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We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one. [ABSTRACT FROM AUTHOR]

Comparing the deformations of different beating hearts is a challenging operation. As in clinics the impaired condition is often recognized upon (local and global) deformation parameters, the particular nature of heart deformation during one beat can be compared among different individuals in the same ordination space more effectively if initial inter-individual form (shape + size) differences are filtered out. This is even more true if the shape of cardiac trajectory itself is under consideration. This need is satisfied by applying a geometric machinery named "parallel transport" in the field of differential geometry. In recent years several parallel transport methods have been applied to cardiological data acquired via echocardiography, CT scan or magnetic resonance. Concomitantly, some efforts were made for comparing different parallel transport algorithms applied to a variety of toy examples and real deformational data. Here we face the problem of comparing the heavily used LDDMM parallel transport with the recently proposed Riemannian "TPS space" in the context of the deformation of the right ventricle. Using local tensors diagnostics and global energy-based and shape distance-based parameters, we explored the maintenance of original deformations in transported data in four systo-diastolic deformations belonging to one healthy subject and three individuals affected by tetralogy of Fallot, atrial septal defect and pulmonary hypertension. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. In particular, we contrasted the TPS space with classic LDDMM and a modified LDDMM able to manage spherical differences. Our results point toward a neat superiority of TPS space over classic LDDMM. The modified LDDMM performs similarly as it maintains better the chosen diagnostics. [ABSTRACT FROM AUTHOR]

A geometrically exact Kirchhoff beam formulation (GEKBF) established on the Lie group SO(3) for simulating the dynamics of a slender beam is proposed. The kinematic description, dynamic equilibrium equations and their linearization are derived in this framework. Then, a second-order interpolation function for the torsion angle is introduced to improve the convergence of the element. Next, the rotation vector parameterization is developed to reduce the geometric nonlinearity caused by the rotation of the rigid body. In addition, the influence of the reference frame is considered, and the derivation of the elastic force and Jacobian matrix is simplified using the spatial-parallel transport. The semi-discrete equations of motion are in the form of a second-order ordinary differential equation on Lie group, which are solved using the Lie group generalized α-method. Finally, the accuracy of the proposed formulation is verified using several numerical examples. [ABSTRACT FROM AUTHOR]

We continue to study the cocongruence of -dimensional planes us­ing the Cartan — Laptev method. In an -dimensional projective space , the cocongruence of -dimensional planes can be given by the following equations . Compositional clothing of a given cocongruence by fields of ()-planes : and points allows one to define connections of three types in the associated bundle. In the present paper, parallel transports of an analogue of Cartan plane are studied in the connections of three types. It is proved 4 theo­rems: 1. Parallel transport of the analogue of the Cartan plane in an arbitrary connection is freely degenerate, i. e., in general, there are no spe­cial transports of this clothing plane. 2. In the group connection of the first type, the parallel transport of an analog of the Cartan plane is connected degenerate, i. e., the plane will be fixed under parallel transport in this connection. 3. In the group connections of the second and third types, the parallel transport of the analogue of the Cartan plane is freely degenerate. 4. The analogue of the Cartan plane is transferred in parallel in a line­ar combination of the first type connection if and only if it is displaced in the plane .

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric. [ABSTRACT FROM AUTHOR]

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point. [ABSTRACT FROM AUTHOR]

In this paper we study geometric aspects of dynamics generated by Young differential equations (YDE) driven by α -Hölder trajectories with α ∈ (1 / 2 , 1) . We present a number of properties and geometrical constructions in this context: Young Itô geometrical formula, horizontal lift in principal fibre bundles, parallel transport, among others. Our main application here is a geometrical decomposition of flows generated by YDEs according to diffeomorphisms generated by complementary distributions (integrable or not). The proof of existence of this decomposition is based on an Itô-Wentzel type formula for Young integration along α -Hölder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022). [ABSTRACT FROM AUTHOR]

A quantum system, when subjected to an adiabatic cyclic evolution of parameters governing its Hamiltonian, gains a phase factor known as the geometric phase. One of its most celebrated formulations in the regime of quantum optics is the Pancharatnam-Berry phase. It is said to be acquired by a beam of light undergoing an evolution of its polarization state. This article builds on the background and notion of the Pancharatnam-Berry phase in quantum optics. A convenient tool for visualizing the evolution of polarization states is the Poincaré sphere, which has been briefly discussed here. The close relation of the phase to a few other important concepts, such as Bargmann invariants and geodesics, has also been discussed in this article. [ABSTRACT FROM AUTHOR]

The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we use the geometry of tangent bundles of the space of random measures endowed with the Wasserstein metric for mapping distributions to tangent spaces. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space-valued data in functional data analysis. In simulations, it performs better than an alternative transformation approach where one maps distributions to a Hilbert space through the log quantile density transformation and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression operator and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributional time series of house prices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article. [ABSTRACT FROM AUTHOR]

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric.

We investigate a model for collective behaviour with intrinsic interactions on smooth Riemannian manifolds. For regular interaction potentials, we establish the local well-posedness of measure-valued solutions defined via optimal mass transport. We also extend our result to the global well-posedness of solutions for manifolds with nonpositive bounded sectional curvature. The core concept underlying the proofs is that of Lipschitz continuous vector fields in the sense of parallel transport. [ABSTRACT FROM AUTHOR]

Manifold-valued functional data analysis (FDA) has become an active area of research motivated by the rising availability of trajectories or longitudinal data observed on nonlinear manifolds. The challenges of analyzing such data come from many aspects, including infinite dimensionality and nonlinearity, as well as time domain or phase variability. In this paper, we study the amplitude part of manifold-valued functions on S 2 , which is invariant to random time warping or re-parameterization. We represent a smooth function on S 2 using a pair of components: a starting point and a transported square-root velocity curve (TSRVC). Under this representation, the space of all smooth functions on S 2 forms a vector bundle, and the simple L 2 norm becomes a time-warping invariant metric on this vector bundle. Utilizing the nice geometry of S 2 , we develop a set of efficient and accurate tools for temporal alignment of functions, geodesic computing, and sample mean and covariance calculation. At the heart of these tools, they rely on gradient descent algorithms with carefully derived gradients. We show the advantages of these newly developed tools over its competitors with extensive simulations and real data and demonstrate the importance of considering the amplitude part of functions instead of mixing it with phase variability in manifold-valued FDA. [ABSTRACT FROM AUTHOR]

Quantum computing is captured in the formalism of the monoidal subcategory of Vect C generated by C 2 — in particular, quantum circuits are diagrams in Vect C — while topological quantum field theories, in the sense of Atiyah, are diagrams in Vect C indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call F Vect C. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to F Vect C , which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category. [ABSTRACT FROM AUTHOR]

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild's ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild's ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space. [ABSTRACT FROM AUTHOR]

We correct the final section of [On the monodromy of irreducible symplectic manifolds, Algebr. Geom. 3 (2016), no. 3, 385-391] and provide some further details. [ABSTRACT FROM AUTHOR]

Convolution plays a crucial role in various applications in signal and image processing, analysis, and recognition. It is also the main building block of convolution neural networks (CNNs). Designing appropriate convolution neural networks on manifold-structured point clouds can inherit and empower recent advances of CNNs to analyzing and processing point cloud data. However, one of the major challenges is to define a proper way to “sweep”filters through the point cloud as a natural generalization of the planar convolution and to reflect the point cloud’s geometry at the same time. In this paper, we consider generalizing convolution by adapting parallel transport on the point cloud. Inspired by a triangulated surface-based method [46], we propose the Narrow-Band Parallel Transport Convolution (NPTC) using a specifically defined connection on a voxel-based narrow-band approximation of point cloud data. With that, we further propose a deep convolutional neural network based on NPTC (called NPTC-net) for point cloud classification and segmentation. Comprehensive experiments show that the proposed NPTC-net achieves similar or better results than current state-of-the-art methods on point cloud classification and segmentation. [ABSTRACT FROM AUTHOR]

Tokamak edge (scrape-off layer (SOL)) plasmas can exhibit non-local transport in the direction parallel to the magnetic field due to steep temperature gradients. This effect along with its consequences has been explored at equilibrium for a range of conditions, from sheath-limited to detached, using the 1D kinetic electron code SOL-KiT, where the electrons are treated kinetically and compared to a self-consistent fluid model. Line-averaged suppression of the kinetic heat flux (compared to Spitzer-Härm) of up to 50% is observed, contrasting with up to 98% enhancement of the sheath heat transmission coefficient, γ _e . Simple scaling laws in terms of basic SOL parameters for both effects are presented. By implementing these scalings as corrections to the fluid model, we find good agreement with the kinetic model for target electron temperatures. It is found that the strongest kinetic effects in γ _e are observed at low-intermediate collisionalities, and tend to increase (keeping upstream collisionality fixed) at increasing upstream densities and temperatures. On the other hand, the heat flux suppression is found to increase monotonically as upstream collisionality decreases. The conditions simulated encompass collisionalities relevant to current and future tokamaks.

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. [ABSTRACT FROM AUTHOR]

In this paper, the Cartan tensors of the (α, β)-norms are investigated in detail. Then an equivalence theorem of (α, β)-norms is proved. As a consequence in Finsler geometry, general (α, β)-metrics on smooth manifolds of dimension n ⩾ 4 with vanishing Landsberg curvatures must be Berwald manifolds. [ABSTRACT FROM AUTHOR]

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an energy functional that depends on the "tangential" component of the derivative of the tantrix of the curve. We show that the total intrinsic curvature of irregular curves agrees with such an energy functional. By exploiting isometric embeddings, the previous results are then extended to irregular curves contained in Riemannian surfaces. Finally, the relationship with the notion of displacement of a smooth curve is analyzed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Tractography is an important technique that allows the in vivo reconstruction of structural connections in the brain using diffusion MRI. Although tracking algorithms have improved during the last two decades, results of validation studies and international challenges warn about the reliability of tractography and point out the need for improved algorithms. In propagation-based tracking, connections have traditionally been modeled as piece-wise linear segments. In this work, we propose a novel propagation-based tracker that is capable of generating geometrically smooth (${C}^{{1}}$) curves using parallel transport frames. Notably, our approach does not increase the complexity of the propagation problem that remains two-dimensional. Moreover, our tracker has a novel mechanism to reduce noise related propagation errors by incorporating topographic regularity of connections, a neuroanatomic property of many brain pathways. We ran extensive experiments and compared our approach against deterministic and other probabilistic algorithms. Our experiments on FiberCup and ISMRM 2015 challenge datasets as well as on 56 subjects of the Human Connectome Project show highly promising results both visually and quantitatively. Open-source implementations of the algorithm are shared publicly. [ABSTRACT FROM AUTHOR]

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, unwrapping and wrapping originally proposed by Jupp and Kent 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. [ABSTRACT FROM AUTHOR]

Accurate navigation of flexible medical instruments like catheters require the knowledge of its pose, that is its position and orientation. In this paper multi-core fibers inscribed with fiber Bragg gratings (FBG) are utilized as sensors to measure the pose of a multi-segment catheter. A reconstruction technique that provides the pose of such a fiber is presented. First, the measurement from the Bragg gratings are converted to strain then the curvature is deduced based on those strain calculations. Next, the curvature and the Bishop frame equations are used to reconstruct the fiber. This technique is validated through experiments where the mean error in position and orientation is observed to be less than 4.69 mm and 6.48 degrees, respectively. The main contributions of the paper are the use of Bishop frames in the reconstruction and the experimental validation of the acquired pose. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Hilbert bundle of generalized statistical manifold with the introduction of a family of α-connections. The expressions of parallel transport for the exponential and mixture connections are derived, and we prove that those two connections are dual and curvature-free. [ABSTRACT FROM AUTHOR]

We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application, we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone. [ABSTRACT FROM AUTHOR]

The dynamics of a closed quantum system is often studied with the direct evolution of the Schrödinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant to physical observables) of the Schrödinger equation can be generally nonoptimal 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 and is 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 Schrödinger 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 Schrödinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse. [ABSTRACT FROM AUTHOR]

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