Riemannian Warped Product Research Articles

Geometric analysis of Riemannian doubly warped product submersions

This paper introduces the concept of Riemannian doubly warped product submersions as a natural generalization of Riemannian warped product submersions. We study some fundamental properties of such submersions and derived the conditions under which the Riemannian doubly warped product submersions have totally geodesic, totally umbilical, and minimal fibers. Furthermore, we analyze the curvature properties associated with Riemannian doubly warped product submersions and establish the Einstein conditions. Some non-trivial examples are explored to demonstrate the existence of such submersions.

Chen inequality for general warped product submanifold of Riemannian warped products I×fSm(c)

In the present paper, we investigate the geometry and topology of warped product submanifolds in Riemannian warped product I×fSm(c) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Our main object is to apply geometrically to number structures and obtained applications of Dirichlet eigenvalues problems. Some new results on mean curvature vanishing are presented, and a partial solution can be found to the well-known problem given by S S Chern. The family of Riemannian warped products generalized to Robertson-Walker space-times

Inequalities on Riemannian Warped Product Submersions for Vertical Casorati Curvatures

Inequalities on Riemannian Warped Product Submersions for Vertical Casorati Curvatures

Gradient Einstein-type structures immersed into a Riemannian warped product

In this paper, we study gradient Einstein-type structure immersed into a Riemannian warped product manifold. We obtain some triviality results for the potential function and smooth map u. We investigate conditions for a gradient Einstein-type structure to be totally umbilical, or totally geodesic immersed into a warped product I×fMn. Furthermore, we study rotational hypersurface into R×fRn has a gradient Einstein-type structure.

Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρ M f n whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.

Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

Riemannian Warped Product Submersions

In this paper, we introduce Riemannian warped product submersions and construct examples and give fundamental geometric properties of such submersions. On the other hand, a necessary and sufficient condition for a Riemannian warped product submersion to be totally geodesic, totally umbilic and minimal is given.

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products

In a Riemannian warped product I×fMn, where I⊂R is an open interval, f is a positive real function defined on I and Mn is a compact Riemannian manifold without boundary, we use equivariant bifurcation theory in order to establish sufficient conditions, in terms of f and the spectrum of the Laplacian on Mn, that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are H2-hypersurfaces, namely, whose boundaries are hypersurfaces with constant second mean curvature H2. For each of our results, we have provided a considerable number of examples that verify all the assumptions under consideration.

A Moser–Bernstein problem for Riemannian warped products

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has classically been studied in the the Euclidean n-dimensional space and it is known as the Moser–Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precisely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $$M=P\times _h{\mathbb {R}}$$, where P is a complete Riemannian parabolic manifold and h a positive smooth function on P.

On a non flat Riemannian warped product manifold with respect to quarter-symmetric connection

Abstract In this paper, we study generalized quasi-Einstein warped products with respect to quarter symmetric connection for dimension n ≥ 3 and Ricci-symmetric generalized quasi-Einstein manifold with quarter symmetric connection. We also investigate that in what conditions the generalized quasi-Einstein manifold to be nearly Einstein manifold with respect to quarter symmetric connection. Example of warped product on generalized quasi-Einstein manifold with respect to quarter symmetric connection are also discussed.

An Extension of Calabi’s Correspondence between the Solutions of Two Bernstein Problems to More General Elliptic Nonlinear Equations

A new correspondence between the solutions of theminimal surface equation in a certain 3-dimensional Riemannian warped product and the solutions of the maximal surface equation in a 3-dimensional standard static space-time is given. This widely extends the classical duality between minimal graphs in 3-dimensional Euclidean space and maximal graphs in 3-dimensional Lorentz–Minkowski space-time. We highlight the fact that this correspondence can be restricted to the respective classes of entire solutions. As an application, a Calabi–Bernstein-type result for certain static standard space-times is proved.

Inverse curvature flows in Riemannian warped products

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers 0<p≤1 of a curvature function satisfying few common properties.

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$ M × e ψ N

If a graph submanifold $(x,f(x))$ of a Riemannian warped product space $(M^m\times_{e^{\psi}}N^n,\tilde{g}=g+e^{2\psi}h)$ is immersed with parallel mean curvature $H$, then we obtain a Heinz type estimation of the mean curvature. Namely, on each compact domain $D$ of $M$, $m\|H\|\leq \frac{A_{\psi}(\partial D)}{V_{\psi}(D)}$ holds, where $A_{\psi}(\partial D)$ and $V_{\psi}(D)$ are the ${\psi}$-weighted area and volume, respectively. In particular, $H=0$ if $(M,g)$ has zero weighted Cheeger constant, a concept recently introduced by D.\ Impera et al.\ (\cite{[Im]}). This generalizes the known cases $n=1$ or $\psi=0$. We also conclude minimality using a closed calibration, assuming $(M,g_*)$ is complete where $g_*=g+e^{2\psi}f^*h$, and for some constants $\alpha\geq \delta\geq 0$, $C_1>0$ and $\beta\in [0,1)$, $\|\nabla^*\psi\|^2_{g_*}\leq \delta$, $\mathrm{Ricci}_{\psi,g_*}\geq \alpha$, and $det_g(g_*)\leq C_1 r^{2\beta}$ holds when $r\to +\infty$, where $r(x)$ is the distance function on $(M,g_*)$ from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.

Parabolicity and uniqueness of complete two-sided hypersurfaces immersed in a Riemannian warped product

In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type $$I\times _fM^n$$ , where $$M^{n}$$ is a connected n-dimensional oriented Riemannian manifold and $$f:I\rightarrow \mathbb {R}$$ is a positive smooth function. As applications, we obtain several uniqueness results concerning these hypersurfaces with constant mean curvature, under standard constraints on the Ricci curvature of $$M^n$$ and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity.

Parabolicity of minimal graphs in Riemannian warped products and rigidity theorems

We provide natural geometric assumption for a minimal graph in a Riemannian warped product to be parabolic. As a consequence, we deal with several Bernstein-type problems.

Stable Minimal Surfaces in Riemannian Warped Products

We obtain several stability results for minimal two-sided surfaces immersed in a wide class of 3-dimensional Riemannian warped products, which includes the class of Riemannian products. As a consequence, some Bernstein-type results are provided. Moreover, a non-existence theorem for a certain non-linear elliptic equation is obtained.

Uniqueness of entire graphs in Riemannian warped products

In this paper, by applying the generalized Omori-Yau maximum principle for complete spacelike hypersurfaces in warped product spaces, we obtain the sign relationship between the derivative of warping function and support function. Afterwards, by using this result and imposing suitable restrictions on the higher order mean curvatures, we establish uniqueness results for the entire graph in a Riemannian warped product space, which has a strictly monotone warping function. Furthermore, applications to such a space are given.

THE EXISTENCE OF SOME METRICS ON RIEMANNIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLD OF CLASS (B)

In this paper, we prove the existence of warping functions on Riemannian warped product manifolds with some prescribed scalar curvatures according to the fiber manifolds of class (B).

THE EXISTENCE OF WARPING FUNCTIONS ON RIEMANNIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLDS OF CLASS (A)

Abstract. In this paper, we prove the existence and the nonex-istence of warping functions on Riemannian warped product mani-folds with some prescribed scalar curvatures according to the bermanifolds of class (A). 1. IntroductionOne of the basic problems in the di erential geometry is studying theset of curvature functions which a given manifold possesses.The well-known problem in di erential geometry is that of whetherthere exists a warping function of warped metric with some prescribedscalar curvature function. One of the main methods of studying di er-ential geometry is by the existence and the nonexistence of a warpedmetric with prescribed scalar curvature functions on some Riemannianwarped product manifolds. In order to study these kinds of problems,we need some analytic methods in di erential geometry.For Riemannian manifolds, warped products have been useful in pro-ducing examples of spectral behavior, examples of manifolds of negativecurvature (cf. [2, 3, 4, 5, 7, 13, 14]), and also in studying L