Lorentzian Space Forms Research Articles

On the behavior at infinity of complete spacelike hypersurfaces in a Lorentzian space form

On the behavior at infinity of complete spacelike hypersurfaces in a Lorentzian space form

Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

Biconservative surfaces with constant mean curvature in lorentzian space forms

Biconservative surfaces with constant mean curvature in lorentzian space forms

On generalized biconservative spacelike surfaces in Lorentz space forms

On generalized biconservative spacelike surfaces in Lorentz space forms

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N^{m+1}_t(c) of dimension m+1 and index t (mge 2 and 1 le tle m). We shall study hypersurfaces {M}^{m}_{t'} which are polyharmonic of order r (briefly, r-harmonic), where rge 3 and either t'=t or t'=t-1. Let A denote the shape operator of {M}^{m}_{t'}. Under the assumptions that {M}^{m}_{t'} is CMC and {{rm{Tr}}},{A}^{2} is a constant, we shall obtain the general condition which determines that {M}^{m}_{t'} is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r ge 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

Main Curvatures Identities on Lightlike Hypersurfaces of Statistical Manifolds and Their Characterizations

In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms.

Conformal geometry of isotropic curves in the complex quadric

Let [Formula: see text] be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in [Formula: see text]. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into [Formula: see text], null with respect to the conformal structure of [Formula: see text]. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

A note on the nonexistence of spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form

We obtain nonexistence results concerning complete noncompact spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form, under the assumption that the support functions with respect to a fixed nonzero vector are linearly related. Our approach is based on a suitable maximum principle recently established by Alías, Caminha and do Nascimento [3].

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.

Complete spacelike hypersurfaces with constant scalar curvature: Descriptions and gaps

We provide sharp lower and upper bounds for the supremum of the norm of the total umbilicity tensor of complete spacelike hypersurfaces with constant scalar curvature immersed in a Lorentzian space form and satisfying a suitable Okumura-type inequality, which corresponds to a weaker hypothesis when compared with the geometric condition of the hypersurface having two distinct principal curvatures. Furthermore, we give a complete description and the gaps of the spacelike hypersurfaces which realize our estimates, obtaining as a consequence new characterizations of totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms. Our approach is based on a version of Omori–Yau’s maximum principle for trace-type differential operators defined on a complete Riemannian manifold.

Comparison Geometry for an Extension of Ricci Tensor

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.

Chen-like inequalities on null hypersurfaces with closed rigging of a Lorentzian manifold

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms

Biharmonic curves in $3$-dimensional Lorentzian Sasakian space forms

Gauss-Bonnet theorem in Lorentzian Sasakian space forms

<abstract> In this paper, we use a Lorentzian approximation scheme to compute the sub-Lorentzian limit of curvatures for curves and Lorentzian surfaces in the Lorentzian Bianci-Cartan-Vranceanu model of $ 3 $-dimensional Lorentzian Sasakian space forms. Based on these results, we get a Gauss-Bonnet theorem in the Lorentzian Sasakian space forms. </abstract>

Conformal homogeneous spacelike surfaces in 3-dimensional Lorentz space forms

Let M13(c) be an 3-dimensional Lorentz space form and C(M13(c)) denote the conformal transformation group of M13(c). A spacelike surface x:M2→M13(c) is called a conformal homogeneous spacelike surface. If there exists a subgroup G⊂C(M13(c)) such that the orbit G(p)=x(M2),p∈x(M2). In this paper, we classify completely conformal homogeneous spacelike surfaces up to a conformal transformation of M13(c).

Some characterizations of timelike rectifying curves in de sitter 3 space

De Sitter space is a non-flat Lorentzian space form with positive constant curvature which plays an important role in the theory of relativity. In this paper, we define the notions of timelike rectifying curve and timelike conical surface in De Sitter 3-space as Lorentzian viewpoint. Moreover, we give some nice characterizations and results of a timelike rectifying curves with respect to curve-hypersurface frame in De Sitter 3-space which is a three dimensional pseudo-sphere in Minkowski 4-space.

Null screen quasi‐conformal hypersurfaces in semi‐Riemannian manifolds and applications

AbstractWe introduce a class of null hypersurfaces of a semi‐Riemannian manifold, namely, screen quasi‐conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.

Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with different previously known definitions of such singularities.

On a Lorentzian Complex Space Form

The aim of the present paper is to study the theory of general relativity in a Lorentzian Kahler space. In this paper, we have considered the Lorentzian complex space form with constant sectional curvature and proved that a Lorentzian complex space form satisfying Einstein’s field equation is a Ricci semi-symmetric space and the energy–momentum tensor is covariant constant. Such space reduces to a flat space for a purely electromagnetic distribution. The existence of the Killing vector field and conformal Killing vector field has been discussed and proved that the cosmological term is proportional to sectional curvature. Further, it is shown that the perfect fluid in a Lorentzian complex space form satisfying Einstein’s field equation does not admit heat flux.