Narrow Results

If we consider a sequence of warped product length spaces, what conditions on the sequence of warping functions imply compactness of the sequence of distance functions? In particular, we want to know when a subsequence converges to a well-defined metric space on the same manifold with the same topology. What conditions on the sequence of warping functions imply Lipschitz bounds for the sequence of distance functions and/or the limiting distance function? In this paper we give answers to both of these questions as well as many examples which elucidate the theorems and show that our hypotheses are necessary.

In this paper, we deal with $n$-dimensional submanifolds immersed in a slab of a warped product of the type $I{\times }_{\rho }{M}^{n+p-1}$. Under suitable constraints on the warping function $\rho$ and assuming that such a submanifold ${\mathrm{\Sigma }}^n$ is either complete or stochastically complete, we apply some maximum principles in order to show that ${\mathrm{\Sigma }}^n$ must be contained in a slice of $I{\times }_{\rho }{M}^{n+p-1}$. In particular, from our results we guarantee the nonexistence of $n$-dimensional closed minimal submanifolds immersed in ${ℝ}^m\times {ℍ}^{n+1}$. Furthermore, we construct a nontrivial duo-graph in $ℝ\times {ℍ}^2\times ℝ$ which illustrates the importance of our rigidity results.

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product ℝ × N^n-1, or globally conformally equivalent to the Euclidean space ℝⁿ or to the round sphere 𝕊ⁿ. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.

In this paper, we show that para-Kahler–Norden (or paraholomorphic) metrics do not exist on warped manifolds.

The purpose of this paper is to study the $W_2$-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the $W_2$-curvature tensor on a warped product manifold in terms of its relation with $W_2$-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate $W_2$-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a $W_2$-curvature flat warped product manifold are derived. Finally, we study the $W_2$-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are $W_2$-curvature flat.

In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

It is known from [K. Yano and M. Kon, Structures on Manifolds (World Scientific, 1984)] that the integration of the Laplacian of a smooth function defined on a compact orientable Riemannian manifold without boundary vanishes with respect to the volume element. In this paper, we find out the some potential applications of this notion, and study the concept of warped product pointwise semi-slant submanifolds in cosymplectic manifolds as a generalization of contact CR-warped product submanifolds. Then, we prove the existence of warped product pointwise semi-slant submanifolds by their characterizations, and give an example supporting to this idea. Further, we obtain an interesting inequality in terms of the second fundamental form and the scalar curvature using Gauss equation and then, derive some applications of it with considering the equality case. We provide many trivial results for the warped product pointwise semi-slant submanifolds in cosymplectic space forms in various mathematical and physical terms such as Hessian, Hamiltonian and kinetic energy, and generalize the triviality results for contact CR-warped products as well.

In this paper, we study bi-warped product submanifolds of the form $M=M_{𝜃}{\times }_{f_1}{M}_T{\times }_{f_2}{M}_{\perp }$ in a Kenmotsu manifold. We obtain a lower bound for the squared norm of the second fundamental form of a bi-warped product submanifold such as $\parallel h{\parallel }^2\ge m_1{csc}^2𝜃(1+{cos}^2𝜃)(\parallel \overrightarrow{\nabla }(ln{f}_1){\parallel }^2-1)+m_2{cot}^2𝜃(\parallel \overrightarrow{\nabla }(ln{f}_2){\parallel }^2-1)$, where $m_1=dim(M_T)$ and $m_2=dim(M_{\perp })$ and $f_1,f_2$ are the warping functions on $M$. The equality case is also considered.

There are two types of warped product pseudo-slant submanifolds, $M_{𝜃}{\times }_f{M}_{\perp }$ and $M_{\perp }{\times }_f{M}_{𝜃}$, in a nearly Kaehler manifold. We derive an optimization for an extrinsic invariant, the squared norm of second fundamental form, on a nontrivial warped product pseudo-slant submanifold $M_{\perp }{\times }_f{M}_{𝜃}$ in a nearly Kaehler manifold in terms of a warping function and a slant angle when the fiber $M_{𝜃}$ is a slant submanifold. Moreover, the equality is verified for depending on what $M_{𝜃}$ and $M_{\perp }$ are, and also we show that if the equality holds, then $M_{\perp }{\times }_f{M}_{𝜃}$ is a simply Riemannian product. As applications, we prove that the warped product pseudo-slant submanifold has the finite Kinetic energy if and only if $M_{\perp }{\times }_f{M}_{𝜃}$ is a totally real warped product submanifold.

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J.41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl.29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

We show in this paper that many well-known theorems about the geometry of warped product submanifolds of Kaehler manifolds and itself nearly Kaehler manifolds can be generalized to CR-slant warped products in nearly Kaehler manifolds.

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold $M$ in the presence of field $\phi$, given by a smooth map with source $M$ and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called $\phi$-Bach tensor, that in absence of the field $\phi$ reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of $\phi$. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map $\phi$ solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of $\phi$-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

Being motivated by a well-known Nash’s embedding theorem, Chen introduced a method to discover the relationship for the extrinsic invariants controlled by the intrinsic one. In this paper, we extend Chen’s inequality for the intrinsic and extrinsic invariants for pointwise bi-slant warped products in locally conformal Kaehler space forms with quarter-symmetric and semi-symmetric connections. The equality case of the inequality is also investigated. Several applications of the inequality are given. Furthermore, we provide two non-trivial examples of such immersions.

A generalized Chen-type inequality and corresponding equality consequences for sequential warped products are proved in this paper. These results, which are based on such warped products having factors holomorphic, totally real and pointwise slant, extend Chen-type inequality for various warped products on nearly Kähler manifolds. Moreover, we also examine the related special cases.

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold.

This paper is intended to present the main results of the author's PhD Thesis² concerning the geometry of biharmonic maps:

(i) we present several new methods, inspired by the Baird-Kamissoko method¹, for constructing proper biharmonic maps starting with harmonic maps and using warped product manifolds;

(ii) we obtain classification and non-existence results for proper biharmonic submanifolds in space forms;

(iii) we study the biharmonicity of the Gauss map for submanifolds in the Euclidean space, in the intent of generalizing the celebrated Ruh Vilms Theorem²⁴.