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The (pseudo-)Riemann-metrizability and Ricci-flatness of Finsler spaces with m-Kropina metric F = α1+mβ−m of Berwald type are investigated. We prove that the affine connection of F can locally be understood as the Levi–Civita connection of some (pseudo-)Riemannian metric if and only if the Ricci tensor of the canonical affine connection is symmetric. We also obtain a third equivalent characterization in terms of the covariant derivative of the 1-form β. We use these results to classify all locally metrizable m-Kropina spaces whose 1-forms have a constant causal character. In the special case where the first de Rham cohomology group of the underlying manifold is trivial (which is true of simply connected manifolds, for instance), we show that global metrizability is equivalent to local metrizability and hence, in that case, our necessary and sufficient conditions also characterize global metrizability. In addition, we further obtain explicitly all Ricci-flat, locally metrizable m-Kropina metrics in (3 + 1)D whose 1-forms have a constant causal character. In fact, the only possibilities are essentially the following two: either α is flat and β is α-parallel, or α is a pp-wave and β is α-parallel. [ABSTRACT FROM AUTHOR]

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply‐connected 6‐manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain 6k$6k$‐dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of 6k$6k$‐dimensional manifolds with a metric of positive Ricci curvature. [ABSTRACT FROM AUTHOR]

In this paper, firstly we study geodesic vectors for the m-th root homogeneous Finsler space admitting (α, β)-type. Then we obtain the necessary and sufficient condition for an arbitrary non-zero vector to be a geodesic vector for the m-th root homogeneous Finsler metric under mild conditions. Finally, we consider a quartic homogeneous Finsler metric on a simply connected nilmanifold of dimension five equipped with an invariant Riemannian metric and an invariant vector field. We study its geodesic vectors and classify the set of all the homogeneous geodesics on 5-dimensional nilmanifolds. [ABSTRACT FROM AUTHOR]

Given an autonomous second-order ordinary differential equation (ODE), we define a Riemannian metric on an open subset of the first-order jet bundle. A relationship is established between the solutions of the ODE and the geodesic curves with respect to the defined metric. We introduce the notion of energy foliation for autonomous ODEs and highlight its connection to the classical energy concept. Additionally, we explore the geometry of the leaves of the foliation. Finally, the results are applied to the analysis of Lagrangian mechanical systems. In particular, we provide an autonomous Lagrangian for a damped harmonic oscillator. [ABSTRACT FROM AUTHOR]

Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the |$k$| -th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär's theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric. [ABSTRACT FROM AUTHOR]

To a Riemannian manifold |$(M,g)$| endowed with a magnetic form |$\sigma $| and its Lorentz operator |$\Omega $| we associate an operator |$M^{\Omega }$|⁠ , called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric |$g$| together with terms of perturbation due to the magnetic interaction of |$\sigma $|⁠. From |$M^{\Omega }$| we derive the magnetic sectional curvature |$\textrm{Sec}^{\Omega }$| and the magnetic Ricci curvature |$\textrm{Ric}^{\Omega }$| that generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of |$\textrm{Ric}^{\Omega }$| being positive on an energy level below the Mañé critical value, with a Bonnet–Myers argument, we establish the existence of a contractible periodic orbit. In particular, when |$\sigma $| is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions that appear when one requires |$\textrm{Sec}^{\Omega }$| to be positive. [ABSTRACT FROM AUTHOR]

Given an open neighborhood of the zero section in the cotangent bundle of N we define a distance-like function ρ on N using certain symplectic embeddings from the standard ball B 2 n (r) to . We show that when is the unit-disk cotangent bundle of a Riemannian metric on N , ρ recovers the metric. As an intermediate step, we give a new construction of a symplectic embedding of the ball of capacity 4 to the product of Lagrangian disks P L : = B n (1) × B n (1) , and we give a new proof of the strong Viterbo conjecture about normalized capacities for P L . We also give bounds of the symplectic packing number of two balls in a unit-disk cotangent bundle relative to the zero section N. [ABSTRACT FROM AUTHOR]

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh's dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh's potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb's law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin's principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control. [ABSTRACT FROM AUTHOR]

AbstractManifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to examine harmonic and generalized magnetic vector fields as mappings from an anti-paraKählerian manifold to its associated tangent bundle, endowed with a ciconia metric. Initially, the conditions under which a vector field is harmonic (or magnetic) concerning a ciconia metric are investigated. Subsequently, the mappings between any given Riemannian manifold and the tangent bundle of an anti-paraKählerian manifold are explored. The paper delves into identifying the circumstances under which vector fields exhibit harmonicity or magnetism within the framework of a ciconia metric. Additionally, it explores the relationships between specific harmonic and magnetic vector fields, particularly emphasizing their behaviour under conformal transformations of metrics. [ABSTRACT FROM AUTHOR]

We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the "singular Yamabe problem". [ABSTRACT FROM AUTHOR]

We consider an analytic family of Riemannian metrics on a compact smooth manifoldM. We assume the Dirichlet boundary condition for the η-Laplacian and obtain Hadamardtype variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all Cr Riemannian metrics Mr on M, all eigenvalues of the η-Laplacian are generically simple, for 2 ≤ r - ∞. This implies the existence of a residual set of metrics in Mr that makes the spectrum of the η-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions η in the space Fr of all Cr functions on M, that again makes the spectrum of the η-Laplacian simple, for 2 ≤ r < ∞. Besides, we provide a precise information about the complement of these residual sets as well as about the structure of the set of deformations of a Riemannian metric (respectively, of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard-type formulas for the eigenvalues of the η-Laplacian in this case. We also establish generic properties of eigenvalues in this context. [ABSTRACT FROM AUTHOR]

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature. [ABSTRACT FROM AUTHOR]

In this paper we formulate and solve the initial-boundary value problem of accreting circular cylindrical bars under finite extension. We assume that the bar grows by printing stress-free cylindrical layers on its boundary cylinder while it is undergoing a time-dependent finite extension. Accretion induces eigenstrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar. This metric explicitly depends on the history of deformation during the accretion process. For a displacement-control loading during the accretion process we find the exact distribution of stresses. For a force-control loading, a nonlinear integral equation governs the kinematics. After unloading there are, in general, a residual stretch and residual stresses. For different examples of loadings we numerically find the axial stretch during loading, the residual stretch, and the residual stresses. We also calculate the stress distribution, residual stretch, and residual stresses in the setting of linear accretion mechanics. The linear and nonlinear solutions are numerically compared in a few accretion examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of Finsler metrics that generalize the well-known (α, β)-metrics. These metrics are defined by a Riemannian metric α and two 1-forms β = bi(x)yi and γ = γi (x)yi. This new class of metrics not only generalizes (α, β)-metrics, but also includes other important Finsler metrics, such as all (generalized) γ-changes of generalized (α, β)-metrics, (α, β)-metrics, and spherically symmetric Finsler metrics in Rn. We find a necessary and sufficient condition for this new class of metrics to be locally projectively flat. Furthermore, we prove the conditions under which these metrics are of Douglas type. [ABSTRACT FROM AUTHOR]

Consider a finite connected 2-complex X endowed with a piecewise Riemannian metric, and whose fundamental group is freely indecomposable, of rank at least 3, and in which every 2-generated subgroup is free. In this paper, we show that we can always find a connected graph Γ⊂X such that π1​ Γ≃F2 ​ ↪π1 X (in short, a 2-incompressible graph) whose length satisfies the following curvature-free inequality: ℓ(Γ)≤4 √2Area(X). This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence, we obtain that the volume entropy of such 2-complexes with unit area is always bounded away from zero. [ABSTRACT FROM AUTHOR]

We study the behavior of a three-dimensional dynamical system with respect to some set S given in 3-dimensional euclidean space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter a ∈ (0 , 1 / 2) , as for S it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that S is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and S for all a ∈ (0 , 1 / 2) . These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases a ∈ { 1 / 9 , 1 / 8 , 1 / 6 } of generalized Wallach spaces. [ABSTRACT FROM AUTHOR]

In the present paper sets related to invariant Riemannian metrics of positive sectional and (or) Ricci curvature on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the influence of the normalized Ricci flow. For invariant Riemannian metrics of the Wallach spaces which admit positive sectional curvature and belong to a given invariant surface of the normalized Ricci flow equation we establish that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces with coincided parameters the set of Riemannian metrics which belong to the invariant surface of the normalized Ricci flow and admit positive Ricci curvature is bounded by three space curves each consisting of exactly two connected components as regular curves. Mutual intersections and asymptotical behaviors of these components are studied as well. We also establish that curves corresponding to Ka¨hler metrics of spaces under consideration form separatrices of saddles of a three-dimensional system of nonlinear autonomous ordinary differential equations obtained from the normalized Ricci flow equation.

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory. [ABSTRACT FROM AUTHOR]

Let M be a closed differentiable manifold of dimension at least 3. Let \Lambda _0 (M) be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on M can have. We prove that for any k greater than or equal to \Lambda _0 (M), there exists a Riemannian metric on M such that its conformal Laplacian has exactly k negative eigenvalues. Also, we discuss upper bounds for \Lambda _0 (M). [ABSTRACT FROM AUTHOR]

If the small spheres of a Finslerian projective metric are quadrics, then it is a Riemannian projective metric of constant curvature. [ABSTRACT FROM AUTHOR]

Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric g α β on M such that Y β = g α β X α ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds. [ABSTRACT FROM AUTHOR]

Introduction: Brain computer interfaces (BCI), which establish a direct interaction between the brain and the external device bypassing peripheral nerves, is one of the hot research areas. How to effectively convert brain intentions into instructions for controlling external devices in real-time remains a key issue that needs to be addressed in brain computer interfaces. The Riemannian geometry-based methods have achieved competitive results in decoding EEG signals. However, current Riemannian classifiers tend to overlook changes in data distribution, resulting in degenerated classification performance in cross-session and/or cross subject scenarios. Methods: This paper proposes a brain signal decoding method based on Riemannian transfer learning, fully considering the drift of the data distribution. Two Riemannian transfer learning methods based log-Euclidean metric are developed, such that historical data (source domain) can be used to aid the training of the Riemannian decoder for the current task, or data from other subjects can be used to boost the training of the decoder for the target subject. Results: The proposed methods were verified on BCI competition III, IIIa, and IV 2a datasets. Compared with the baseline that without transfer learning, the proposed algorithm demonstrates superior classification performance. In contrast to the Riemann transfer learning method based on the affine invariant Riemannian metric, the proposed method obtained comparable classification performance, but is much more computationally efficient. Discussion: With the help of proposed transfer learningmethod, the Riemannian classifier obtained competitive performance to existingmethods in the literature. More importantly, the transfer learning process is unsupervised and time-efficient, possessing potential for online learning scenarios. [ABSTRACT FROM AUTHOR]

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C²-smoothable. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic symmetry and local behavior of positive solutions at infinity to the equation outside a bounded set in Rn, where Rn/3, Laplacian with asymptotically flat Riemannian metric g. We prove that the solution, at1, either converges to a fundamental solution of the Laplace operator on the Euclidean space, or is asymptotically close to a Fowler-type solution defined on Rn/3. [ABSTRACT FROM AUTHOR]

In this treatise, we discuss existence and uniqueness questions for parametric minimal surfaces in Riemannian spaces, which represent minimal graphs. We choose the Riemannian metric suitably such that the variational solution of the Riemannian Dirichlet integral under Dirichlet boundary conditions possesses a one-to-one projection onto a plane. At first we concentrate our considerations on existence results for boundary value problems. Then we study uniqueness questions for boundary value problems and for complete minimal graphs. Here we establish curvature estimates for minimal graphs on discs, which imply Bernstein-type results. [ABSTRACT FROM AUTHOR]

We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets—movie ratings, hyperspectral images, and high-dimensional functions—suggest that the proposed algorithms have better or favorably comparable performance to other candidates. [ABSTRACT FROM AUTHOR]

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list several properties of such foliations and propose two conjectures about the topological types of their leaves, both of which coincide with the classical Weinstein conjecture in the case of contact flows. We give positive partial results for our conjectures in particular cases—when the holonomy of the contact foliation preserves a Riemannian metric, for instance—extending already established results in the field of Contact Dynamics. [ABSTRACT FROM AUTHOR]

We developed a new Riemannian color difference metric based on Friele's line element and the results of psychophysical color discrimination experiments. Visual adaptation effects are incorporated into the model. This new color difference metric was validated against various data sets available in the literature. We found adequate agreement with experimentally determined threshold ellipsoids/ellipses and a better threshold predictability compared with other color difference metrics. The new Riemannian color difference metric was applied for the calculation of the color gamut volume and the maximum number of mutually discernible colors. [ABSTRACT FROM AUTHOR]

Let |$K$| be a smooth, origin-symmetric, strictly convex body in |${\mathbb{R}}^{n}$|⁠. If for some |$\ell \in \textrm{GL}(n,{\mathbb{R}})$|⁠ , the anisotropic Riemannian metric |$\frac{1}{2}D^{2} \left \Vert \cdot \right \Vert_{\ell K}^{2}$|⁠ , encapsulating the curvature of |$\ell K$|⁠ , is comparable to the standard Euclidean metric of |${\mathbb{R}}^{n}$| up-to a factor of |$\gamma> 1$|⁠ , we show that |$K$| satisfies the even |$L^{p}$| -Minkowski inequality and uniqueness in the even |$L^{p}$| -Minkowski problem for all |$p \geq p_{\gamma }:= 1 - \frac{n+1}{\gamma }$|⁠. This result is sharp as |$\gamma \searrow 1$| (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all |$\gamma < \infty $|⁠. In particular, whenever |$\gamma \leq n+1$|⁠ , the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold. [ABSTRACT FROM AUTHOR]

Interpolation between camera positions is a standard problem in computer graphics and can be considered the foundation of camera path planning. As the basis for a new interpolation method, we introduce a new Riemannian metric in camera space, which measures the 3D image flow under a small movement of the camera. Building on this, we define a linear interpolation between two cameras as shortest geodesic in camera space, for which we provide a closed‐form solution after a mild simplification of the metric. Furthermore, we propose a geodesic Catmull‐Rom interpolant for keyframe camera animation. We compare our approach with several standard camera interpolation methods and obtain consistently better camera paths especially for cameras with extremely varying scales. [ABSTRACT FROM AUTHOR]

This paper explores the groundbreaking contributions of Hermann von Helmholtz and Erwin Schrödinger to the geometry of color space ‐a 3D space that correlates color distances with perceptual differences. Drawing upon his expertise in non‐Euclidean geometry, physics, and psychophysics, Helmholtz introduced the first Riemannian line element in color space between 1891 and 1892, inaugurating a new line of research known as higher color metric, a term coined by Schrödinger in 1920. During his tenure at the University of Vienna, Schrödinger extensively worked on color theory and rediscovered Helmholtz's forgotten line element. In his 1920 papers titled "Grundlinien einer Theorie der Farbmetrik im Tagessehen," published in the Annalen der Physik, Schrödinger elucidated certain shortcomings in Helmholtz's model and proposed his refined version of the Riemannian line element. This study delves into this captivating chapter in the history of color science, emphasizing the profound impact of Helmholtz's and Schrödinger's work on subsequent research in color metrics up to the present day. [ABSTRACT FROM AUTHOR]

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]

We study Brownian motion on the space of distinct landmarks in Rd$\mathbb {R}^d$, considered as a homogeneous space with a Riemannian metric inherited from a right‐invariant metric on the diffeomorphism group. As of yet, there is no proof of long‐time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long‐time existence for configurations of exactly two landmarks, governed by a radial kernel. For low‐order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher‐order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations. [ABSTRACT FROM AUTHOR]

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter κ , for a complete Riemannian metric and a three-form H on a manifold M. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion H. When κ = 0 the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when H = 0 and κ > 0 the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form H. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining, in particular, the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton and leave as open the construction of a Heterotic soliton with non-constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization, and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients—including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries—are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature. Funding: X. Li was partially supported by National Key R&D Program of China [Grants 2020YFA0711900, 2020YFA0711901], the National Natural Science Foundation of China [Grants 12271107, 62141407], the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], the "Chenguang Program" by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission [Grant 19CG02], the Shanghai Science and Technology Program [21JC1400600]. Y. Luo and A. R. Zhang were partially supported by the National Science Foundation [CAREER-2203741]. [ABSTRACT FROM AUTHOR]

Given an autonomous second-order ordinary differential equation (ODE), we define a Riemannian metric on an open subset of the first-order jet bundle. A relationship is established between the solutions of the ODE and the geodesic curves with respect to the defined metric. We introduce the notion of energy foliation for autonomous ODEs and highlight its connection to the classical energy concept. Additionally, we explore the geometry of the leaves of the foliation. Finally, the results are applied to the analysis of Lagrangian mechanical systems. In particular, we provide an autonomous Lagrangian for a damped harmonic oscillator.

We prove the Perelman type W-entropy formula for the geodesic flow on the L 2 -Wasserstein space over a complete Riemannian manifold equipped with Otto's infinite dimensional Riemannian metric. To better understand the similarity between the W-entropy formula for the geodesic flow on the Wasserstein space and the W-entropy formula for the heat flow of the Witten Laplacian on the underlying manifold, we introduce the Langevin deformation of flows on the Wasserstein space over a Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over a Riemannian manifold, and can be regarded as the potential flow of the compressible Euler equation with damping on a Riemannian manifold. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and a compact Riemannian manifold, and prove the convergence of the Langevin deformation for c → 0 and c → ∞ respectively. Moreover, we prove the W-entropy-information formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. The rigidity theorems are proved for the W-entropy for the geodesic flow and the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the CD(0, m)-condition. Our results are new even in the case of Euclidean spaces and complete Riemannian manifolds with non-negative Ricci curvature. [ABSTRACT FROM AUTHOR]

We study the effective front associated with first-order front propagations in two dimensions (⁠|$n=2$|⁠) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous |${\mathbb{Z}}^{2}$| -periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics [ 4 , 6 ]. To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the literature of partial differential equations (PDE). Combining with the sufficiency result in [ 15 ], our result leads to a realization type conclusion: for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior. [ABSTRACT FROM AUTHOR]

A locally conformally product (LCP) structure on compact manifold M is a conformal structure c together with a closed, non-exact and non-flat Weyl connection D with reducible holonomy. Equivalently, an LCP structure on M is defined by a reducible, non-flat, incomplete Riemannian metric h D on the universal cover M ̃ of M , with respect to which the fundamental group π 1 (M) acts by similarities. It was recently proved by Kourganoff that in this case (M ̃ , h D) is isometric to the Riemannian product of the flat space ℝ q and an incomplete irreducible Riemannian manifold (N , g N). In this paper, we show that for every LCP manifold (M , c , D) , there exists a metric g ∈ c such that the Lee form of D with respect to g vanishes on vectors tangent to the distribution on M defined by the flat factor ℝ q , and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds). [ABSTRACT FROM AUTHOR]

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0, 2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes. [ABSTRACT FROM AUTHOR]

We show that every Finsler surface with isotropic main scalar and isotropic flag curvature is Riemannian or relatively constant Landsberg metric. Using it, we prove that every homogeneous Finsler surface with isotropic flag curvature and isotropic main scalar is Riemannian or locally Minkowskian. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-Riemannian quantity χ-curvature of (α, β)-metrics, a special class of Finsler metrics with Riemannian metric a and a 1-form β. We prove that every (a, β)-metric has a vanishing χ-curvature under certain conditions. [ABSTRACT FROM AUTHOR]

This paper presents a memory-efficient, first-order method for low multilinear rank approximation of high-order, high-dimensional tensors. In our method, we exploit the second-order information of the cost function and the constraints to suggest a new Riemannian metric on the Grassmann manifold. We use a Riemmanian coordinate descent method for solving the problem and also provide a global convergence analysis matching that of the coordinate descent method in the Euclidean setting. We also show that each step of our method with the unit step size is actually a step of the orthogonal iteration algorithm. Experimental results show the computational advantage of our method for high-dimensional tensors. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the 3-dimensional Lie group (ℍ2 × ℝ, g) where g is a left-invariant Riemannian metric and we determine the Ricci solitons and Ricci bi-conformal vector fields on it. [ABSTRACT FROM AUTHOR]

We study the Schrödinger equation in quantum field theory (QFT) in its functional formulation. In this approach, quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schrödinger time evolution on Wentzel–Kramers–Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real-time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT. [ABSTRACT FROM AUTHOR]

A contraction metric for a discrete-time dynamical system given by an autonomous iteration is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a fixed point and it is robust to small perturbations of the system, including those varying the position of the fixed point.In this paper, we prove a converse theorem, showing the existence of a contraction metric, which is the solution of a matrix-valued equation. We then develop a construction method based on meshless collocation to approximate the solution of this equation. Error estimates imply that a sufficiently close approximation is itself a contraction metric. The method is applied to several examples. [ABSTRACT FROM AUTHOR]