semi-Riemannian Space Research Articles

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Let M be a hypersurface isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n>3, such that its shape operator A satisfies A3=ϕA2+ψA+ρId, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square S2 of the Ricci operator S of M is not a linear combination of S and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R⋅S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n>3, with three distinct principal curvatures and the Ricci operator S with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

The geometry at infinity of LW-spacelike submanifolds in semi-Riemannian space forms

The geometry at infinity of LW-spacelike submanifolds in semi-Riemannian space forms

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

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

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

On Isometric Immersions of Null Manifolds into Semi-Riemannian Space Forms of Arbitrary Index

A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q

On the rigidity of mean curvature flow solitons in certain semi-Riemannian warped products

We obtain rigidity results concerning complete noncompact solitons of the mean curvature flow related to a nonsingular Killing vector field $K$ globally defined in a semi-Riemannian space, which can be modeled as a warped product whose base corresponds to a fixed integral leaf of the distribution orthogonal to $K$ and the warping function is equal to $|K|$. Our approach is based on a suitable maximum principle dealing with a notion of convergence to zero at infinity. As application, we study the uniqueness of solutions for the mean curvature flow soliton equation in these ambient spaces.

Rotation hypersurfaces in semi-Riemannian space forms of arbitrary index

Let n, ν be integers such that n≥2 and 0≤ν≤n+1. Let Mc,νn+1 be a (n+1)-dimensional semi-Riemannian space form of constant sectional curvature c and index ν. We give the parametrizations of rotation hypersurfaces in Mc,νn+1. Adding the condition for the r-th mean curvature to be constant, we show the study of these hypersurfaces is reduced to the study of a function on two variables. As an illustrative example, we study the particular case of rotation hypersurfaces with constant scalar curvature in Mc,νn+1, determining the values of n,c,ν for which these hypersurfaces are isoparametric. In the final part we study the variation of the second fundamental form of rotation hypersurfaces in order to obtain some rigidity results.

Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.

Normalized null hypersurfaces in nonflat Lorentzian space forms satisfying Lrx = Ux + b

In the present work, we classify normalized null hypersurfaces $x:(M,g,N)\to Q^{n+2}_1(c)$ immersed into one of the two real standard nonflat Lorentzian space-forms and satisfying the equation $L_r x=\mathcal U x+b$ for some field of screen constant matrices $\mathcal U$ and some field of screen constant vectors $b\in\mathbb{R}^{n+2}$, where $L_r$ is the linearized operator of the $(r+1)-$mean curvature of the normalized null hypersurface for $r=0,...,n$. We show that if the immersion $x$ is a solution of the equation $L_r x=\mathcal U x+b$ for $1\leq r\leq n$ and the normalization $N$ is quasi-conformal, then $M$ is either an $(r+1)-$maximal null hypersurface, or a totally umbilical (or geodesic) null hypersurface or an almost isoparametric normalized null hypersurface with at most two non-zero principal curvatures. We also show that a null hypersurface $M$, of a real standard semi-Riemannian nonflat space form $Q_t^{n+2}(c)$, admits a totally umbilical screen distribution (and then $M$ is totally umbilical or totally geodesic) if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$. In particular a null hypersurface $M\to Q_t^{n+2}(c)$ is totally geodesic if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$ passing through the origin.

On the Biconservative Quasi-Minimal Immersions into Semi-Euclidean Spaces

In this paper, we study biconservative immersions into the semi-Riemannian space form $$R^4_2(c)$$ of dimension 4, index 2 and constant curvature $$c\in \{0,-1,1\}$$ . First, we obtain a characterization of quasi-minimal proper biconservative immersions into $$R^4_2(c)$$ . Then we obtain the complete classification of quasi-minimal biconservative surfaces in $$R^4_2(0)={\mathbb {E}}^4_2$$ . We also obtain a new class of biharmonic quasi-minimal surfaces in $${\mathbb {E}}^4_2$$ .

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

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.

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.

Pseudo null curve variations for Bishop frame in 3D semi-Riemannian manifold

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

On Quasi-Minimal Isometric Immersions into Non-Flat Semi Riemannian Space Forms of Index Two

In this paper, first we gave a summary of recent results on biconservative immersions. Then, we obtained a necessary and sufficient condition for the existence of biconservative quasi-minimal immersions into a four dimensional semi-Riemannian space form of index two with non-zero sectional curvatures. We also constructed an explicit example of biconservative quasi-minimal surface.

On the definition of spacetimes in noncommutative geometry: Part I

Abstract In this paper we consider semi-Riemannian time and space oriented manifolds of even dimension, and characterize the Lorentzian and antilorentzian signatures in terms of a time-orientation 1-form and a natural Krein product on spinor fields. It turns out that all the data available in Noncommutative Geometry (the algebra of functions, the Krein space of spinor fields, the representation of the algebra on it, the Dirac operator, charge conjugation and chirality), but nothing more, play a role in this characterization. It thus yields a possible definition extending Connes’ notion of even spectral triple to the Lorentzian setting.

On the Gauss map of minimal Lorentzian surfaces in 4-dimensional semi-Riemannian space forms with index 2

On the Gauss map of minimal Lorentzian surfaces in 4-dimensional semi-Riemannian space forms with index 2

Spacelike isoparametric hypersurfaces

We generalise Ferus' work to study isoparametric hypersurfaces in semi-Riemannian space forms focusing, in this particular case, on anti-De Sitter spaces. We will show that two is an upper bound for the number of principal curvatures in a spacelike isoparametric hypersurface in the anti-De Sitter space. This fact will lead us to deduce a partial classification of isoparametric hypersurfaces in anti-De Sitter spaces.

Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

We study constant angle semi-Riemannian hypersurfaces M immersed in semi-Riemannian space forms, where the constant angle is defined in terms of a closed and conformal vector field Z in the ambient space form. We show that such hypersurfaces belong to the class of hypersurfaces with a canonical principal direction. This property is a type of rigidity. We further specialize to the case of constant mean curvature (CMC) hypersurfaces and characterize them in two relevant cases: when the hypersurface is orthogonal to Z then it is totally umbilical, whereas if Z is tangent to the hypersurface then it has zero Gauss–Kronecker curvature and either its mean curvature vanishes or the ambient is a semi-Euclidean space. We also treat in detail the surface case, giving a full characterization of the constant angle CMC surfaces immersed in all three dimensional space forms. They are isoparametric surfaces with constant principal curvatures when the ambient is flat. If the mean curvature of the surface is not ±2/3 they are either totally umbilic or totally geodesic. In particular when the surface has zero mean curvature it is totally geodesic.