semi-Riemannian Space Research Articles

On complete spacelike submanifolds with second fundamental form locally timelike in a semi-Riemannian space form

In this work we obtain a Simon’s type inequality for a suitable tensor and apply it in order to obtain some results on a complete n -dimensional spacelike submanifold M n with parallel mean curvature vector and second fundamental form locally timelike in an ( n + p ) -dimensional semi-Riemannian space form, N q n + p ( c ) of index q ( 1 ≤ q ≤ p ).

New characterizations of complete spacelike submanifolds in semi-Riemannian space forms

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau's technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

A CHARACTERIZATION OF SCREEN CONFORMAL HALF LIGHTLIKE SUBMANIFOLDS

We study the geometry of screen conformal half lightlike submanifolds of a semi-Riemannian manifod. The main result is a classification theorem for screen conformal half lightlike submanifolds of a semi-Riemannian space form with a Killing co-screen distribution.

Partially Null Curves of Constant Breadth in Semi-Riemannian Space

In this work, some characterizations of partially null curves of constant breadth in Semi-Riemannian space are presented.

Isometric Immersions with Geodesic Normal Sections in Semi-Riemannian Geometry

We study an isometric immersion $f:M \to \bar{M}$ with geodesic normal sections, where $\bar{M}$ is a semi-Riemannian space form. In Riemannian geometry, it is known that $f$ is helical, in particular, all geodesics of $M$ have the same proper order in $\bar{M}$. However this does not hold in general, when $\bar{M}$ is indefinite semi-Riemannian. We give sufficient conditions for an isometric immersion with geodesic normal sections to be helical.

On quasi-Einstein Cartan type hypersurfaces

We investigate curvature properties of quasi-Einstein Cartan type hypersurfaces in semi-Riemannian space forms.

Some applications of a Simons' type formula for complete spacelike submanifolds in a semi-Riemannian space form

In this work we obtain a Simons' type inequality for a suitable tensor and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in a semi-Riemannian space form.

A note on the geometry of Lie groups

The degenerate semi-Riemannian geometry of a Lie group is studied. Then a naturally reductive homogeneous semi-Riemannian space is obtained from the Lie group in a natural way.

Branes, moduli spaces and smooth transition from big crunch to big bang

We consider branes N in a Schwarzschild-AdS(n+2) bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map φ : N → S with potential V , where S is a semiRiemannian moduli space. By transforming the field equation appropriately, we get an equivalent field equation that is smooth across the singularity r = 0, and which has smooth and uniquely determined solutions which exist across the singularity in an interval (− , ). Restricting a solution to (− , 0) resp. (0, ), and assuming n odd, we obtain branes N resp. N which together form a smooth hypersurface. Thus a smooth transition from big crunch to big bang is possible both geometrically as well as physically.

On the mean curvature of spacelike submanifolds in semi-Riemannian manifolds

In this paper we establish some estimates for the higher-order mean curvature of a complete spacelike hypersurface in spacetimes with sectional curvature satisfying certain condition. We also obtain the estimate for the mean curvature of a complete spacelike submanifold in semi-Riemannian space forms.

ON A CURVATURE CHARACTERIZATION OF RICCI-PSEUDOSYMMETRIC HYPERSURFACES

The author presents a curvature characterization of Ricci-pseudosymmetric hypersurfaces in semi-Riemannian space forms Nsn+1(c),n≥4.

Harmonic maps and isometric embeddings of the spacetime

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell–Magaard theorem we show that any n-dimensional ( n⩾3) Lorentzian manifold can be isometrically and harmonically embedded in a ( n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann–Robertson–Walker spacetime in a five-dimensional Ricci-flat space and proceed to obtain a general scheme to minimally embed any vacuum solution of general relativity in Ricci-flat spaces with codimension one.

Osculating spheres and osculating circles of a curve in semi-riemannian space

Ilı the Euclidean 3-space, t,here is a uııigue sphere for a curve o : / —» R^ such that the sphere touchc.s a at. the third order at ct(0). The iııtersectioıı of the splıere with oseulatiııg plaııe is a eircle which touches a at the secoud order at a(0) [5j. In this paper, the osculating sphere and the osculating circle of the curve are studied for each of timelike, spacelike and null (Ughtlike) curves in Semi-Riemannian Spaces; Rj, R| and R^.

On hypersurfaces in a locally affine Riemannian Banach manifold II

In our previous work (2002), we proved that an essential second‐order hypersurface in an infinite‐dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse, in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi‐Riemannian Banach space is an essential hypersurface of second order.

On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms

Solutions of the P. J. Ryan problem as well as investigations of curva- ture properties of Cartan hypersurfaces and Ricci-pseudosymmetric hypersurfaces lead to curvature identities holding on every hypersurface M isometrically immersed in a semi- Riemannian space form. These identities, under some assumptions, give rises to new gen- eralized Einstein metric conditions on M. We investigate hypersurfaces satisfying such curvature conditions.

CONFORMAL VECTOR FIELDS AND TOTALLY UMBILIC HYPERSURFACES

In this article, we show that if a semi-Riemannian space form carries a conformal vector field V of which the tangential part <TEX>$V^T$</TEX> on a connected hypersurface <TEX>$M^N$</TEX> ecomes a conformal vector field and the normal part <TEX>$V^N</TEX> on <TEX>$M^N$</TEX> does not vanish identically, then <TEX>$M^N$</TEX> is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

Symmetric Soliton Configurations of Bosonic String Theories

We obtain a reduction of the symmetry holographic principle for symmetric configurations of Nambu–Goto–Polyakov string theories in a semi-Riemannian space. The argument reduces the search of string configurations with a certain degree of symmetry to that for elastic curves in a corresponding orbit space. These solutions are solitons which are holographically related to particles that evolve along elastic worldlines in the orbit space. We also exhibit examples and applications to obtain soliton string shapes with cylindrical, rotational, toroidal etc. symmetry. In most of the cases we can determine the whole moduli space of symmetric solitons.

Quasi-Einstein hypersurfaces in semi-Riemannian space forms

We investigate curvature properties of hypersurfaces of a semi-Rieman- nian space form satisfyingR·C = LQ(S,C), which is a curvature condition of pseu- dosymmetry type. We prove that under some additional assumptions the ambient space of such hypersurfaces must be semi-Euclidean and that they a re quasi-Einstein Ricci- semisymmetricmanifolds.

P.J. Ryan's problem in semi-Riemannian space forms

We prove that the conditions R·R=0 and R·S=0 are equivalent for hypersurfaces of a 5-dimensional semi-Riemannian space form N5(c). This solves a problem by P.J. Ryan in the case of hypersurfaces of dimension 4 in semi-Riemannian space forms.