{1}(r)\times {\mathbb {R}}$$ , $$\mathbb S^{1}(r)\times {\mathbb {S}}^1(\sqrt{{\tilde{r}}^2-r^2})$$ or $$\mathbb S^{1}(r)\times {\mathbb {H}}^1(-\sqrt{{\tilde{r}}^2+r^2})$$ . Furthermore, we obtain a version of the classical Liebmann’s Theorem (Liebmann in Math Phys Klasse 44–55, 1899) showing that the only compact (without boundary) surfaces having positive constant Gaussian curvature, immersed with flat normal bundle in the hyperbolic space $${\mathbb {H}}^{2+p}$$ , are the totally umbilical round spheres.

Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

Linear Weingarten submanifolds in the hyperbolic space

We use suitable maximum principles in order to obtain characterization results concerning n-dimensional linear Weingarten submanifolds immersed with globally flat normal bundle and parallel normalized mean curvature vector field in the $$(n+p)$$ -dimensional hyperbolic space $${\mathbb {H}}^{n+p}$$ . In particular, when $$p=2$$ , we present complete descriptions of these submanifolds.

Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds. Also, we prove that such submanifolds have parallel mean curvature vector field under the assumption that they have diagonalizable shape operator with at most two distinct principal curvatures in the direction of the mean curvature vector field, and apply it to obtain a partial classification result.

Characterizations of linear Weingarten submanifolds

In this paper, we deal with complete linear Weingarten submanifolds Mn immersed with parallel normalized mean curvature vector field in a Riemannian space form Qcn+p of constant sectional curvature c and with n≥4. We recall that a submanifold is called linear Weingarten when its mean and scalar curvatures are linearly related. In this setting, we establish a suitable extension of the generalized maximum principle of Omori–Yau in order to show that such a submanifold Mn must be either totally umbilical or isometric to a Clifford torus S1(1−r2)×Sn−1(r), when c=1, a circular cylinder R×Sn−1(r), when c=0, or a hyperbolic cylinder H1(−1+r2)×Sn−1(r), when c=−1. We also study the parabolicity of these submanifolds with respect to a Cheng–Yau modified operator.

Linear Weingarten submanifolds immersed in a space form

In this paper, we deal with complete linear Weingarten submanifolds $M^n$ immersed with parallel normalized mean curvature vector field in a Riemannian space form $\mathbf{Q}_c^{n+p}$ of constant sectional curvature $c$. Under an appropriated restriction on the norm of the traceless part of the second fundamental form, we show that such a submanifold $M^n$ must be either totally umbilical or isometric to a Clifford torus, if $c = 1$, a circular cylinder, if $c = 0$, or a hyperbolic cylinder, if $c = −1$. We point out that our results are natural generalizations of those ones obtained in [2] and [6].

Submanifolds with constant scalar curvature in a space form

We deal with complete submanifolds Mn having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form Qcn+p of constant sectional curvature c∈{1,0,−1}. In this setting, we show that such a submanifold Mn must be either totally umbilical or isometric to a Clifford torus S1(1−r2)×Sn−1(r), when c=1, a circular cylinder R×Sn−1(r), when c=0, or a hyperbolic cylinder H1(−1+r2)×Sn−1(r), when c=−1. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli [2], Cheng [4] and Guo and Li [6].

General Rotational Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classification of minimal general rotational surfaces of elliptic and hyperbolic type, general rotational surfaces with parallel normalized mean curvature vector field, flat general rotational surfaces, and general rotational surfaces with flat normal connection.

ON THE THEORY OF LORENTZ SURFACES WITH PARALLEL NORMALIZED MEAN CURVATURE VECTOR FIELD IN PSEUDO-EUCLIDEAN 4-SPACE

We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.