Warped Product Metric Research Articles

Gradient Ricci-Yamabe solitons on warped product manifolds

We give the necessary and sufficient conditions for a gradient Ricci-Yamabe soliton with warped product metric. As physical applications, we consider gradient Ricci-Yamabe solitons on generalized Robertson-Walker space-times and standard static space-times.

ON WARPED PRODUCT MANIFOLDS ADMITTING τ-QUASI RICCI-HARMONIC METRICS

In this paper, we study warped product manifolds admitting $\tau$-quasi Ricci-harmonic(RH) metrics. We prove that the metric of the fibre is harmonic Einstein when warped product metric is $\tau$-quasi RH metric. We also provide some conditions for $M$ to be a harmonic Einstein manifold. Finally, we provide necessary and sufficient conditions for a metric $g$ to be $\tau$-quasi RH metric by using a differential equation system.

Finsler warped product metrics with almost vanishing H-curvature

In this paper, we show that a Finsler warped product metric is of almost vanishing [Formula: see text]-curvature if and only if it is of almost vanishing [Formula: see text]-curvature. Furthermore, the corresponding one form is exact.

Stability estimates for an inverse Steklov problem in a class of hollow spheres

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem.

Uniqueness results in the inverse spectral Steklov problem

This paper is devoted to an inverse Steklov problem for a particular class of \begin{document}$ n $\end{document} -dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.

Stability in the Inverse Steklov Problem on Warped Product Riemannian Manifolds

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in $${\mathbb {R}}^d$$ . We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

Global existence of the harmonic map heat flow into Lorentzian manifolds

We investigate a parabolic–elliptic system for maps (u,v) from a compact Riemann surface M into a Lorentzian manifold N×R with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the v-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic–elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.

On the Hidden Mechanism Behind Non-uniqueness for the Anisotropic Calderón Problem with Data on Disjoint Sets

We show that there is generically non-uniqueness for the anisotropic Calderon problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n ≥ 3, there exist in the conformal class of g an infinite number of Riemannian metrics˜gmetrics˜ metrics˜g such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data ΓD and Neumann data ΓN are measured on disjoint sets and satisfy ΓD ∪ ΓN = ∂M. The conformal factors that lead to these non-uniqueness results for the anisotropic Calderon problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (M, g) and are associated to a natural but subtle gauge invariance of the anisotropic Calderon problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension n ≥ 3 to the anisotropic Calderon problem at fixed frequency with data on disjoint sets and modulo this gauge invariance. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.

A global weak solution to the Lorentzian harmonic map flow

We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric. We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.

Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra $ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.

On Brane Solutions with Intersection Rules Related to Lie Algebras

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M0 of dimension d0 ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented.

The Burghelea-Friedlander-Kappeler–gluing formula for zeta-determinants on a warped product manifold and a product manifold

The Burghelea-Friedlander-Kappeler (BFK)-gluing formula for the regularized zeta-determinants of Laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta-determinants of a one-parameter family of the Dirichlet-to-Neumann operators. When the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface, this constant is well known. In this paper, we discuss this constant in two cases: one is when a warped product metric is given near a cutting hypersurface, and the other is when a manifold is a product manifold. Especially in the first case, we use the result of Fucci and Kirsten [Commun. Math. Phys. 317, 635-665 (2013)] in which the regularized zeta-determinant of the Laplacian defined on a warped product manifold is computed.

A Convergence Result of Ricci Flow on $\mathbb{R}^{3}$ with Warped Product Metric

In this paper, we show that the Ricci flow on \(\mathbb{R}^{3}\) with warped product metric may converge to the product of the cigar with the real line under certain assumptions.

Generalized formalism in gauge-invariant gravitational perturbations

By use of the gauge-invariant variables proposed by Kodama and Ishibashi, we obtain the most general perturbation equations in the $(m+n)$-dimensional spacetime with a warped product metric. These equations do not depend on the spectral expansions of the Laplace-type operators on the $n$-dimensional Einstein manifold. These equations enable us to have a complete gauge-invariant perturbation theory and a well-defined spectral expansion for all modes and the gauge invariance is kept for each mode. By studying perturbations of some projections of Weyl tensor in the case of $m=2$, we define three Teukolsky-like gauge-invariant variables and obtain the perturbation equations of these variables by considering perturbations of the Penrose wave equations in the $(2+n)$-dimensional Einstein spectime. In particular, we find the relations between the Teukolsky-like gauge-invariant variables and the Kodama-Ishibashi gauge-invariant variables. These relations imply that the Kodama-Ishibashi gauge-invariant variables all come from the perturbations of Weyl tensor of the spacetime.

Ricci curvature bounds for warped products

We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning Euclidean and spherical cones (Bacher and Sturm [3]). On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below (Alexander and Bishop, 2004 [2]). For the proof we combine techniques developed in these papers. Because the Finslerian warped product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott–Villani/Sturm.

Ricci flow of warped product metrics with positive isotropic curvature on S p+1 × S 1

We study the asymptotic behaviour of the ODE associated to the evolution of curvature operator in the Ricci flow of a doubly warped product metric on S p + 1 ×S 1 with positive isotropic curvature.

Perturbation of a warped product metric of an end and the growth property of solutions to eigenvalue equations

When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to $ -\frac{1}{r}$, where $r$ stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rotationally symmetric one and derive growth estimates of solutions to the eigenvalue equation, from which the absence of eigenvalues will follows.

Remarks on the product of harmonic forms

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in the presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is constant and derive further topological obstructions to the existence of formal metrics. In particular, we determine the necessary and sufficient conditions for a Vaisman metric to be formal.

THE BFK-GLUING FORMULA FOR ZETA-DETERMINANTS AND THE VALUE OF RELATIVE ZETA FUNCTIONS AT ZERO

The purpose of this paper is to discuss the constant term appearing in the BFK-gluing formula for the zeta-determinants of Laplacians on a complete Riemannian manifold when the warped product metric is given on a collar neighborhood of a cutting compact hypersurface. If the dimension of a hypersurface is odd, generally this constant is known to be zero. In this paper we describe this constant by using the heat kernel asymptotics and compute it explicitly when the dimension of a hypersurface is 2 and 4. As a byproduct we obtain some results for the value of relative zeta functions at s=0.

Spectrum of the Laplacian on quaternionic Kähler manifolds

Spectrum of the Laplacian on quaternionic Kähler manifolds