Search Results (38)

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

This paper investigates compact Riemannian hypersurfaces immersed in $(n+1)$-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, Euclidean sphere $S^{n+1}$, and de Sitter space $S_1^{n+1}$. Full article

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere. Full article

We reformulate holographic space–time models in terms of Hilbert bundles over the space of the time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite-dimensional Hilbert spaces. Following Jacobson, I view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein–Hawking relation between area and entropy, generalized to arbitrary causal diamonds. The time-like geodesics are equivalent to the nested sequences of causal diamonds, and the area of the holoscreen (The holoscreen is the maximal $d-2$ volume (“area”) leaf of a null foliation of the diamond boundary. I use the term area to refer to its volume.) encodes the entropy of a certain density matrix on a finite-dimensional Hilbert space. I review arguments that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator $L_0$ of a $1+1$-dimensional CFT of a large central charge, living on an interval in the longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is $\ln {\mathrm{D}}_{\diamond },$ up to subleading corrections, in the limit of a large-dimension diamond Hilbert space. I also connect those arguments to the derivation of the ’t Hooft commutation relations for horizon fluctuations. I present a tentative connection between the ’t Hooft relations and $U\left(1\right)$ currents in the CFTs on the past and future diamond boundaries. The ’t Hooft relations are related to the Schwinger term in the commutator of the vector and axial currents. The paper in can be read as evidence that the near-horizon dynamics for causal diamonds much larger than the Planck scale is equivalent to a topological field theory of the ’t Hooft CR plus small fluctuations in the transverse geometry. Connes’ demonstration that the Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of the sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy and makes the geometry both “fuzzy” and quantum. Following the analysis of Carlip and Solodukhin, I model the expansion coefficients as two-dimensional fermionic fields. I argue that the local excitations in the interior of a diamond are constrained states where the spinor variables vanish in the regions of small area on the holoscreen. This leads to an argument that the quantum gravity in asymptotically flat space must be exactly supersymmetric. Full article

In this paper, we investigate warped products on super quasi-Einstein manifolds under affine connections. We explore their fundamental properties, establish conditions for their existence, and prove that these manifolds can also be nearly quasi-Einstein and pseudo quasi-Einstein. To illustrate, we provide examples in both Riemannian and Lorentzian geometries, confirming their existence. Finally, we construct and analyze an explicit example of a warped product on a super quasi-Einstein manifold with respect to affine connections. Full article

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form. Full article

This paper focuses on some geometrical and physical properties of a conformal η-Ricci soliton (Cη-RS) on a four-dimension Lorentzian Para-Sasakian (LP-S) manifold. The first section presents an introduction to Cη-RS on LP-S manifolds, followed by a discussion of preliminary ideas about the LP-Sasakian manifold. In the subsequent sections, we establish several results pertaining to four-dimension LP-S manifolds that exhibit Cη-RS. Additionally, we consider certain conditions associated with Cη-RS on four-dimension LP-S manifolds. Besides these geometrical points of view, we consider this soliton in a perfect fluid spacetime and obtain some interesting physical properties. Finally, we present a case study of a Cη-RS on a four-dimension LP-S manifold. Full article

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric $\omega$-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric $\omega$-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric $\omega$-connection is Bach flat. Full article

In the current article, we examine Lorentzian para-Kenmotsu (shortly, LP-Kenmotsu) manifolds with regard to the generalized symmetric metric connection ${\nabla }^{\mathcal{G}}$ of type $(\alpha ,\beta )$. First, we obtain the expressions for curvature tensor, Ricci tensor and scalar curvature of an LP-Kenmotsu manifold with regard to the connection ${\nabla }^{\mathcal{G}}$. Next, we analyze LP-Kenmotsu manifolds equipped with the connection ${\nabla }^{\mathcal{G}}$ that are locally symmetric, Ricci semi-symmetric, and $\phi$-Ricci symmetric and also demonstrated that in all these situations the manifold is an Einstein one with regard to the connection ${\nabla }^{\mathcal{G}}$. Moreover, we obtain some conclusions about projectively flat, projectively semi-symmetric and $\phi$-projectively flat LP-Kenmotsu manifolds concerning the connection ${\nabla }^{\mathcal{G}}$ along with several consequences through corollaries. Ultimately, we provide a 5-dimensional LP-Kenmotsu manifold example to validate the derived expressions. Full article

In this paper, we investigate Ricci solitons on spacelike hypersurfaces in a special Lorentzian warped product manifold, the so-called generalized Robertson–Walker (GRW) spacetimes. Such spacetimes admit a natural form of symmetry which is represented by the conformal vector field $f\partial t$, where f is the warping function and $\partial t$ is the unit timelike vector field tangent to the base (which is here a one-dimensional manifold). We use this symmetry to introduce some fundamental formulas related to the Ricci soliton structures and the Ricci curvature of the fiber, the warping function, and the shape operator of the immersion. We investigate different rigidity results for Ricci solitons on the slices, in addition to the totally umbilical spacelike supersurfaces of GRW. Furthermore, our study is focused on significant GRW spacetimes such as Einstein GRW spacetimes and those which obey the well-known null convergence condition (NCC). Full article

In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold $({ }_{f}B{\times }_{b}F,\tilde{g},\mathrm{\Pi }={\mathrm{\Pi }}_B+{\mathrm{\Pi }}_F)$ equipped with a product Poisson structure $\mathrm{\Pi }$, using warping functions and sectional curvatures of its factor manifolds $(B,{\tilde{g}}_B,{\mathrm{\Pi }}_B)$ and $(F,{\tilde{g}}_F,{\mathrm{\Pi }}_F)$. Qualar and null sectional contravariant curvatures of $({ }_{f}B{\times }_{b}F,\tilde{g},\mathrm{\Pi })$ are also given. As an example, we construct a four-dimensional Lorentzian doubly warped product Poisson manifold where qualar and sectional curvatures are obtained. Full article

In this article, we investigate Ricci solitons occurring on spacelike hypersurfaces of Einstein Lorentzian manifolds. We give the necessary and sufficient conditions for a spacelike hypersurface of a Lorentzian manifold, equipped with a closed conformal timelike vector field $\overline{\xi }$, to be a gradient Ricci soliton having its potential function as the inner product of $\overline{\xi }$ and the timelike unit normal vector field to the hypersurface. Moreover, when the ambient manifold is Einstein and the hypersurface is compact, we establish that, under certain straightforward conditions, the hypersurface is an extrinsic sphere, that is, a totally umbilical hypersurface with a non-zero constant mean curvature. In particular, if the ambient Lorentzian manifold has a constant sectional curvature, we show that the compact spacelike hypersurface is essentially a round sphere. Full article

The de-Sitter spacetime is a maximally symmetric Lorentzian manifold with constant positive scalar curvature that plays a fundamental role in modern cosmology. Here, we investigate bulk-viscosity-assisted quasi de-Sitter inflation, that is the period of accelerated expansion in the early universe during which $-\dot{H}\ll H^2$, with $H\left(t\right)$ being the Hubble expansion rate. We do so in the framework of a causal theory of relativistic hydrodynamics, which takes into account non-equilibrium effects associated with bulk viscosity, which may have been present as the early universe underwent an accelerated expansion. In this framework, the existence of a quasi de-Sitter universe emerges as a natural consequence of the presence of bulk viscosity, without requiring introducing additional scalar fields. As a result, the equation of state, determined by numerically solving the generalized momentum-conservation equation involving bulk viscosity pressure turns out to be time dependent. The transition timescale characterising its departure from an exact de-Sitter phase is intricately related to the magnitude of the bulk viscosity. We examine the properties of the new equation of state, as well as the transition timescale in the presence of bulk viscosity pressure. In addition, we construct a fluid description of inflation and demonstrate that, in the context of the causal formalism, it is equivalent to the scalar field theory of inflation. Our analysis also shows that the slow-roll conditions are realised in the bulk-viscosity-supported model of inflation. Finally, we examine the viability of our model by computing the inflationary observables, namely the spectral index and the tensor-to-scalar ratio of the curvature perturbations, and compare them with a number of different observations, finding good agreement in most cases. Full article

We prove the upper and lower bounds of the diameter of a compact manifold $(M,g(t\left)\right)$ with ${dim}_{\mathbb{R}}M=n(n\ge 3)$ and a family of Riemannian metrics $g\left(t\right)$ satisfying some geometric flows. Except for Ricci flow, these flows include List–Ricci flow, harmonic-Ricci flow, and Lorentzian mean curvature flow on an ambient Lorentzian manifold with non-negative sectional curvature. Full article

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature. Full article