Parallel Mean Curvature Vector Research Articles

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.

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

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.

On the rigidity of spacelike submanifolds with parallel Gaussian mean curvature vector

On the rigidity of spacelike submanifolds with parallel Gaussian mean curvature vector

Spacelike submanifolds with parallel Gaussian mean curvature vector: rigidity and nonexistence

Spacelike submanifolds with parallel Gaussian mean curvature vector: rigidity and nonexistence

Gap type results for spacelike submanifolds with parallel mean curvature vector

We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).

Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

Under the assumption that the second fundamental form is locally timelike, we establish new nonexistence and umbilicity results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the (n+p)-dimensional de Sitter space mathbb {S}^{n+p}_q of index q, such that 1le qle p. Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21].

Stochastically Complete, Parabolic and L1-Liouville Spacelike Submanifolds with Parallel Mean Curvature Vector

Stochastically Complete, Parabolic and L1-Liouville Spacelike Submanifolds with Parallel Mean Curvature Vector

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

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.

Bonnet and Isotropically Isothermic Surfaces in 4-Dimensional Space Forms

We study the Bonnet problem for surfaces in 4-dimensional space forms, namely, to what extent a surface is determined by the metric and the mean curvature. Two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. We deal with the structure of the moduli space of congruence classes of isometric surfaces with the same mean curvature and with properties inherited on a surface by this structure. The study of this problem led us to a new conformally invariant property, called isotropic isothermicity, that coincides with the usual concept of isothermicity for surfaces lying in totally umbilical hypersurfaces, and is related to lines of curvature and infinitesimal isometric deformations that preserve the mean curvature vector field. The class of isotropically isothermic surfaces includes the one of surfaces with a vertically harmonic Gauss lift and particularly the minimal surfaces, and overlaps with that of isothermic surfaces without containing the entire class. We show that if a simply connected surface is not proper Bonnet, which means that the moduli space is a finite set, then it admits either at most one, or exactly three Bonnet mates. For simply connected proper Bonnet surfaces, the moduli space is either 1-dimensional with at most two connected components diffeomorphic to the circle, or the 2-dimensional torus. We prove that simply connected Bonnet surfaces lying in totally geodesic hypersurfaces of the ambient space as surfaces of non-constant mean curvature always admit Bonnet mates that do not lie in any totally umbilical hypersurface. Such surfaces either admit exactly three Bonnet mates, or they are proper Bonnet with moduli space the torus. We show that isotropic isothermicity characterizes the proper Bonnet surfaces, and we provide relevant conditions for non-existence of Bonnet mates for compact surfaces. Moreover, we study compact surfaces that are locally proper Bonnet, and we prove that the existence of a uniform substructure on the local moduli spaces characterizes surfaces with a vertically harmonic Gauss lift that are neither minimal, nor superconformal. In particular, we show that the only compact, locally proper Bonnet surfaces with moduli space the torus, are those with nonvanishing parallel mean curvature vector field and positive genus.

Special Liouville metric with the Ricci condition

Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dimensional Riemannian manifold with the special type of the Liouville metric satisfying the Ricci condition to the complex hyperbolic plane with parallel mean curvature vector.

On Slant Curves in Sasakian Lorentzian 3-Manifolds

In this paper, we study $C$-parallel mean curvature vector field and $C$-proper mean curvature vector field along a slant Frenet curve in a Sasakian Lorentzian 3-manifold. In particular, we prove that a slant Frenet curve $\gamma$ in a Sasakian Lorentzian $3$-manifold $M$ satisfying $\Delta_{\dot{\gamma}} H =0$ is a geodesic or pseudo-helix with $\kappa^2=\tau^2$. For example, we find slant pseudo-helix in Lorentzian Heisenberg 3-space.

Spacelike submanifolds with parallel mean curvature vector in a de Sitter space S^{n+p}_{q}(c)

Spacelike submanifolds usually appear in the study of questions related to causality in general relativity. In this paper, we study an n-dimensional spacelike submanifold in (n + p)-dimensional connected de Sitter space S^{n+p}_{q}(c) of index q (1leq q leq p) and of constant curvature c, and we obtain some integral inequalities of Simons type and rigidity theorems.

Pointwise Slant Curves in Quasi-paraSasakian 3-Manifolds

This paper is devoted to the study of pointwise slant Frenet curves in a quasi-paraSasakian metric 3-manifold. We give a characterization result for the existence of such curves and determine their curvature and torsion in this class of manifold. Further, the characterizations for these curves having harmonic and $$\mathbf {C}$$ -parallel mean curvature vector field are derived.

Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes—An Overview

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.

Self-shrinker Type Submanifolds in the Euclidean Space

In this paper, we devote to investigating a new class of submanifolds, named the self-shrinker type submanifolds. We define a functional and prove its critical point is a self-shrinker type submanifold. We also show that a compact self-shrinker type submanifold with the parallel mean curvature vector in the Euclidean Space $$\mathbb {R}^{n+p}$$ is a minimal submanifold in a hypersphere $$\mathbb {S}^{n+p-1}$$ of $$\mathbb {R}^{n+p}$$ .

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.

ON WEAK BIHARMONIC GENERALIZED ROTATIONAL SURFACE IN E⁴

In the present paper we consider weak biharmonic rotational surfaces in Euclidean 4-space E⁴. We have proved that the general rotational surface of parallel mean curvature vector field is weak biharmonic then either it is minimal or a constant mean curvature. Further, we show that if Vranceanu surface of constant mean curvature is weak-biharmonic then it is a Clifford torus in E⁴.

Submanifolds with parallel mean curvature vector in η-Einstein Sasaki manifold

In this note, we prove that for a 3-dimensional submanifold in an η-Einstein 5-Sasaki manifold with Einstein constant a≥4, if the Reeb vector field is tangent to the submanifold and it has parallel mean curvature form, then it is a slant submanifold. Furthermore, if a>4, then it is either an invariant submanifold or an anti-invariant submanifold.