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Complex curved composite components often rely on multiple reference curve algorithms for path planning in automated fiber placement. However, the reference curves are typically manually drawn. Moreover, designing the reference curves follows an iterative planning-analysis-improvement process, which can be inefficient. A new approach for the automatic pre-analysis and optimized generation of reference curves for fiber placement is proposed in this paper to enhance the efficiency of reference curve analysis and generation. Firstly, a pre-analysis algorithm for reference curves based on triangular meshes is proposed. This algorithm analyzes the theoretical geodesic curvature and angular deviation of the path before its planning. Subsequently, a comprehensive evaluation index for reference curve generation is formulated based on the pre-analysis algorithm, and the reference curve is optimized using genetic algorithms. The results demonstrate that the pre-analysis algorithm accurately computes the steering radius distribution of the path. Areas with over-limit steering radius can be eliminated while maintaining angular deviations within 10° by utilizing optimized reference curves for path planning. [ABSTRACT FROM AUTHOR]

This paper is a case study and historical account of nineteenth-century surveying of the Texas and Pacific Railway Eighty-Mile Reserve, and the consequences from ignoring curvature of the earth allowances in large surveys extending beyond the horizon. The methods used in the survey of the Eighty-Mile Reserve in Texas are compared to the way surveyors of the Public Lands of the United States carried out their work by making allowances for the curvature of the earth. When Texas became a republic in 1836, it was rich in land extending as far north as present-day Wyoming and Colorado, but financially broke. The first governmental agency in the Republic of Texas was the General Land Office and its instructions in the survey of the Texas public domain required boundary corners to be square, when practicable. The first land grants in Texas were very large, with no formal system of surveying to make allowances for the curvature of the earth. Consequently in later years, large errors were found, creating boundary misclosures, overlaps, gaps, and huge excesses in acreage over what was patented; and vacancies of unsurveyed, unsold land were not listed in the General Land Office as permanent school fund lands. When Texas was annexed by the United States in 1845, it still had enormous outstanding debts. By 1850, the State of Texas sold 67-million-acres (37,113,938 ha) of its public domain to the United States in order to clear its public debts. Ignoring curvature of the earth corrections in surveys continued until around 1930 when prominent surveyors, geodesists, and engineers convinced the Texas General Land Office to embrace a Texas coordinate system that would project a survey onto a flat map and accurately represent the surface of the earth in the least distorted way by creating the state plane coordinate system of Texas. [ABSTRACT FROM AUTHOR]

We prove that a globally hyperbolic smooth spacetime endowed with a C1-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even C1,1, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics. [ABSTRACT FROM AUTHOR]

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit partial differential equation (PDE), in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards Evolutional Variational Inequality flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time. [ABSTRACT FROM AUTHOR]

We continue the study of the analogue of Thurston's metric on the Teichmüller space of Euclidean triangles which was started by Saglam–Papadopoulos (Minimal stretch maps between Euclidean triangles, 2022). By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace of acute triangles. We deduce from the form of the Finsler unit sphere a result on the infinitesimal rigidity of the metric. We give a description of the maximal stretching loci for a family of extreme Lipschitz maps. [ABSTRACT FROM AUTHOR]

It is commonly accepted that superluminal travel may be used to facilitate time travel. This is a purely special-relativistic argument, using the fact that for observers in two frames of reference, separated by a spacelike interval, the non-causal (spacelike) future of one observer includes part of the causal past of the other. In this paper we provide a concrete realization of this argument in a curved general-relativistic spacetime, using warp drives as the means of faster-than-light travel. By generalizing the usual warp drive metric to allow for a non-unit lapse function, we allow the warp drive to switch between reference frames in a purely geometric way. With an additional modification allowing the warp drive to have compact support, this permits us to glue two warp drives together to construct a closed timelike geodesic, such that a test particle following the geodesics of the two warp drives travels back to its own past. This provides a precise mathematical model for the connection between faster-than-light travel and time travel in general relativity, and the first such model to be explicitly formulated using two warp drives. We also give a detailed discussion of weak energy condition violations in the non-unit-lapse warp drive. [ABSTRACT FROM AUTHOR]

In this work, we mainly focus on testing the spacetime properties around black holes surrounded by a dark fluid, which are potential candidates for dark energy described by the Chaplygin-like equation of state through its shadow. To do this, we first study the black hole's horizon structure and shadow for the non-rotating case. Then, we obtain a rotating black hole solution in the presence of a dark fluid using the generalized Newman–Janis algorithm and study the effects of the black hole spin and the fluid parameters on the black hole horizons. Also, we obtained the shadow cast of the rotating black hole using celestial coordinates and showed that the presence of the dark fluid causes an increase in shadow size. Moreover, we use the shadow size of supermassive black holes Sagittarius A* and M87* from Event Horizon Telescope observations to obtain constraints on the spin, black hole charge, and dark fluid parameters. Lastly, we investigate the energy emission rate of a charged black hole surrounded by a Chaplygin-like dark fluid, comparing it to both rotating and non-rotating cases. [ABSTRACT FROM AUTHOR]

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective. [ABSTRACT FROM AUTHOR]

We define compactness of a gravitational lens as the scaled closest distance of approach (i.e., r0/M) of the null geodesic giving rise to an image. We model 40 supermassive dark objects as Schwarzschild lenses and compute compactness of lenses (determined by the formation of the first-order relativistic image). We then obtain a novel formula for the compactness of a lens as a function of mass to the distance ratio (M/Dd) and the ratio of lens–source to the observer–source distances (Dds/Ds). This formula yields a very important result: Just an observation of a relativistic image would give an incredibly accurate upper bound to the physical compactness (the ratio of the radius to mass) of the lens without having any knowledge of mass of the lens, angular source position, and observer–source and lens–source distances. Similarly, we show that the observation of the second-order relativistic image would give a lower value of upper bound to the physical compactness. These results, though obtained for supermassive dark objects at galactic centers, are valid for any object compact enough to give rise to relativistic images. [ABSTRACT FROM AUTHOR]

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C^2 open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli [Duke Math. J. 173 (2024), pp. 347–390]. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are C^2 stably ergodic if and only if they are Anosov. [ABSTRACT FROM AUTHOR]

We classify and enumerate all rational numbers with approximation constant at least 1 3 using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions , which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed. [ABSTRACT FROM AUTHOR]

We construct compact Lorentz manifolds without closed geodesics. [ABSTRACT FROM AUTHOR]

We introduce refined Morse gauges to correct the proof and statement of Lemma 2.10 of [Groups Geom. Dyn. 11, 1281–1306 (2017)] written by the first author. [ABSTRACT FROM AUTHOR]

We consider the 3-sphere S³ seen as the boundary at infinity of the complex hyperbolic plane H²C. It comes equipped with a contact structure and two classes of special curves. First, R-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, C-circles are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere S³ is near to being an R-circle. We analyse the classical foliation of the complement of an R-circle by arcs of C-circles. Next, we consider deformations of this situation where the R-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of PU.2; 1/. As an application, we describe a class of spherical CR uniformisations of certain cusped 3-manifolds. [ABSTRACT FROM AUTHOR]

This study explores the deflection angle of photon rays or light-like geodesics within the framework of Eddington-inspired Born–Infeld (EiBI) gravity background space-time, taking into account the influence of cosmic strings. The primary focus lies in deriving the effective potential of the system applicable to both null and time-like geodesics, as well as determining the angle of deflection for light-like geodesics. Our analysis shows that the presence of cosmic strings induces modifications in these physical quantities, leading to shifts in their respective values. [ABSTRACT FROM AUTHOR]

In this contribution, we use Gaussian posterior probability densities to characterize local estimates from distributed sensors, and assume that they all belong to the Riemannian manifold of Gaussian distributions. Our starting point is to introduce a proper Lie algebraic structure for the Gaussian submanifold with a fixed mean vector, and then the average dissimilarity between the fused density and local posterior densities can be measured by the norm of a Lie algebraic vector. Under Gaussian assumptions, a geodesic projection based algebraic fusion method is proposed to achieve the fused density by taking the norm as the loss. It provides a robust fixed point iterative algorithm for the mean fusion with theoretical convergence, and gives an analytical form for the fused covariance matrix. The effectiveness of the proposed fusion method is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, one obtains rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, this leads to a classification of totally geodesic subvarieties of dimension at least two in strata with at most two zeros. [ABSTRACT FROM AUTHOR]

In this paper, we explore the characteristics of two novel regular spacetimes that exhibit a nonzero vacuum energy term, under the form of a (quasi) anti-de Sitter phase. Specifically, the first metric is spherical, while the second, derived by applying the generalized Newman–Janis algorithm to the first, is axisymmetric. We show that the equations of state of the effective fluids associated with the two metrics asymptotically tend to negative values, resembling quintessence. In addition, we study test particle motions, illustrating the main discrepancies among our models and more conventional metrics exhibiting non-vanishing anti-de Sitter phase. [ABSTRACT FROM AUTHOR]

A covariant, global, variational framework for perturbations in field theories is presented. Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions. The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series with respect to the flow parameter, hence it contains perturbations at any order. Mechanics is included as a special case. As a simple application, we recover the well-known discussion about stability of geodesics on a sphere S 2 . [ABSTRACT FROM AUTHOR]

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian 2-spheres. On any Riemannian 2-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index 1, 2 and 3. In particular, for an orientable Riemannian surface, we prove strong Morse inequalities for the length functional applied to the space of simple closed curves. [ABSTRACT FROM AUTHOR]

The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces (M, F), that is, F (x , y) ⟼ F ― (x , y) = e ϕ (x , y) F (x , y) , where the function ϕ (x , y) depends on both position x and direction y, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If F is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface (M, F) to be transformed to a (conic) pseudo-Finsler surface (M , F ―) under the transformation F ― = e ϕ (x , y) F . In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function F ― . In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor ϕ (x , y) that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for F ― to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor ϕ (x , y) . Various examples are provided whenever the situation needs. [ABSTRACT FROM AUTHOR]

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss–Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic. [ABSTRACT FROM AUTHOR]

Geodesic domes are structures which deliver effective solutions, associated with cost savings, due to the lack of intermediate supports when roofing large-sized objects. The multitude of advantages of this type of construction was translated into the shaping of innovative lightweight objects of geodesic domes, constructed on a regular octahedron. In this article, the use of strut sections was applied to covers generated due to a regular octahedron. Two families of domes were compared, resulting from the use of two methods that differ from each other in their topology. Each family consist of 8 structures thus finally 16 geodesic domes were considered. The generated domes were compared in terms of the same section for all strut elements, and the optimized section for each group of struts. To find the design focused on obtaining optimal solution, a number of comparative analysis were carried out. The presented analysis is extremely important in the context of environmental impact, because it shows the steel consumption and the sense of optimization. It was indicated that by optimizing the cross-sections of the strut elements, the steel consumption, as well as their weight, decreased by approx. 10–25% for dome structures which were created using method 1 and 20–40% for domes generated using method 2. The article aims to provide suggestion on the selection of the appropriate innovative geodesic dome mesh. A mesh based on various methods of shaping the covers of geodesic domes created using a regular octahedron. [ABSTRACT FROM AUTHOR]

This paper aims to study the Berger type deformed Sasaki metric gBS on the second order tangent bundle T2M over a bi-Kählerian manifold M. The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric gBS and calculate all forms of Riemannian curvature tensors of this metric. Also, they study geodesics on the second order tangent bundle T2M and bi-unit second order tangent bundle T 1 , 1 2 M , and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base. Finally, they present some conditions for a section σ: M → T2M to be harmonic and study the harmonicity of the different canonical projections and inclusions of (T2M, gBS). Moreover, they search the harmonicity of the Berger type deformed Sasaki metric gBS and the Sasaki metric gS with respect to each other. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optical appearance of a regular static spherically symmetric hairy black hole within the context of Minkowski deformation governed by the parameter α . This black hole describes a hairy black hole with geometric deformations in the radial and temporal metric components, parameterized by α . The optical appearance of the black hole, illuminated by a static thin accretion disk in three toy emission function models, exhibits distinctive shadows and photon rings. Our findings reveal that for static spherically symmetric hairy Minkowski-deformed regular black holes, the event horizon radius r h , photon sphere radius r ph , critical impact parameter b ph , and innermost stable circular orbit radius r isco all have a positive correlation with α . These parameters affect the null geodesic trajectories, shadows, and the optical appearance of the photon rings illuminated by a static thin accretion disk around the black hole. Utilizing this, we obtained the optical appearance of this black hole when observed in the forward direction of the optical and geometric thin accretion disk models. In all three models, the radius of the photon ring images also shows a positive correlation with the parameter α . Therefore, under the static spherically symmetric hairy Minkowski-deformed regular black hole, there is no degeneracy of photon rings dependent on the parameter α . Theoretically, this allows for the distinction of different spacetime metrics of hairy Minkowski-deformed regular black holes, providing a potential method for testing the no-hair theorem through future observations of black hole photon rings. [ABSTRACT FROM AUTHOR]

Velocity, density, and magnetic fluctuations of the geodesic acoustic mode (GAM) have been measured using the Doppler backscattering system, Faraday-effect polarimeter-interferometer, and external pick-up coils in the Experimental Advanced Superconducting Tokamak. Simultaneous measurements of density and velocity fluctuations at the midplane and top of plasmas demonstrate that m = 1 density fluctuations are quantitatively balanced by the compression of perpendicular flow fluctuations. Furthermore, internal magnetic fluctuations associated with GAM have now been directly measured by laser-based Faraday-effect polarimetry for the first time. Line-averaged magnetic fluctuations (up to 16 Gauss, B ̃ ¯ R , GAM B T ∼ 0.066 %) are significantly larger than those extrapolated from edge coils (a few Gauss) and that magnetic fluctuations increase with β. The observed discrepancy between finite β theory and experimental data indicates the need for further theoretical investigations. [ABSTRACT FROM AUTHOR]

We have studied the packing of congruent disks on a spherical cap, for caps of different size and number of disks, N. This problem has been considered before only in the limit cases of circle packing inside a circle and on a sphere (Tammes problem), whereas all intermediate cases are unexplored. Finding the preferred packing configurations for a domain with both curvature and border could be useful in the description of physical and biological systems (for example, colloidal suspensions or the compound eye of an insect), with potential applications in engineering and architecture (e.g., geodesic domes). We have carried out an extensive search for the densest packing configurations of congruent disks on spherical caps of selected angular widths ( θ max = π / 6 , π / 4 , π / 2 , 3 π / 4 , and 5 π / 6) and for several values of N. The numerical results obtained in the present work have been used to establish (at least qualitatively) some general features for these configurations, in particular the behavior of the packing fraction as function of the number of disks and of the angular width of the cap, or the nature of the topological defects in these configurations (it was found that as the curvature increases, the overall topological charge on the border tends to become more negative). Finally, we have studied the packing configurations for N = 19 , 37, 61, and 91 (hexagonal numbers) for caps ranging from the flat disk to the whole sphere, to observe the evolution (and eventual disappearance) of the curved hexagonal packing configurations while increasing the curvature. [ABSTRACT FROM AUTHOR]

Studying the Gödel universe as a Lie group with left-invariant Lorentz metric and using the methods of geometric theory of the optimal control for the search of geodesics on Lie groups with left-invariant (sub-)Lorentz metrics, we find the expressions for timelike and isotropic geodesics in elementary functions. Also, we pay special attention to proving that the Gödel universe has no closed timelike or isotropic geodesics. [ABSTRACT FROM AUTHOR]

Nonlinear electrodynamics has been frequently employed in the study of black holes. Some of these black holes may be regular, while others are singular. In this work, we consider a nonlinear electrodynamics model known as inverse electrodynamics to obtain black hole solutions. We demonstrate that, in addition to these solutions not being regular, they are also not asymptotically flat. We investigate which energy conditions are violated in the presence of this type of source. Furthermore, we calculate thermodynamic quantities and find that there are two phases, with one of them being thermodynamically stable. Finally, we derive the geodesics for massive and massless particles in this spacetime. For the massive case, we observe that there are no bound orbits. [ABSTRACT FROM AUTHOR]

This note discusses the computation of the distance function with respect to Finsler metrics. To this end, we show how the Finsler variants of the Eikonal equation can be solved by a primal-dual algorithm exploiting the variational structure. We also discuss the acceleration of the algorithm by preconditioning techniques, and illustrate the flexibility of the proposed method through a series of numerical examples. [ABSTRACT FROM AUTHOR]

This paper is committed to studying Karush-Kuhn-Tucker (in short, KKT) type necessary and sufficient optimality conditions for non-smooth quasi-convex (geodesic sense) optimization problems on Riemannian manifolds. Recently, Ansari et al. [Ansari QH, Babu F, Zeeshan M. Incremental quasi-subgradient method for minimizing geodesic quasi-convex function on Riemannian manifolds with applications. Numer Funct Anal Optim. 2022;42(13):1492–1521. doi: ] defined the quasi-subdifferential on Riemannian manifolds and established the existence results of the quasi-subdifferential. We provide several auxiliary results for the quasi-subdifferential in the current study. We offer the KKT optimality conditions for the quasi-convex optimization problems on Riemannian manifolds with or without the Slater-constraint qualifications. To verify the suggested outcomes, we formulate numerical examples. In addition, we also provide our results in the Euclidean spaces, which are original and distinct from earlier findings in the Euclidean spaces. [ABSTRACT FROM AUTHOR]

Centrality measures are scalar values given to each vertex in the graph to quantify its importance based on an assumption. Stress and tension are the two centrality measures which depends on shortest paths. Stress of a vertex v is the number of shortest paths passing through vertex v whereas, tension on an edge e is the number of shortest paths passing through the edge e. In this paper, authors obtain formulae for the evaluation of stress and tension of certain graphs and graph operations with diameter less than or equal to two. Also obtain tension of some wheel related graphs such as gear graph, helm graph, flower graph and sunflower graph. [ABSTRACT FROM AUTHOR]

For every complete and minimally immersed submanifold f : M n → S n + p whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / (2 ⁢ p − 1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some results about almost Yamabe soliton and almost Ricci-Bourguignon soliton with special soliton vector field. In fact, we prove that every compact non-trivial almost Ricci-Bourguignon soliton with constant scalar curvature is isometric to a Euclidean sphere. Then we show that every compact almost Ricci-Bourguignon soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Finally, we prove that a complete almost Ricci-Bourguignon soliton (M, g, V, λ, ρ) has V as the contact vector field of a contact manifold M with metric g and its Reeb vector field is geodesic, then it becomes a Ricci-Bourguignon soliton and g has constant scalar curvature. In particular, if V is strict, then g is a compact Sasakian Einstein. [ABSTRACT FROM AUTHOR]

We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows. [ABSTRACT FROM AUTHOR]

In this paper, for a given spray S on an n-dimensional manifold M, we investigated the geometry of S-invariant functions. For an S-invariant function P, we associated a vertical subdistribution VP and found the relation between the holonomy distribution and VP by showing that the vertical part of the holonomy distribution is the intersection of all spaces VFS associated with FS where FS is the set of all Finsler functions that have the geodesic spray S . As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with S-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is S-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is H-invariant if and only if it is constant. [ABSTRACT FROM AUTHOR]

Let G be a graph and X ⊆ V (G). Then X is a mutual-visibility set if each pair of vertices from X is connected by a geodesic with no internal vertex in X. The mutual-visibility number μ (G) of G is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given. [ABSTRACT FROM AUTHOR]

We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in ℝ². The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point (0, 0). In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus. [ABSTRACT FROM AUTHOR]

The research presented in this paper concerns the determination of the attraction basins of Newton's iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton's method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton's method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton's method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points. [ABSTRACT FROM AUTHOR]

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Robertson–Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the cosmological expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow. [ABSTRACT FROM AUTHOR]

The non-abelian Hodge correspondence maps a polystable |$\textrm{SL}(2, {\mathbb{R}})$| -Higgs bundle on a compact Riemann surface |$X$| of genus |$g \geq 2$| to a connection that, in some cases, is the holonomy of a branched hyperbolic structure. Gaiotto's conformal limit maps the same bundle to a partial oper, that is, to a connection whose holonomy is that of a branched complex projective structure compatible with |$X$|⁠. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with |$X$|⁠. We also show that, for zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichmüller's space. [ABSTRACT FROM AUTHOR]

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop. [ABSTRACT FROM AUTHOR]

Bratteli–Vershik models of compact, invertible zero-dimensional systems have been well studied. We take up such a study for polygonal billiards on the hyperbolic plane, thus considering these models beyond zero-dimensions. We describe the associated Bratteli models and show that these billiard dynamics can be described by Vershik maps. [ABSTRACT FROM AUTHOR]

In this research, we investigate the intriguing realm of pointwise bislant Riemannian submersions, a generalization of many previous submersions, such as antiinvariant, slant, semislant, pointwise slant, pointwise semislant, and bislant submersions, within the framework of almost product manifolds. After giving an original example, we delve into the submersion’s integrability conditions and geodesics. We explore the concept of φ−pluriharmonicity and φ−invariance within this context. The study sheds light on the profound interplay between pointwise bislant submersions’ fibers and their being either geodesic or mixed geodesic, offering valuable insights into these intriguing mappings’ geometric properties. [ABSTRACT FROM AUTHOR]

In this paper we use the Homotopy analysis method to obtain an analytic approximation for the entire photon trajectory in the Schwarzschild spacetime. This is usually expressed exactly in terms of an elliptic integral. We compare our approximation with other formulae found in the literature, which were specifically obtained for the Schwarzschild solution. Unlike some of these formulae, our approximation can be applied and maintains a good accuracy for emission point close to the event horizon and also for emission angles close to and greater than π / 2 . We show that our method can easily be applied to other spherically symmetric solutions such as the Reissner-Nordström solution. Such an approximation would be useful when accurate determination of the light trajectories around compact objects is required without the need to revert to time consuming numerical integration of elliptic integrals. [ABSTRACT FROM AUTHOR]

The smoothing of surface curves is an essential tool in mesh processing, important to applications that require segmenting and cutting surfaces such as surgical planning. Surface curves are typically designed by professionals to match certain surface features. For this reason, the smoothed curves should be close to the original and easily adjustable by the user in interactive tools. Previous methods achieve this desired behavior, e.g., by utilizing energy‐minimizing splines or generalizations of Bézier splines, which require a significant number of control points and may not provide interactive frame rates or numerical stability. This paper presents a new algorithm for robust smoothing of discrete surface curves on triangular surface meshes. By using a scalar penalty potential as the fourth coordinate, the given surface mesh is embedded into the 4D Euclidean space. Our method is based on finding geodesics in this lifted surface, which are then projected back onto the original 3D surface. The benefits of this approach include guaranteed convergence and good approximation of the initial curve. We propose a family of penalty potentials with one single parameter for adjusting the trade‐off between smoothness and similarity. The implementation of our method is straightforward as we rely on existing methods for computing geodesics and penalty fields. We evaluate our implementation and confirm its robustness and efficiency. [ABSTRACT FROM AUTHOR]

The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime. [ABSTRACT FROM AUTHOR]

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings. [ABSTRACT FROM AUTHOR]