Warped Product Structure Research Articles

Some characterizations of Quasi-Einstein and doubly product manifold

This research paper explores the properties of quasi-Einstein manifolds with a unit concircular generator vector field within doubly warped product structures. The paper begins by investigating the characteristics of quasi-Einstein manifolds that possess a unit concircular generator vector field. Subsequently, it analyzes the behavior of the Hessian, Riemann, and Ricci vector fields in the context of a doubly warped product. Moreover, the impact of the quasi-Einstein condition on both the doubly warped product and its factor manifolds is studied. Finally, conditions are derived to determine whether a doubly warped product satisfying the quasi-Einstein condition can be classified as a warped or a direct product. By presenting these findings, this study contributes to the comprehension of quasi-Einstein manifolds and their interactions within doubly warped product structures.

Electrostatic system with divergence-free Bach tensor and non-null cosmological constant

Abstract We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provided that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.

On a class of Finsler metrics of quadratic Weyl curvature

The class of warped product manifolds plays an important role in differential geometry and physics. In this paper, we shall study product manifolds $\mathbb{R} \times M$ with Finsler metrics arising from warped product structure. We give an equivalent condition for those metrics to be of quadratic Weyl curvature.

Hyperbolic compactification of M-theory and de Sitter quantum gravity

We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.

On the cross curvature flow

In this paper, we study the cross curvature soliton. We study the cross curvature soliton with a warped product structure. On the other hand, we show that the volume entropy is decreasing along the cross curvature flow.

Conformal gradient fields on Finsler manifolds

In this note, some basic structural conditions that are imposed on a Finsler manifold as a consequence of existence of a conformal gradient field are presented. Then, after reiterating the definition of warped product Finsler manifolds and the constituents of its related variational problem, the correlation between geodesics of this warped product structure and geodesics of its constructing Finsler manifolds is studied.

Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold

AbstractIn this paper, we study Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. First, we prove that if a Kenmotsu metric is a Yamabe soliton, then it has constant scalar curvature. Examples has been provided on a larger class of almost Kenmotsu manifolds, known asβ-Kenmotsu manifold. Next, we study quasi Yamabe soliton on a complete Kenmotsu manifoldMand proved that it has warped product structure with constant scalar curvature in a region Σ where ∣Df∣ ≠ 0.

Remarks on the Warped Product Structure from the Hessian of a Function

The warped product structure of a gradient Yamabe soliton and a Ricci soliton with a concircular potential field is proved in another way.

Constructions of Einstein Finsler metrics by warped product

The warped product structures of Finsler metrics are studied in this paper. We give the formulae of the flag curvature and Ricci curvature of these metrics, and obtain the characterization of such metrics to be Einstein. Some Einstein Finsler metrics of this type are constructed.

High-frequency resolvent estimates on asymptotically Euclidean warped products

We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of only finitely many components. We prove a new dichotomy: the high-frequency resolvent is either bounded by for any , or blows up faster than any polynomial (at least along a subsequence). A stronger result holds if the manifold is analytic. The method of proof is to exploit the warped product structure to separate variables, obtaining a one-dimensional semiclassical Schrödinger operator. We then classify the microlocal resolvent behavior associated to every possible type of critical value of the potential, and translate this into the associated resolvent estimates. Weakly stable trapping admits highly concentrated quasimodes and fast growth of the resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely based on the classical positive commutator argument, we show that any weakly unstable trapping forces at least some spreading of quasimodes. Several applications are given for spreading of quasimodes in partially rectangular billiards, decay rates for damped wave equations.

Classification of Einstein equations with cosmological constant in warped product space-time

We classify all warped product space-times in three categories as (i) generalized twisted product structures, (ii) base conformal warped product structures and (iii) generalized static space-times and then we obtain the Einstein equations with the corresponding cosmological constant by which we can determine uniquely the warp functions in these warped product space-times.

Rigidity of (m,ρ)-quasi Einstein manifolds

This paper deals with the study on -quasi Einstein manifolds. First, we give some characterizations of an -quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an -quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an -dimensional Riemannian manifold and q is a smooth function on I. Finally, a non-trivial example of an -quasi Einstein manifold verifying our results in terms of the potential function is presented.

Warped product Finsler manifolds from Hamiltonian point of view

In this paper, the Finslerian warped product structures are introduced as Hamiltonian formalism without restricting Finsler functions to be absolutely homogeneous. Afterwards, the constituents of the related variational problem and Finslerian connections of this warped product are obtained according to those of its constructing Finsler manifolds.

Generalized quasi Yamabe gradient solitons

We prove that a nontrivial complete generalized quasi Yamabe gradient soliton (Mn,g) must be a quasi Yamabe gradient soliton on each connected component of M and that a nontrivial complete locally conformally flat generalized quasi Yamabe gradient soliton has a special warped product structure.

A Note on (Anti-)Self Dual Quasi Yamabe Gradient Solitons

In this note we prove that a (anti-)self dual quasi Yamabe gradient soliton with positive sectional curvature is rotationally symmetric. We also prove that half conformally flat gradient Yamabe soliton has a special warped product structure provided that the potential function has no critical point.

The ADM mass of asymptotically flat hypersurfaces

We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.

Euclidean hypersurfaces with a totally geodesic foliation of codimension one

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.

Warped product rigidity

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. This warped product structure will be used to study warped product Einstein structures in (HPW3).

On the structure of gradient Yamabe solitons

We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we prove a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion $$f : M^n \to {\mathbb{R}}^{n+1}$$ of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso 0(M n ) of the isometry group Iso(M n ) of M n admits an orthogonal representation $$\Phi\colon\,Iso^0(M^n) \to SO(n + 1)$$ such that $$f \circ g = \Phi(g)\circ f$$ for every $$g \in Iso^0(M^n)$$ . If G is a closed connected subgroup of Iso(M n ) acting polarly on M n , we prove that Φ(G) acts polarly on $${\mathbb{R}}^{n+1}$$ , and we obtain that f(M n ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.