Mean Curvature Vector Research Articles

Ricci flow of Kaehlerian slant submanifolds in complex space forms and its applications

AbstractThe normalized Ricci flow converges to a constant curvature metric for a connected Kaehlerian slant submanifold in a complex space form if the squared norm of the second fundamental form satisfies certain upper bounds. These bounds include the constant sectional curvature, the slant angle, and the squared norm of the mean curvature vector. Additionally, we demonstrate that the submanifold is diffeomorphic to the sphere $$\mathbb {S}^{n_1}$$ S n 1 under some restriction on the mean curvature. We claim that some of our previous results are rare cases.

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Our investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, we delve into the condition that determines when equality is achieved within the inequality. Furthermore, we explore a number of implications of our findings.

Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with n=2, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely.

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 $\mathbb{E}^5_1$.

A New Representation for Slant Curves in Sasakian 3-Manifolds

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

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

Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms

Let M be a Lorentz surface and F:M→N a time-like and conformal immersion of M into a 4-dimensional neutral space form N with zero mean curvature vector. We show that the curvature K of the induced metric on M by F is identically equal to the constant sectional curvature L0 of N if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If K≡L0, then the normal connection ∇⟂ of F is flat, while the converse is not necessarily true. We also prove that a holomorphic paracomplex quartic differential Q on M defined by F is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we get that K is identically equal to L0 if and only if not only ∇⟂ is flat but also Q is zero or null

Ricci Curvature Inequalities for Contact CR-Warped Product Submanifolds of an Odd Dimensional Sphere Admitting a Semi-Symmetric Metric Connection

The primary objective of this paper is to explore contact CR-warped product submanifolds of Sasakian space forms equipped with a semi-symmetric metric connection. We thoroughly examine these submanifolds and establish various key findings. Furthermore, we derive an inequality relating the Ricci curvature to the mean curvature vector and warping function.

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}.

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

The lifts of surfaces in neutral 4-manifolds into the 2-Grassmann bundles

A twistor lift of a space-like or time-like surface in a neutral hyperKähler 4-manifold with zero mean curvature vector is given by a (para)holomorphic function, which yields (para)holomorphicity of the Gauss maps of space-like or time-like surfaces in E24 with zero mean curvature vector. For a space-like or time-like surface in an oriented neutral 4-manifold with zero mean curvature vector such that both twistor lifts belong to the kernel of the curvature tensor, its (para)complex quartic differential is holomorphic. If both twistor lifts of a time-like surface with zero mean curvature vector have light-like or zero covariant derivatives, then either the shape operator with respect to a light-like normal vector field vanishes or all the shape operators of the surface are light-like or zero. Examples with the former (resp. latter) property are given by the conformal Gauss maps of time-like surfaces of Willmore type with zero paraholomorphic quartic differential (resp. time-like surfaces in 4-dimensional neutral space forms based on the Gauss-Codazzi-Ricci equations).

Spacelike submanifolds with parallel mean curvature vector in the de Sitter space: characterizations and gaps

Spacelike submanifolds with parallel mean curvature vector in the de Sitter space: characterizations and gaps

Correction to: Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector

Correction to: Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector

Surfaces in Non-flat 3-Space Forms Satisfying square vec {textbf{H}}=lambda vec {textbf{H}}

In this paper, we locally classify the surfaces immersed into the non-flat (Riemannian or Lorentzian) 3-space forms satisfying the condition Box vec {textbf{H}}=lambda vec {textbf{H}} for a real number lambda , where vec {textbf{H}} is the mean curvature vector field and Box denotes the Cheng–Yau operator of the surface. We obtain the classification result by proving, at a first step, that the mean curvature function must be constant and, in a second step, we complete the classification.

A convergent finite element algorithm for mean curvature flow in arbitrary codimension

Optimal-order uniform-in-time H^1 -norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulas. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least 2 and backward difference formulas of orders 2 to 5. Numerical experiments in codimension 2 illustrate and complement our theoretical results.

On complete trapped submanifolds in globally hyperbolic spacetimes

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and generalized Robertson–Walker spacetimes.

Parallel Codazzi tensors with submanifold applications

AbstractA decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

Parallel mean curvature biconservative surfaces in complex space forms

AbstractIn this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non‐flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons‐type formula for a well‐chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two‐dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non‐flat complex space forms. We conclude by constructing examples of CMC non‐PMC biconservative submanifolds from the Segre embedding and discuss when they are proper‐biharmonic.

A new approach to the study of spacelike submanifolds in a spherical Friedmann–Lemaître–Robertson–Walker spacetime: characterization of the stationary spacelike submanifolds as an application

A natural codimension one isometric embedding of each -dimensional spherical Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime in the -dimensional Lorentz–Minkowski spacetime permits to contemplate as a rotation Lorentzian hypersurface in . After a detailed study of such Lorentzian hypersurfaces, any k-dimensional spacelike submanifold of such an FLRW spacetime can be contemplated as a spacelike submanifold of . Then, we use that situation to study k-dimensional stationary (i.e. of zero mean curvature vector field) spacelike submanifolds of the FLRW spacetime. In particular, we prove a wide extension of the Lorentzian version of the classical Takahashi theorem, giving a characterization of stationary spacelike submanifolds of when contemplating them as spacelike submanifolds of .