Parallel Normalized Mean Curvature Vector Research Articles

On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector

In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2.

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

Planar Pseudo-geodesics and Totally Umbilic Submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion ι:M↪Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota :M \\hookrightarrow Q$$\\end{document} is totally umbilic if and only if the extrinsic shape of every geodesic of M is planar. This extends a well-known result about surfaces in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{3}$$\\end{document}.

A Classification of Parallel Normalized Biconservative Submanifold in the Minkowski Space in Arbitrary Dimension

IIn this paper, we examine PNMCV-MCGL biconservative submanifold in a Minkowski space $\mathbb{E}_1^{n+2}$ with nondiagonalizable shape operator, where PNMCV-MCGL submanifold denotes a submanifold with parallel normalized mean curvature vector and the mean curvature whose gradient is lightlike ($\langle\nabla H,\nabla H\rangle=0$). We obtain some conditions about connection forms, principal curvatures and some results about them. Then we use them to obtain a classification of such submanifolds. Finally, we showed that there is no biconservative such submanifold in Minkowski space of arbitrary dimension.

Biharmonic PNMCV submanifolds in Euclidean 5-space

In this article, we study 3-dimensional biconservative and biharmonic submanifolds of $\mathbb{E}^5$ with parallel normalized mean curvature vector (PNMCV). First, we prove that the principal curvartures and principal directions of biconservative PNMCV isometric immersions into $\mathbb{E}^5$ can be determined intrinsically. Then, we complete the proof of Chen's biharmonic conjecture for PNMCV submanifolds of $\mathbb{E}^5$.

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.

$L$-parabolic linear Weingarten spacelike submanifolds immersed in an Einstein manifold

In this paper, we deal with complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in an Einstein manifold $\smash{\mathcal E^{n+p}\_p}$ of index $p$. Under some curvature constraints on the ambient space, we establish an $L$-parabolicity criterion related to a suitable modified Cheng–Yau's operator $L$ and we apply it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be an isoparametric hypersurface of $\smash{\mathcal E^{n+1}\_1}$ with two distinct principal curvatures one of which is simple.

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.

On the parabolicity of linear Weingarten spacelike submanifolds in the de Sitter space

In this paper, we deal with complete linear Weingarten spacelike sub-manifold immersed with parallel normalized mean curvature vector in de Sitter space of index p. Imposing appropriate restrictions on the values of the mean curvature function H, we establish a parabolicity criterion related to a suitable Cheng-Yau’s modified operator and we use it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be either totally umbilical or isometric to certain hyperbolic cylinders of .

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

A Characterization of Linear Weingarten Submanifolds in a Semi-Riemannian Space Form with Arbitrary Index

In this paper, we deal with the spacelike linear Weingarten submanifolds with parallel normalized mean curvature vector in an $$(n+p)$$ -dimensional semi-Riemannian space form $$N^{n+p}_{q}(c)$$ of constant sectional curvature c with index q, where $$1\le q\le p$$ . In this setting, we obtain an important inequality and apply some appropriated generalized maximum principles to a suitable Cheng–Yau-modified operator to obtain some characterizations of the linear Weingarten submanifolds.

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.

Complete spacelike submanifolds in a locally symmetric semi-Riemannian space

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold [Formula: see text] of [Formula: see text].

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.

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.