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In this paper, we investigate Zermelo's navigation problem for some special rotation surfaces. In this respect, we find some Randers-type metrics for these rotation surfaces. Furthermore, we get the H-distortion for the metric induced by surfaces.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms D2n+1(ϵ) 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. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern.

In this study, we seek to establish new upper bounds for the mean curvature and constant sectional curvature of the first positive eigenvalue of the α-Laplacian operator on Riemannian manifolds. More precisely, various methods are used to determine the first eigenvalue for the α-Laplacian operator on the closed oriented pseudo-slant submanifolds in a generalized Sasakian space form. From our findings for the Laplacian, we extend many Reilly-like inequalities to the α-Laplacian on pseudo slant submanifold in a unit sphere.

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold Σβ1+β2 in a unit sphere Sd that provides trivial homology and fundamental groups. We also validate the instability of current flows in π1(Σβ1+β2). The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures.

The objective of this paper is to achieve the inequality for Ricci curvature of a semi-slant warped product submanifold isometrically immersed in a generalized complex space form admitting a nearly Kaehler structure in the expressions of the squared norm of mean curvature vector and warping function. In addition, the equality case is likewise discussed. We provide numerous physical applications of the derived inequalities. Later, we proved that under a certain condition the base manifold $ N_T^{n_1} $ is isometric to a $ n_1 $-dimensional sphere $ S^{n_1}(\frac{\lambda_1}{n_1}) $ with constant sectional curvature $ \frac{\lambda_1}{n_1}. $

This study attempts to establish new upper bounds on the mean curvature and constant sectional curvature of the first positive eigenvalue of the ψ − Laplacian operator on Riemannian manifolds. Various approaches are being used to find the first eigenvalue for the ψ − Laplacian operator on closed oriented bi-slant submanifolds in a Sasakian space form. We extend different Reilly-like inequalities to the ψ − Laplacian on bi-slant submanifolds in a unit sphere depending on our results for the Laplacian operator. The conclusion of this study considers some special cases as well.

The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral $p$-currents in a compact oriented warped product pointwise semi-slant submanifold $M^n$ in the Euclidean space $\mathbb{R}^{p+2q}$ which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold $M^n$, and prove that $M^n$ is homeomorphic to $\mathbb{S}^n$ if $n=4$, and $M^n$ is homotopic to $\mathbb{S}^n$ if $n=3$. Furthermore, the same results are generalized for CR-warped products and our results recovered \cite{S4}.

Our main objective of this paper is to derive the relationship between the main extrinsic invariant, and the contact CR δ-invariant (new intrinsic invariant) on a generic submanifold in trans-Sasakian generalized Sasakian space forms. Further, we find a lower bound of the squared norm of the mean curvature (main extrinsic invariant) in terms of a CR δ-intrinsic invariant, and the Laplacian of the warping function for CR-warped products in the same ambient space forms. We also investigate the classifications and triviality of connected, compact CR-warped product manifolds isometrically immersed into the trans-Sasakian generalized Sasakian space forms.

In this work, we introduce a new type of generalized mixed-type quadratic-additive functional equation and obtain its general solution. The main goal of this work is to investigate Ulam stability of this mixed type of quadratic-additive functional equation in the setting of non-Archimedean fuzzy φ-2-normed space and non-Archimedean Banach space using the direct and fixed point approaches by taking into our account two cases: even mapping and odd mapping.

In this work, we investigate the refined stability of the additive, quartic, and quintic functional equations in modular spaces with and without the Δ2-condition using the direct method (Hyers method). We also examine Ulam stability in 2-Banach space using the direct method. Additionally, using a suitable counterexample, we eventually demonstrate that the stability of these equations fails in a certain case.

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

This paper aims to formulate exact solutions of anisotropic gravitating source with the cosmological constant in (n+2) dimensional spacetime, which may lead to collapse or expansion. We have considered the expansion scalar to be negative and positive to achieve the collapsing and expansion scenario. These assumptions enabled us to investigate the range of a particular parameter for which the expansion scalar becomes negative and positive. Further, we used the definition of mass function and found the condition for the trapped surface. The plots of pressures, anisotropy parameter, and energy density for both collapse and expansion are also presented.

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

In this study, we develop a general inequality for warped product semi-slant submanifolds of type Mn=NTn1×fNϑn2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field ξ. Furthermore, if the potential vector field ξ of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of η-Ricci-Yamabe soliton on an imperfect fluid GRW spacetime with a harmonic potential function ψ. Finally, we examine necessary and sufficient conditions for a 1-form η, which is the g-dual of the vector field ξ on imperfect fluid GRW spacetime to be a solution of the Schrödinger-Ricci equation.

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

Abstract We introduce new classes of generalized Orlicz-Garling sequences and Orlicz-Lorentz sequences by using a sequence of Orlicz functions and difference operator. We show that the Orlicz-Garling sequence space admits a unique 1-subsymmetric basis and a 1-dominated block basic sequence in g(M,Δ(m),v,p) $g(\mathcal{M},\Delta^{(m)}, v, p)$. We also make an effort to prove that every symmetric normalized block Orlicz-Garling sequence dominates an Orlicz-Lorentz sequence. Finally, we study some geometric properties of these spaces and establish some inclusion relations between spaces.

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional metric dimension for all the connected networks is an NP-hard problem. In this note, we find the sharp bounds of the fractional metric dimensions of all the connected networks under certain conditions. Moreover, we have calculated the fractional metric dimension of grid-like networks, called triangular and polaroid grids, with the aid of the aforementioned criteria. Moreover, we analyse the bounded and unboundedness of the fractional metric dimensions of the aforesaid networks with the help of 2D as well as 3D plots.

In the present work, we consider two types of bi-warped product submanifolds, M=MT×f1M⊥×f2Mϕ and M=Mϕ×f1MT×f2M⊥, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.

Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper.

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M ⊥ × f M θ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a basic inequality become equality. The next purpose of this paper to find the necessary condition admitting gradient Ricci soliton, that the warped product semi-slant submanifold of Kenmotsu space form, is an Einstein warped product. We also discuss some obstructions to these constructions in more detail.

In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.