scholarly journals Osserman Lightlike Hypersurfaces on a Foliated Class of Lorentzian Manifolds

2016 ◽  
Vol 8 (2) ◽  
pp. 55
Author(s):  
C. Atindogbe ◽  
K. L. Duggal
Keyword(s):  
Curvature Tensor ◽  
Constant Curvature ◽  
Open Subset ◽  
Normal Vector ◽  
Lorentzian Manifolds ◽  
Schwarzschild Spacetime ◽  
Osserman Manifold ◽  
The Family ◽  
Characterization Theorems ◽  
Lightlike Hypersurfaces

This paper deals with a family of Osserman lightlike hypersurfaces $(M_u)$ of a  class of Lorentzian manifolds $\bar{M}$ such that its each null normal vector is defined on some open subset of $\bar{M}$ around $M_u$.  We prove that a totally umbilical family of lightlike hypersurfaces of a connected Lorentzian pointwise Osserman manifold of constant curvature is locally Einstein and pointwise ${\cal F}-$Osserman, where our foliation approach provides the required algebraic symmetries of the induced curvature tensor. Also we prove two new characterization theorems for the family  of Osserman lightlike hypersurfaces, supported by a physical example of Osserman lightlike hypersurfaces of the Schwarzschild spacetime.

scholarly journals Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Dae Ho Jin
Keyword(s):  
Vector Field ◽  
Space Form ◽  
Sasakian Manifold ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Characterization Theorems ◽  
Lightlike Hypersurfaces

We study lightlike hypersurfacesMof an indefinite generalized Sasakian space formM-(f1,f2,f3), with indefinite trans-Sasakian structure of type(α,β), subject to the condition that the structure vector field ofM-is tangent toM. First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type(α,β). Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

2001 ◽  
Vol 440 ◽  
pp. 269-291 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Curvature Tensor ◽  
Deformation Gradient ◽  
Bending Stiffness ◽  
Boundary Element Methods ◽  
Simple Shear Flow ◽  
Normal Vector ◽  
Membrane Thickness ◽  
Bending Moments ◽  
Balance Equations ◽  
Capsule Deformation

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

On convergence, admissibility and attractors for damped wave equations on squeezed domains

2002 ◽  
Vol 132 (4) ◽  
pp. 765-791
Author(s):  
M. C. Carbinatto ◽  
K. P. Rybakowski
Keyword(s):  
Wave Equations ◽  
Upper Semicontinuity ◽  
Conley Index ◽  
Normal Vector ◽  
Thin Domains ◽  
Local Flow ◽  
Normal Vector Field ◽  
Continuation Principle ◽  
Bounded Lipschitz Domain ◽  
The Family

Let be an arbitrary non-empty bounded Lipschitz domain in RM RN. Given > 0, squeeze by the factor in the y-direction to obtain the squeezed domain := {(x, y) | (x, y) }. Let and be positive constants. Consider the following semilinear damped wave equation on , where is the exterior normal vector field on and G is an appropriate nonlinearity, which ensures that (W) generates a (local) flow ̃ on X := H1() L2(). We show that there is a closed subspace X0 of X and a flow ̃0 on X0 that is the limit flow of the family ̃, > 0. We show that, as 0, the family ̃ converges in some singular sense to ̃ and establish a technical singular asymptotic compactness property. As a corollary, we obtain an upper-semicontinuity result for global attractors of the family ̃, 0, generalizing results obtained previously by Hale and Raugel for domains that are ordinate sets of a positive function.The results obtained here are also applied in our paper On a general Conley index continuation principle for singular perturbation problems to establish a singular Conley index continuation principle for damped wave equations on thin domains.

scholarly journals Lightlike Einstein hypersurfaces in Lorentzian manifolds with constant curvature

2006 ◽  
Vol 29 (1) ◽  
pp. 58-71 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
J. Tossa
Keyword(s):  
Constant Curvature ◽  
Lorentzian Manifolds

scholarly journals Hyperbolicity for Log Smooth Families with Maximal Variation

2021 ◽  
Author(s):  
Chuanhao Wei ◽  
Lei Wu
Keyword(s):  
Open Subset ◽  
Minimal Model ◽  
General Type ◽  
Canonical Model ◽  
Base Space ◽  
Zariski Open Subset ◽  
Smooth Family ◽  
Log Canonical ◽  
The Family

Abstract We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

scholarly journals Curvature Estimates in Asymptotically Flat Lorentzian Manifolds

2005 ◽  
Vol 57 (4) ◽  
pp. 708-723 ◽  
Author(s):  
Felix Finster ◽  
Margarita Kraus
Keyword(s):  
Fundamental Form ◽  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Second Fundamental Form ◽  
Riemannian Curvature ◽  
Lorentzian Manifolds ◽  
Asymptotically Flat ◽  
Riemannian Curvature Tensor ◽  
Curvature Estimates

AbstractWe consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzianmanifold becomes flat in the limit when the ADM energy tends to zero.

Some characterizations of Lorentzian manifolds

2019 ◽  
Vol 16 (01) ◽  
pp. 1950016
Author(s):  
Uday Chand De ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

On weakly conformally symmetric pseudo-Riemannian manifolds

2017 ◽  
Vol 29 (03) ◽  
pp. 1750007
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Integral Curves ◽  
Pontryagin Form ◽  
Lorentzian Case

In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the [Formula: see text]-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in [Formula: see text]. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric [Formula: see text]-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

Compact null hypersurfaces in Lorentzian manifolds

2021 ◽  
Vol 21 (2) ◽  
pp. 251-263
Author(s):  
C. Atindogbé ◽  
M. Gutiérrez ◽  
R. Hounnonkpe
Keyword(s):  
Global Symmetries ◽  
Lorentzian Manifold ◽  
Lorentzian Manifolds ◽  
Geometric Properties ◽  
Totally Geodesic ◽  
Null Hypersurfaces ◽  
Ambient Manifold ◽  
The Family

Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

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