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In this article, we present a new Cucker–Smale model on smooth Riemannian manifolds using the concepts of covariant derivative and parallel transport, and we also study its emergent dynamics under an a priori assumption on the energy functional. For Euclidean space, our proposed model coincides with the original Cucker–Smale model. As concrete examples, we consider three Riemannian manifolds: the unit 2-sphere, the unit circle, and the Poincare half-plane, and provide explicit reductions from the proposed general model to aforementioned manifolds via explicit formulas for the covariant derivative and parallel transport.

We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group $G$, we prove that the smooth 2-group extensions of $G$ arising from our construction provide new models for the string group of $G$., Comment: 79 pages, 2 figures; v2: minor corrections, comments and references added; Final version to be published in Communications in Mathematical Physics

First-order non-convex Riemannian optimization algorithms have gained recent popularity in structured machine learning problems including principal component analysis and low-rank matrix completion. The current paper presents an efficient Riemannian Stochastic Path Integrated Differential EstimatoR (R-SPIDER) algorithm to solve the finite-sum and online Riemannian non-convex minimization problems. At the core of R-SPIDER is a recursive semi-stochastic gradient estimator that can accurately estimate Riemannian gradient under not only exponential mapping and parallel transport, but also general retraction and vector transport operations. Compared with prior Riemannian algorithms, such a recursive gradient estimation mechanism endows R-SPIDER with lower computational cost in first-order oracle complexity. Specifically, for finite-sum problems with $n$ n components, R-SPIDER is proved to converge to an $\epsilon$ e -approximate stationary point within $\mathcal {O}\big (\min \big (n+\frac{\sqrt{n}}{\epsilon ^2},\frac{1}{\epsilon ^3}\big)\big)$ O min n + n e 2 , 1 e 3 stochastic gradient evaluations, beating the best-known complexity $\mathcal {O}\big (n+\frac{1}{\epsilon ^4}\big)$ O n + 1 e 4 ; for online optimization, R-SPIDER is shown to converge with $\mathcal {O}\big (\frac{1}{\epsilon ^3}\big)$ O 1 e 3 complexity which is, to the best of our knowledge, the first non-asymptotic result for online Riemannian optimization. For the special case of gradient dominated functions, we further develop a variant of R-SPIDER with improved linear rate of convergence. Extensive experimental results demonstrate the advantage of the proposed algorithms over the state-of-the-art Riemannian non-convex optimization methods.

The objective of the paper is to contribute to the theory of error-based control systems on Riemannian manifolds. The present study focuses on system where the control field influences the covariant derivative of a control path. In order to define error terms in such systems, it is necessary to compare tangent vectors at different points using parallel transport and to understand how the covariant derivative of a vector field along a path changes after such field gets parallely transported to a different curve. It turns out that such analysis relies on a specific map, termed principal pushforward map. The present paper aims at contributing to the algebraic theory of the principal pushforward map and of its relationship with the curvature endomorphism of a state manifold.

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real., 28 pages, 16 figures, 2 tables

In this article, we consider the Schrodinger flow of maps from two dimensional hyperbolic space \begin{document}$ {{\mathbb{H}}}^2 $\end{document} to sphere \begin{document}$ {{\mathbb{S}}}^2 $\end{document} . First, we prove the local existence and uniqueness of Schrodinger flow for initial data \begin{document}$ u_0\in\mathbf{H}^3 $\end{document} using an approximation scheme and parallel transport introduced by McGahagan [ 32 ]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrodinger flow to that of a system of coupled Schrodinger equations with potentials. Then we prove the global existence of equivariant Schrodinger flow for small initial data \begin{document}$ u_0\in\mathbf{H}^1 $\end{document} by Strichartz estimates and perturbation method.

The dynamics of a closed quantum system is often studied with the direct evolution of the Schrodinger equation. In this paper, we propose that the gauge choice (i.e. degrees of freedom irrelevant to physical observables) of the Schrodinger equation can be generally non-optimal for numerical simulation. This can limit, and in some cases severely limit the time step size. We find that the optimal gauge choice is given by a parallel transport formulation. This parallel transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The parallel transport dynamics can be derived from a Hamiltonian structure, thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the parallel transport dynamics in the context of the singularly perturbed linear Schrodinger equation, and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrodinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse., Comment: SIAM Multiscale Model. Simul. accepted

Optimizing Hungarian parallel public transport is an actual and returning question, which needs to be investigated from the customers, operators, and the users point of view. Previous works of authors are based on the analysis of supply, present topic concentrates on travelers mode choice in long distance public transport. This research is based on a novel survey method, called stated preference, which questionnaire method uses real data, puts the interviewed travelers in real traffic situations. The passengers have to make decisions according to their own preference in mode choice. Present research highlights the sensitivity of users – depending on motivation and travel distance – of travel time, travel costs, and comfort in different public transport modes. The analysis is the next chapter of the research in parallel transport.

International audience; This paper presents a derivation of the parallel transport equation expressed in the Lie algebra of a Lie group endowed with a left-invariant metric.The use of this equation is exemplified on the group of rigid body motions SE(3), using basic numerical integration schemes, and compared to the pole ladder algorithm. This results in a stable and efficient implementation of parallel transport. The implementation leverages the python package geomstats and is available online.

Weyl particles exhibit chiral transport property under external curved space-time geometry. This effect is called the chiral gravitational effect, which plays an important role in quantum field theory. However, the absence of real Weyl particles in nature hinders the observation of such interesting phenomena. In this paper, we show that chiral gravitational effect can be manifested in Weyl metamaterials with spatially controlled nonlocality. This inhomogeneous modulation results in a spatially dependent group velocity in the Weyl cone dispersion, which is equivalent to introducing a curved background space-time (or gravitational field) for Weyl pseudospinors according to the Clifford algebra. The Weyl equation describing the Weyl cone dispersions is reformulated into the covariant form that contains a nonAbelian gauge field by considering the parallel transport of pseudospinors. The synthetic gravitational field (i.e., space-time curvature) leads to the quantization of energy levels, including chiral zeroth order energy modes (or simply chiral zero modes) that determine the chiral transport property of pseudospinors. Inhomogeneous Weyl metamaterials can potentially become a platform for investigating the interaction between Weyl particles and gravitational fields in future table-top experiments. The theoretical model and the associated results may serve as a guidance for experimentalists to observe such chiral transport effects in photonic Weyl systems.

We investigate a simple and robust scheme for choosing the phases of adiabatic electronic states smoothly (as a function of geometry) so as to maximize the performance of ab initio non-adiabatic dynamics methods. Our approach is based upon consideration of the overlap matrix (U) between basis functions at successive points in time and selecting the phases so as to minimize the matrix norm of log(U). In so doing, one can extend the concept of parallel transport to cases with sharp curve crossings. We demonstrate that this algorithm performs well under extreme situations where dozens of states cross each other either through trivial crossings (where there is zero effective diabatic coupling), or through non-trivial crossings (when there is a non-zero diabatic coupling), or through a combination of both. In all cases, we compute the time-derivative coupling matrix elements (or equivalently non-adiabatic derivative coupling matrix elements) that are as smooth as possible. Our results should be of interest to all who are interested in either non-adiabatic dynamics, or more generally, parallel transport in large systems.

We propose a general notion of parallel transport on RCD spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space. peerReviewed

We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.

We present a unified method to simulate deformable elastic bodies consisting of mixed-dimensional components represented with potentially non-manifold simplicial meshes. Building on well-known simplicial rod, shell, and solid models for elastic continua, we categorize and define a comprehensive palette expressing all possible constraints and elastic energies for stiff and flexible connections between the 1D, 2D, and 3D components of a single conforming simplicial mesh. This palette consists of three categories: point connections, in which simplices meet at a single vertex around which they may twist and bend; curve connections in which simplices share an edge around which they may rotate (bend) relative to one another; and surface connections, in which a shell is embedded on or into a solid. To define elastic behaviors across non-manifold point connections, we adapt and apply parallel transport concepts from elastic rods. To address discontinuous forces that would otherwise arise when large accumulated relative rotations wrap around in the space of angles, we develop an incremental angle-update strategy. Our method provides a conceptually simple, flexible, and highly expressive framework for designing complex elastic objects, by modeling the geometry with a single simplicial mesh and decorating its elements with appropriate physical models (rod, shell, solid) and connection types (point, curve, surface). We demonstrate a diverse set of possible interactions achievable with our method, through technical and application examples, including scenes featuring complex aquatic creatures, children's toys, and umbrellas.

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold ℳ = M × 𝕊 N × I {\mathcal{M}=M\times\mathbb{S}^{N}\times I} , whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for N → ∞ {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on ℳ {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle 𝒪 ⁢ ℳ {\mathcal{OM}} . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fr\'{e}chet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing., Comment: 19 pages

Three dimensional (3D) distribution of impurity emission intensity is obtained from two dimensional (2D) image using newly developed single field-of-view tomography scheme. Spectrometer with multi-channel fiber array views the entire edge region of stochastic magnetic layer of LHD, including divertor plates, divertor legs, the stochastic layer, and the last closed flux surface. The tomography scheme introduces new regularization terms in Lagrangian function, based on the transport feature in magnetized plasma that the emission assumed to be smooth along the magnetic field due to the very fast parallel transport compared to the transport across the magnetic field. Feasibility of the scheme is checked with test emission distributions. The scheme is applied to the experimental data in LHD, where the impurity emission distribution of different charge states and of different electron densities are obtained in relation to the magnetic field structure in the edge stochastic layer.

Brain atrophy as measured from structural MR images, is one of the primary imaging biomarkers used to track neurodegenerative disease progression. In diseases such as frontotemporal dementia or Alzheimer's disease, atrophy can be observed in key brain structures years before any clinical symptoms are present. Atrophy is most commonly captured as volume change of key structures and the shape changes of these structures are typically not analysed despite being potentially more sensitive than summary volume statistics over the entire structure. In this paper we propose a spatiotemporal analysis pipeline based on Large Diffeomorphic Deformation Metric Mapping (LDDMM) to detect shape changes from volumetric MRI scans. We applied our framework to a cohort of individuals with genetic variants of frontotemporal dementia and healthy controls from the Genetic FTD Initiative (GENFI) study. Our method, take full advantage of the LDDMM framework, and relies on the creation of a population specific average spatiotemporal trajectory of a relevant brain structure of interest, the thalamus in our case. The residuals from each patient data to the average spatiotemporal trajectory are then clustered and studied to assess when presymptomatic mutation carriers differ from healthy control subjects. We found statistical differences in shape in the anterior region of the thalamus at least five years before the mutation carrier subjects develop any clinical symptoms. This region of the thalamus has been shown to be predominantly connected to the frontal lobe, consistent with the pattern of cortical atrophy seen in the disease., Graphical abstract Image 1, Highlights • Clustering shape parametrisation allows local shape analysis. • Thalamic shape changes appear 5 years before onset of fronto temporal dementia. • Shape changes seem to occur before volume changes. • Pre-symptomatic shape changes in thalamus are dorsofrontal, where connecting to temporal lobes.

We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Fr\'echet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form., Comment: 19 pages, 1 figure

In this paper, we present the Fermi-Walker parallel transport and the generalized Fermi-Walker parallel transport according to quasi frame in three dimensional Minkowski space.

New analytical expressions for parallel transport coefficients in multicomponent collisional plasmas are presented in this paper. They are improved versions of the expressions written in [V. M. Zhdanov. Transport Processes in Multicomponent Plasma, vol. 44. 10 2002.], based on Grad's 21N-moment method. Both explicit and approximate approaches for transport coefficients calculation are considered. Accurate application of this closure for the Braginskii transport equations is discussed. Viscosity dependence on the heat flux is taken into account. Improved expressions are implemented into the SOLPS-ITER code and tested for deuterium and neon ITER cases. Some typos found in [V. M. Zhdanov. Transport Processes in Multicomponent Plasma, vol. 44. 10 2002.] are corrected., This is the pre-submission version. The published version can be found using DOI

Studying the changes of shape is a common concern in many scientific fields. We address here two problems: (1) quantifying the deformation between two given shapes and (2) transporting this deformation to morph a third shape. These operations can be done with or without point correspondence, depending on the availability of a surface matching algorithm, and on the type of mathematical procedure adopted. In computer vision, the re-targeting of emotions mapped on faces is a common application. We contrast here four different methods used for transporting the deformation toward a target once it was estimated upon the matching of two shapes. These methods come from very different fields such as computational anatomy, computer vision and biology. We used the large diffeomorphic deformation metric mapping and thin plate spline, in order to estimate deformations in a deformational trajectory of a human face experiencing different emotions. Then we use naive transport (NT), linear shift (LS), direct transport (DT) and fanning scheme (FS) to transport the estimated deformations toward four alien faces constituted by 240 homologous points and identifying a triangulation structure of 416 triangles. We used both local and global criteria for evaluating the performance of the 4 methods, e.g., the maintenance of the original deformation. We found DT, LS and FS very effective in recovering the original deformation while NT fails under several aspects in transporting the shape change. As the best method may differ depending on the application, we recommend carefully testing different methods in order to choose the best one for any specific application.

We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with Androulidakis-Skandalis holonomy groupoids in the regular case. Finally we prove functoriality of all our constructions under appropriate morphisms., Comment: Changes from V1: Definition 4.7 corrected to Definitions 4.7 and 4.8. Theorem 4.9 expanded to Theorem 4.10, and detail added to proof. Minor typos fixed. Changes from V2: Minor editing changes, Example 4.6 revised and generalised. Changes from V3: coordinate representations of prolongations revised. Changes from V4: hypotheses of Theorem 4.15 changed. Changes from V5: Published version

We present a consistent and complete description of the coupling to matter in the Teleparallel Equivalent to General Relativity (TEGR) theory built from a Cartan connection, as we proposed in previous works. A first theorem allows us to obtain parallel transport from the Cartan connection into a proper Ehresmann connection, while a second ensures to link the TEGR-Cartan connection to the Ehresmann one-form that contains the Levi-Civita connection. This yields a coupling to matter in agreement with observations and the Equivalence Principle. As the fundamental fields proceed from the Cartan connection, if one insists on interpreting TEGR as a gauge theory of translations, such translation gauge field can be extracted from the consistent theory presented. However, this would entail a fundamental change in the structures known for gauge theory and a split between gauge field and connection is imperative. The willingness to take such a step is left to the reader., 9pp no Figs

There has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling and unwrapping originally proposed in Jupp and Kent (1987) for spherical data. In particular we develop such a fitting procedure for shapes of configurations in general $m$-dimensional Euclidean space, extending our previous work for two dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first-order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data., 35 pages, 12 figures. Revised version

A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality $\chi$ which defines the supertrace of the superalgebra: $STr(...) = Tr (\chi ...)$, we construct a covariant differential: $D = \chi (d + A) + \Phi$, where A is the standard even Lie-subalgebra connection 1-form and $\Phi$ a scalar field valued in the odd module. Despite the fact that $\Phi$ is a scalar, $\Phi$ anticommutes with $(\chi A)$ because $\chi$ anticommutes with the odd generators hidden in $\Phi$. Hence the curvature $F = DD$ is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure., Comment: 10 pages, 24 references. This is the revised version published in JHEP. Our new definition of the superconnection is unchanged relative to version 1 of this preprint, but thanks to the careful help of the referee, the discussion of the literature is streamlined and much improved

International audience; In cases of pressure or volume overload, probing cardiac function may be difficult because of the interactions between shape and deformations.In this work, we use the LDDMM framework and parallel transport to estimate and reorient deformations of the right ventricle. We then propose a normalization procedure for the amplitude of the deformation, and a second-order spline model to represent the full cardiac contraction. The method is applied to 3D meshes of the right ventricle extracted from echocardiographic sequences of 314 patients divided into three disease categories and a control group. We find significant differences between pathologies in the model parameters, revealing insights into the dynamics of each disease.

We present a three-dimensional metric affine theory of gravity whose field equations lead to a connection introduced by Schr\"odinger many decades ago. Although involving nonmetricity, the Schr\"odinger connection preserves the length of vectors under parallel transport, and appears thus to be more physical than the one proposed by Weyl. By considering solutions with constant scalar curvature, we obtain a self-duality relation for the nonmetricity vector which implies a Proca equation that may also be interpreted in terms of inhomogeneous Maxwell equations emerging from affine geometry., Comment: 5 pages