In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in $3$-spheres with positive sectional curvatures.
In this paper, we study the doubly warped product manifolds with semisymmetric metric connection. We derive the curvatures formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.
In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds.
In this article, we present a complete study of two disjoint classes of conformal vector fields on doubly warped product manifolds as well as on doubly warped space-times. Then we study Ricci solitons on doubly warped product manifollds admitting these types of conformal vector fields.
In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. Also, we obtain some conditio…
In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generalized $(κ,μ)$-space forms.
In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$ into the doubly warped product $_{f}B\times_{b}F$ can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections $_{f}B\times_{b}F\to B$ and $_{f}B\times_{b}F\to F$, respectively. Some characterizations for non-harmonic biharmonic maps are given by using product of harmonic maps and warping metric. Specially, in the case of $f=1$, the results for warped product in \cite{Balmus-mont} are obtained.
Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.
We investigate manifolds obtained as a quotient of a doubly warped product. We show that they are always covered by the product of two suitable leaves. This allows us to prove, under regularity hypothesis, that these manifolds are a doubly warped product up to a zero measure subset formed by an union of leaves. We also obtain a necessary and sufficient condition which ensures the decomposition of the whole manifold and use it to give sufficient conditions of geometrical nature. Finally, we study the uniqueness of direct product decomposition in the non simply connected case.
In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian if and only it is Berwaldian. In continue, we prove that every relatively isotropic Landsberg DWP-manifold is a Landsberg manifold. We show that a relatively isotropic mean Landsberg warped product manifold is a weakly Landsberg manifold. Finally, we show that there is no any lo…
For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.
We give a characterization {\sl à la Obata} for certain families of K\''ahler manifolds. These results are in the same line as other extensions of the well-known Obata's rigidity theorem from \cite{Obata62}, like for instance the generalizations in \cite{RanjSant97} and \cite{Santhanam07}. Moreover, we give a complete description of the so-called K\''ahler doubly-warped product structures whose underlying metric is Einstein.
We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the $\left(R_1,R_2\right)$-doubly warped products of smoo…
A. Gray presented an interesting $O\left( n\right) $ invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold $M_{i},i=1,2$ of a doubly warped product manifold $M=_{f_{2}}M_{1}\times _{f_{1}}M_{2}$ lie in the same Einstein-like class of $M$? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is…
In this paper, we mainly study gradient $ρ$-Einstein solitons on doubly warped product manifolds. More explicitly, we obtain necessary and sufficient conditions for a doubly warped product manifold to be a gradient $ρ$-Einstein soliton. We also apply our main result to warped product spacetime models such as generalized Robertson-Walker and standard static spacetimes as well as 3-dimensional Walker manifolds. We finally establish that there is no 3-dimensional esentially conformally symmetric gradient $ρ$-Einstein soliton.
Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with a velocity vector field and its acceleration. Their structure dictates the general form of the energy-momentum tensor in the Einstein equations in DW spherical metrics. The anisotropic pressure and the heat current of an imperfect fluid descend from the gradient of the acc…
We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $\mathbb{S}^{d_1+1}\times \mathbb{S}^{d_2}$, where $d_i \geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.
Any compact spacelike hypersurface immersed in a doubly warped product spacetime $I{}_{h} \times_ρ \mathbb{P}$ with nondecreasing warping factor $ρ$ must be a spacelike slice, provided that the mean curvature satisfies $H\geqρ'/hρ$ everywhere on the hypersurface. The conclusion also holds, under suitable assumptions on the immersion, when the hypersurface is complete and noncompact. A similar rigidity property is shown for compact hypersurfaces in spacetimes carrying a conformal, strictly expanding, timelike vector field.
Some rigidity theorems for spectral curvature bounds
Central China Normal University（华中师范大学）；POSTECH（韩国科学技术院）；Henan University（河南大学）；Center for Applied Mathematics of Henan Province, Henan University（河南省应用数学中心）
