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We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the "stretching+folding=chaos" mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale--Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times. [ABSTRACT FROM AUTHOR]

In this paper, we consider self-similar solutions of the r -mean curvature flow into the (n + 1) -dimensional Euclidean space. By employing some general maximum principles as the main tool, we characterize some self-similar solutions of the r -mean curvature flow under some suitable geometric constraints. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson–Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost-Ricci soliton, almost- η -Ricci soliton, almost-gradient η -Ricci soliton. As a generalization of curvature inheritance [K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys. 33(9) (1992) 2989–2997] and curvature collineation [G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, J. Math. Phys. 10(4) (1969) 617–629], in this paper, we introduce the notion of generalized curvature inheritance and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp., Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate in this paper the Type IV singular bouncing in the framework of f (R ,) theory of gravity where R and mean the curvature scalar and the Gauss–Bonnet invariant, respectively. Cosmological f (R ,) models constrained by the slow-roll evolution is reconstructed and their explicit forms are provided near the bounce and far away from it. One obtains finally two models whose stability is numerically analyzed in this work. Our results show that the stability of the reconstructed models is very affected by their parameters. The model far from the singularity recovers stability quickly than the model near the singularity. [ABSTRACT FROM AUTHOR]

In this paper, we reconstruct a f (R , T) model where R and T are the curvature scalar and the trace of the stress-tensor, respectively, and confront it with Λ CDM model through the cosmographic parameters. The reconstructed f (R , T) model is fundamentally characterized by a input parameter β for which some suitable values are chosen. Our results show that for some values of these parameters the reconstructed f (R , T) fits very well with the Λ CDM regarding the evolution of the cosmographic parameters. [ABSTRACT FROM AUTHOR]

Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold M : ∂ ∂ t g (t) = − (R g (t) − R ¯ g (t)) g (t) in M. In this paper, we prove the corresponding results on manifolds with boundary. More precisely, given a compact manifold M with smooth boundary ∂ M , we study the convergence rate of the Yamabe flow with boundary: ∂ ∂ t g (t) = − (R g (t) − R ¯ g (t)) g (t) in M and H g (t) = 0 on ∂ M and the conformal mean curvature flow: ∂ ∂ t g (t) = − (H g (t) − H ¯ g (t)) g (t) on ∂ M and R g (t) = 0 in M. [ABSTRACT FROM AUTHOR]

In this work, we generalize the theory of perturbations in a (D + 1)-dimensional space–time with cosmological constant, studying scalar, vector and tensor perturbations, as well as its structure in Newtonian and Synchronous gauge. We also show the theory of perturbations in the context of brane cosmology, where branes are embedded in a set of D-spatial dimensions, a temporal dimension, and an additional spatial dimension. In both standard and brane cosmology, an unperturbed space–time is provided with a Friedmann–Lemaitre–Robertson–Walker metric and arbitrary sectional curvature, the matter content has the shape of a perfect fluid. In addition, we consider the arbitrary sectional curvature, obtaining the respective equations in the Newtonian and Synchronous gauge. We highlight that the results presented in this paper can be used to treat brane cosmology with two concentric branes or tackle the σ 8 tension with a braneworld approach. Finally, as an example of the utility of all the technology presented in this paper, we show an application to the energy flux and the repercussions in the framework of the braneworlds. [ABSTRACT FROM AUTHOR]

Let M Σ be an n -dimensional Thom–Mather stratified space of depth 1. We denote by β M the singular locus and by L the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge α -class α w (M Σ) ∈ KO n . In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of M Σ is a homogeneous space of positive scalar curvature, L = G / K , where the semisimple compact Lie group G acts transitively on L by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when M Σ and β M are spin, we reinterpret our obstruction in terms of two α -classes associated to the resolution of M Σ , M , and to the singular locus β M. Finally, when M Σ , β M , L and G are simply connected and dim M is big enough, and when some other conditions on L (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two α -classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature. [ABSTRACT FROM AUTHOR]

This paper presents a low dropout bandgap voltage reference circuit that combines high output voltage accuracy and low temperature-coefficient in a wide temperature range. The low temperature-coefficient is achieved by an advanced high-order curvature compensation technique, which is implemented by a novel exponential piecewise compensation term in a high-temperature range. A supply noise self-regulation technique is proposed to realize an excellent power supply rejection ratio (PSRR) without using any amplifiers in this paper. The proposed high-accuracy voltage reference is designed in a 2 μ m bipolar process. The circuit generates a reference voltage of 3.3 V and can operate under a minimum supply voltage of 3.8 V, while consuming 44.8 μ A. The output voltage accuracy is ± 0.15%. The PSRR at 10 Hz and 1 MHz is − 6 4 dB and − 7 8 dB, respectively. The line regulation is 2.45 ppm/V in a supply voltage range from 3.8 V to 20 V. The average temperature coefficient is 11.9 ppm/∘C in a temperature range from − 5 5 ∘ C to +125∘C. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some (possibly nonsmooth) Gromov–Hausdorff limits at infinity. The Gromov–Hausdorff asymptotic analysis allows us to recover and extend different previous existence theorems. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories. [ABSTRACT FROM AUTHOR]

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed 1 -forms. Another family of well-known examples are projectively flat Finsler metrics on the 2 -sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let g be a Riemannian–Finsler metric on a closed surface Σ , and p be a small antisymmetric potential on Σ that is a natural generalization of 1 -form (see Sec. 1). If the Randers-type Finsler metric g − p is geodesically reversible, and the geodesic flow of g is topologically transitive, then we prove that p must be a closed 1 -form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the 2 -sphere of constant positive curvature. [ABSTRACT FROM AUTHOR]

In this paper, we developed, in detail, the geometric structure of a certain type of geometry usually used in physical applications. This paper contains building blocks (BB), tensors of different orders, curves admitted by the geometry and a comparison between this geometry and others used in the literature. Second-order symmetric and skew-symmetric tensors, usually used for physical applications in the context of the geometrization philosophy, are given. Also, in the framework of the same philosophy, the admitted curves can be used as equations of motion of test particles for any field theory constructed in this geometry. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize almost co-Kähler manifolds and co-Kähler three-manifolds whose metrices are the gradient ρ -Einstein solitons. At first we prove that a proper (κ ̃ , μ ̃) -almost co-Kähler manifold with κ ̃ < 0 does not admit gradient ρ -Einstein soliton. It is also shown that if a proper -Einstein almost co-Kähler manifold with constant coefficients admits a gradient ρ -Einstein soliton, then either the manifold is a K -almost co-Kähler manifold or the soliton is trivial. Next, we prove that in case of co-Kähler three-manifold the manifold is of constant scalar curvature. Moreover, either the manifold is flat or the gradient of the potential function is collinear with the Reeb vector field ξ. Finally, we construct two examples to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Jacobi stability of the Shimizu–Morioka model by using the KCC-theory. First, by associating the nonlinear connection and Berwald connection, five geometrical invariants of the dynamical model are obtained. Furthermore, the Jacobi stability of the Shimizu–Morioka model at equilibrium is studied in terms of the eigenvalues of the deviation curvature tensor. It shows that the three equilibria are always Jacobi unstable. Finally, the dynamical behavior of the components of the deviation vector is discussed, which geometrically characterizes the chaotic behavior of studied model near the origin. It proved the onset of chaos in the Shimizu–Morioka model. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in f (ℛ) modify gravity. The semiconformal flatness of general spacetime and spacetime in f (ℛ) gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure p and energy density σ. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in f (ℛ) gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension n in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases n = 2, 3, 4. Second, supposing further that the scalar curvature is constant for n ≥ 5, we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with n ≥ 3, we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel curvature tensor called the 풦-curvature tensor, which is formed by combining the Weyl tensor and the 풲2-curvature tensor. It is proved that a semi-Riemannian manifold with traceless 풦-curvature tensor is Einstein, while a 풦-curvature flat semi-Riemannian manifold exhibits constant sectional curvature. We show that a space-time with traceless 풦-curvature tensor or 풦-curvature flat represents dark energy era. A semi-Riemannian manifold with divergence-free 풦-curvature tensor is Weyl harmonic. In a space-time where the 풦-curvature tensor is of divergence-free, both the isotropic pressure and energy density are constants. Under certain conditions, it is shown that a pseudo-풦-symmetric semi-Riemannian manifold reduces to a pseudo-Riemannian manifold. It is demonstrated that pseudo-풦-symmetric space-times can be regarded as generalized Robertson–Walker (GRW) space-times. Finally, a concrete example of pseudo-풦-symmetric semi-Riemannian manifolds is presented. [ABSTRACT FROM AUTHOR]

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds M3. Given a conformal vector field V ∈ 픛(M3), a conformal trajectory of V is a regular curve γ in M3 satisfying ∇γ′γ′ = qV × γ′, for some fixed nonzero constant q ∈ ℝ. In this paper, we study the conformal trajectories in the space forms ℝ3, 핊3 and ℍ3. For (non-Killing) conformal vector fields in 핊3 (respectively, in ℍ3), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ℝ3, we obtain the same result for the radial vector field and characterizing all conformal trajectories. [ABSTRACT FROM AUTHOR]

In the past few decades, various techniques for structural damage identification based on dynamic properties have been widely studied, among which modal curvature is the research hotspot, due to its high sensitivity to local damage. To realize story-level structural damage identification of shear buildings, this paper first derives the patterns of variation in modal curvature of the damaged structure. Then, a new damage index sensitive to local damage called the Modal Curvature Change modified with Softmax (MCCSM for short), is proposed to improve the accuracy of structural damage identification. Afterward, the validity of the damage index is verified by a shaking table test of a scaled RC frame structure model and its corresponding numerical model. The research results show that the damage location indicated by MCCSM corresponds well with the damage phenomenon in the shaking table test and numerical simulation; compared with using the theoretical mode shape, the MCCSM index can give comparable damage identification results using the mode shape obtained through operational modal analysis from vibration time-history data, which proves the feasibility in engineering practices; through numerical simulation, it is proved that the MCCSM index is applicable in a wide range of damage cases with different story position, number of damaged story, and damage extent. [ABSTRACT FROM AUTHOR]

In this paper, we give a lot of new non-projectively flat Finsler warped product metrics with zero flag curvature. We also manufacture new generalized Douglas–Weyl (GDW) metrics which are nontrivial in the sense that these metrics are neither Douglas metrics nor Weyl metrics via hyperbolic spaces. [ABSTRACT FROM AUTHOR]

The goal of this paper is to analyze sharp-type inequalities including the scalar and Ricci curvatures of bi-slant Riemannian submersions in complex space forms. Then, for bi-slant Riemannian submersion between a complex space form and a Riemannian manifold, we give inequalities involving the Casorati curvature of the space ker φ ∗ . Also, we mention some examples. [ABSTRACT FROM AUTHOR]

It is known that the Frenet–Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet–Serret apparatus specifies important geometric invariants, including the curvature and the torsion of a curve. It is also acknowledged in quantum information science that low complexity and high efficiency are essential features to achieve when cleverly manipulating quantum states that encode quantum information about a physical system. In this paper, we propose a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by dynamically evolving state vectors. Specifically, we propose a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced by a parallel-transported pure quantum state evolving unitarily under a stationary Hamiltonian specifying the Schrödinger equation. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector | T 〉 to the state vector | Ψ 〉 and represents a useful measure of the bending of the quantum curve. Our proposed constant torsion coefficient, instead, is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector | T 〉 , orthogonal to both | T 〉 and | Ψ 〉. The torsion coefficient provides a convenient measure of the twisting of the quantum curve. Remarkably, we show that our proposed curvature and torsion coefficients coincide with those existing in the literature, although introduced in a completely different manner. Interestingly, not only we establish that zero curvature corresponds to unit geodesic efficiency during the quantum transportation in projective Hilbert space, but we also find that the concepts of curvature and torsion help enlighten the statistical structure of quantum theory. Indeed, while the former concept can be essentially defined in terms of the concept of kurtosis, the positivity of the latter can be regarded as a restatement of the well-known Pearson inequality that involves both the concepts of kurtosis and skewness in mathematical statistics. Finally, not only do we present illustrative examples with nonzero curvature for single-qubit time-independent Hamiltonian evolutions for which it is impossible to generate torsion, but we also discuss physical applications extended to two-qubit stationary Hamiltonians that generate curves with both nonzero curvature and nonvanishing torsion traced by quantum states with different degrees of entanglement, ranging from separable states to maximally entangled Bell states. In the Appendix C, we examine the different curvature and torsion characteristics of the three qubit | GHZ 〉 and | W 〉 states under evolution by a quantum Heisenberg Hamiltonian. [ABSTRACT FROM AUTHOR]

In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector | T 〉 to the state vector | Ψ 〉 and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector | T 〉 to the state vector | Ψ 〉 , orthogonal to | T 〉 and | Ψ 〉 and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian. [ABSTRACT FROM AUTHOR]

In this paper, we review a recently offered notion of quasi-static evolution of the axial self-gravitating structures at large scales and the criterium to characterize the corresponding evolutionary aspects under the influence of strong curvature regimes. In doing so, we examine the axial source's dynamic and quasi-static behavior within the parameters of various modified gravity theories. We address the formalism of these notions and their possible implications in studying the dissipative and anisotropic configuration. We initiate by considering higher-order curvature gravity. The Palatini formalism of f (R) gravity is also taken into consideration to analyze the behavior of the kinematical as well as the dynamical variables of the proposed problem. The set of invariant velocities is defined to comprehend the concept of quasi-static approximation that enhances the stability of the system in contrast to the dynamic mode. It is identified that vorticity and distinct versions of the structure scalars Y I , Y II and Y KL play an important role in revealing the significant effects of a fluid's anisotropy. As another example of evolution, we check the influence of Palatini-based factors on the shearing motion of the object. A comparison-based study of the physical nature of distinct curvature factors on the propagation of the axial source is exhibited. This provides the intriguing platform to grasp the notion of quasi-static evolution together with the distinct curvature factors at current time scenario. The importance of slowly evolving axially symmetric regimes will be addressed through the distinct modified gravitational context. Finally, we share a list of queries that, we believe, deserve to be addressed in the near future. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

A new beam finite element for non-prismatic (tapered and curved) beams with thin rectangular cross-section ("strip" beams) is proposed in this paper. The element is geometrically exact and constitutes the extension, to the large 3D displacement and rotation setting, of those proposed by the author for the 2D [R. Gonçalves, Int. J. Struct. Stab. Dyn. 23 (2022) 2350037] and linearized lateral-torsional buckling [R. Gonçalves, Int. J. Struct. Stab. Dyn. 23(12) (2023) 2350139] cases. The element incorporates torsion-related warping, as well as arbitrary mid-line warping, and allows nonlinear tapered geometries, arbitrary initially curved configurations (including pre-twist) and eccentric loads (loads offset from the centroid). All expressions required to implement the proposed element are given in a straightforward vector/matrix format. Several numerical tests are presented to show that the element has a very good predictive capability and a low computational cost, when compared with solutions obtained with refined shell finite element meshes, even for members with a high taper ratio and initial curvature. [ABSTRACT FROM AUTHOR]

In this paper, we study the cosmic dynamics of varying vacuum models where the dark matter interacts with the vacuum energy. We consider the homogeneous and isotropic spacetime with spatial curvature and apply the dynamical system technique to the varying vacuum models by specifying the form of energy exchange rate (Q) between the dark energy and dark matter. Further, we utilize the cosmographic parameters and statefinder parameters in the terms of dynamical variables of the cosmological dynamical system to explore the cosmic dynamics of the universe. The Milne universe solution exists as a consequence of spatially curved geometry in the model. The models also yield radiation- matter- and dark energy-dominated phases in order and thus explain the late-time accelerated expansion of the universe. The strength of interaction terms will affect the existence of cosmological phases in the model but there will always be an attractor corresponding to the accelerating universe. We numerically solve the system to illustrate the evolution of cosmological quantities and dynamical variables in the models. The role of curvature is visible during the transition from the decelerated phase into the accelerating phase. The numerical solutions affirm that the cosmological parameters in the models are consistent with their corresponding observational values. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Curvature inheritance symmetry on M-projectively flat spacetimes" published in the International Journal of Geometric Methods in Modern Physics. The authors have noted that the results presented in Section 3 of the paper hold for perfect fluid spacetimes in general, not just M-projectively flat spacetimes. Therefore, the term "M-projectively flat" should be removed from the heading, contents of Section 3, abstract, and introduction. The correct form of Equation (26) is also provided in the correction notice. [Extracted from the article]

This paper presents an analysis of the effect of a "slight" curvature of the axis of a thin-walled beam on its eigenfrequencies and forms. By "slight" we understand such a beam curvature at which no theory of curved beams (arches, horizontally curved girders), but classical beam theory is applied. The subject of the analysis is the open nonprismatic thin-walled beams with any geometrical parameters, and prismatic beams. The thin-walled beam model used in the analysis was derived on the basis of the momentless theory of plates, using the Vlasov assumptions. The free vibrations of the beams were analysed. Calculations were performed for four groups of beams. Each of the groups comprised two beams with the same configuration of flanges and identical geometrical cross section characteristics, but differing in web geometry. One of the beams had a symmetrical web (a beam with a rectilinear axis in the web plane), whereas the web of the other beam was asymmetrical (a beam with its axis curved in the web plane). The algorithm based on Chebyshev polynomials was used to solve the variable coefficient equations describing the model's vibration. According to this algorithm, solutions are sought in the form of Chebyshev series and the closed analytical recurrence formulas generated by the algorithm are used to calculate the coefficients of the equations. It has been shown that even a "slight" curvature of the beam's axis has a significant effect on its eigenfrequencies and forms, as confirmed by a comparison of the results with the ones obtained using Finite Element Method (FEM). [ABSTRACT FROM AUTHOR]

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of ℝ n + 1 , all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a "ground state" substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality. [ABSTRACT FROM AUTHOR]

In the process of finding Einstein metrics in dimension , we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer-Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer-Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space can be at most , with equality implying that is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor . In this second paper, we apply our results in the first paper to show that either is a standard sphere or the dimension of the space of Fischer-Marsden solutions can be at most . [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations by considering almost ∗- η -Ricci–Bourguignon soliton as a Kenmotsu metric. It is shown that if a Kenmotsu metric endows a ∗- η -Ricci–Bourguignon soliton, then the curvature tensor R with the soliton vector field V is given by the expression (ℒ V R) (V 1 , ξ) ξ = 2 { V 1 (r) ξ − V 1 (D r) + ξ (D r) − ξ (r) ξ − D r }. Next, we show that if an almost Kenmotsu manifold such that ξ belongs to (κ , − 2) ′ -nullity distribution where κ < − 1 acknowledges a ∗- η -Ricci–Bourguignon soliton satisfying Ω + ψ ≠ [ (r + 4 n 2) + { ξ (ξ (r)) − ξ (D r) } ] , then the manifold is Ricci-flat and is locally isometric to ℍ n + 1 (− 4) × ℝ n . Moreover if the metric admits a gradient almost ∗- η -Ricci–Bourguignon soliton and ξ leaves the scalar curvature r invariant on a Kenmotsu manifold, then the manifold is an η -Einstein. Also, if a Kenmotsu metric represents an almost ∗- η -Ricci–Bourguignon soliton with potential vector field V is pointwise collinear with ξ , then the manifold is an η -Einstein. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a nontrivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study certain types of metric such as ∗-conformal Ricci–Yamabe soliton (RYS), whose potential vector field is torse-forming on Kenmotsu manifold. Here, we establish the conditions for solitons to be expanding, shrinking or steady and find the scalar curvature when the manifold admits ∗-conformal RYS on Kenmotsu manifold. Next, we developed the nature of the vector field when the manifold satisfies ∗-conformal RYS. Also, we have adorned some applications of torse-forming vector field in terms of ∗-conformal RYS on Kenmotsu manifold. We have also studied infinitesimal CL-transformation and Schouten–van Kampen connection on Kenmotsu manifold, whose metric is ∗-conformal RYS. We present an example of ∗-conformal RYS on three-dimensional Kenmotsu manifold, and verify some of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the η -Einstein cosymplectic manifolds with the gradient Einstein solitons and the conformal vector fields. It is proven that if an η -Einstein cosymplectic manifold M 2 n + 1 of dimension 2 n + 1 with n > 1 admits a gradient Einstein soliton, then either M 2 n + 1 is Ricci flat or the gradient of Einstein potential function is pointwise collinear with the Reeb vector field of M 2 n + 1 . Also, we prove that if M 2 n + 1 admits a conformal vector field W , then either M 2 n + 1 is Ricci flat or W is homothetic. We also establish that a conformal vector field W on M 2 n + 1 is a strict infinitesimal contact transformation, provided the scalar curvature of M 2 n + 1 is constant. Finally, the non-trivial examples of cosymplectic manifolds admitting the gradient Einstein solitons are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the obtain Chen inequalities containing the Ricci and scalar inequalities on the vertical and the horizontal distributions for semi-slant Riemannian submersions from complex space forms onto Riemannian manifolds. The equality case of the obtained inequalities is discussed. At the end, some geometric consequences are obtained. Moreover, many examples are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence results of a critical p -Laplacian equation involving critical and subcritical Hardy potential on compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we give some vanishing theorems for harmonic p -forms on complete noncompact Riemannian manifolds satisfying a weighted p -Poincaré inequality under the pointwise curvature pinching conditions which are bounded from above by the weight function. In some of our results, the condition on scalar curvature is no longer required as in previous ones. [ABSTRACT FROM AUTHOR]

A steady Newtonian fluid flow through annular region in curved pipes is studied in this paper. The flow takes place due to an axial pressure gradient and it is three-dimensional in nature. The equations governing the flow are highly coupled and nonlinear. The solutions are obtained, analytically, using a regular perturbation method. The effects of curvature ratio (δ) , Reynolds number (Re) and the radius ratio (l) on the axial velocity and stream function are presented graphically. It is observed that, along with curvature ratio and Reynolds number, radius ratio highly affects the fluid flow in annular curved pipes. [ABSTRACT FROM AUTHOR]

In this paper, we study ∗-Conformal η -Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting ∗-Conformal η -Ricci soliton. We obtain some significant results on ∗-Conformal η -Ricci soliton in Sasakian manifolds satisfying R (ξ , X) ⋅ S = 0 , S (ξ , X) ⋅ R = 0 , P ¯ (ξ , X) ⋅ S = 0 , where P ¯ is Pseudo-projective curvature tensor. The conditions for ∗-Conformal η -Ricci soliton on ϕ -conharmonically flat and ϕ -projectively flat Sasakian manifolds have been obtained in this paper. Lastly we give an example of five-dimensional Sasakian manifolds satisfying ∗-Conformal η -Ricci soliton. [ABSTRACT FROM AUTHOR]

This paper concerns with the characterization of a spacetime and modified gravity such as f (R , G) , f (R , T) and f (R , L m) -gravity equipped with the projective curvature tensor. We establish that a projectively flat perfect fluid spacetime represents dark energy era. Also, we prove that a projectively flat perfect fluid spacetime is either locally isometric to Minkowski spacetime or a de-Sitter spacetime. Furthermore, it is shown that a perfect fluid spacetime admitting harmonic projective curvature tensor becomes a generalized Robertson–Walker spacetime and is of Petrov type I , D or O. Lastly, we investigate the effect of projectively flat perfect fluid spacetime solutions in f (R , G) , f (R , T) and f (R , L m) -gravity, respectively. We also investigate the spacetime as a f (R , G) -gravity solution and use the flat Friedmann–Robertson–Walker metric to establish a relation among jerk, snap, and deceleration parameters. Numerous energy conditions are studied in terms of Ricci scalar with the model f (R , G) = exp (R) + α (6 G) β . For this model, the strong energy condition is violated but the weak, dominant and null energy conditions are fulfilled, which is in excellent accordance with current observational investigations that show the universe is now accelerating. [ABSTRACT FROM AUTHOR]

In this paper, we study some important Remannian and non-Riemannian curvature properties of spherically symmetric Finsler metrics. Under a condition on the geodesic coefficient, we find the necessary and sufficient conditions under which spherically symmetric metrics are of scalar flag curvature, W -quadratic or projectively Ricci-flat. For spherically symmetric metrics of relatively isotropic Landsberg curvature, we find the necessary and sufficient conditions under which these metrics are of constant flag curvature or Ricci-quadratic. Finally, we prove a rigidity result that every spherically symmetric metric of relatively isotropic Landsberg curvature is Ricci-quadratic if and only if it is a Berwald metric. Moreover, if the Finsler metric is negatively complete then it reduces to a Riemannian metric. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the rigidity of self-shrinkers in a Euclidean space ℝ n + 1 . We first prove that any self-shrinker X : M → ℝ n + 1 with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is an open part of a hyperplane ℝ n , a cylinder S k (k) × ℝ n − k   (1 ≤ k ≤ n − 1) or the round sphere S n (n). Then, it can be applied to show that any complete self-shrinker X : M → ℝ n + 1 with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is isometric to a hyperplane ℝ n , a cylinder S k (k) × ℝ n − k (1 ≤ k ≤ n − 1) or the round sphere S n (n). Finally, some characterizations of self-shrinkers with constant mean curvature in ℝ n + 1 are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we show Liouville-type theorems for exponentially harmonic maps and p-harmonic maps between Kähler manifolds. We then study the quadratic form of an exponentially harmonic map of Kähler manifolds and its applications. We also prove that if f : M → N is an exponentially harmonic map from a compact Kähler manifold M into a Kähler manifold N with constant energy density such that the curvature of N is strongly semi-negative, then f is holomorphic or antiholomorphic. [ABSTRACT FROM AUTHOR]

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation. [ABSTRACT FROM AUTHOR]

Sultana–Dyer black hole, obtained by a time-dependent conformal transformation of Schwarzschild black hole, is an exact solution of general relativity in spherical symmetry. This paper provides the investigation of geometrical properties of the Sultana–Dyer spacetime by means of covariant derivative(s) of the geometric quantity "curvature". It is shown that such a spacetime is 2 -quasi-Einstein, Einstein spacetime of level 3 and fulfills the generalized Roter type condition. The spacetime admits pseudosymmetric Weyl curvature as well as pseudosymmetric conharmonic curvature. Also, Weyl compatibility and Riemann compatibility of the Ricci tensor are shown. Finally, a comparison is drawn between Schwarzschild and Sultana–Dyer spacetimes with respect to their curvature restricted geometric properties. [ABSTRACT FROM AUTHOR]

In this paper, we study slant curves and Legendre curves in three-dimensional Walker manifolds. We give a classification of these curves and we present a necessary and sufficient condition for a slant curve to admit a proper mean curvature vector field. We close the study by examples. [ABSTRACT FROM AUTHOR]

In this paper, the projective curvature tensor field of a generalized indefinite Sasakian space form is investigated. Some results dealing with projectively semi-symmetric and ϕ -projectively semi-symmetric generalized indefinite Sasakian space forms are obtained. Furthermore, biharmonic Legendre Frenet curves are discussed on these manifolds. [ABSTRACT FROM AUTHOR]

In this paper, the modified F (R) gravity theory with the function F (R) = − (1 / β) ln (1 − β R) is studied. The action at small coupling β becomes Einstein–Hilbert action. The bound on the parameter β from local tests is β ≤ 2 × 1 0 − 6 cm2. We find the constant curvature solutions and it was shown that the de Sitter space is unstable but a solution with zero Ricci scalar is stable. The potential and the mass of the scalar field (scalaron) are obtained in Einstein's frame. The slow-roll cosmological parameters are studied and e-folds number is evaluated. The critical points of autonomous equations are analyzed. The function m (r) that describes the deviation from the Λ CDM model is calculated. [ABSTRACT FROM AUTHOR]