Theory Of Geodesics Research Articles

Geodesic conformal gradient device based on a torus.

In recent years, optical fields on non-Euclidean geometry have become a hot topic. The geodesic conformal transformation theory, linking curved surfaces with planar gradient refractive indices, holds unique advantages in controlling curved optical fields. However, this theory has not yet addressed surfaces with non-trivial topology with a certain genus. In this work, we design a gradient planar device based on the geodesic conformal transformation theory for toroidal surfaces, which can achieve Gaussian beam focusing. Unlike traditional angle-preserving geodesic theories, the non-zero genus results in the one-to-two discontinuous boundaries in the device, and we utilize inversion transformations to rectify this drawback.

Multiple Lagrangian Jet‐Core Structures in the Malvinas Current

AbstractThe Malvinas Current (MC) in the Southwestern Atlantic is characterized as a system of multiple Lagrangian jets. This is done by applying geodesic theory of Lagrangian Coherent Structures (LCSs) on altimetry‐derived velocity. The number and spatial disposition of jet‐cores organizing the MC are found to vary in time influenced by topographic waves propagating along the current. The Lagrangian analysis reveals that permanent jet‐cores tend to approach one another upstream, yet do not merge into a single jet as the Eulerian analysis of the data suggests. Independent support for the existence of multiple jets is provided by trajectories of surface drifters from a deployment of unparalleled dimensions for the region and ocean color images. The drifters develop the characteristic boomerang‐shaped patterns into which passive tracers straddling jet‐cores deform and exhibit Lagrangian metrics that are largely consistent with those derived from altimetry data. Ocean color imagery reveals similar “chevrons” and suggest upwelling along the jets, which is consistent with a new solution of arrested topographic wave dynamics. This observation and the finding that the MC jet system is influenced by propagating waves suggests that the MC variability may have major implications in the modulation of primary productivity on synoptic timescales. Drifters flowing into the open ocean, exhibit organizing patterns that are largely shaped by mesoscale circulation and essentially differ from those observed along the MC. Evidence of this observation is provided on the basis of LCS analysis and the calculation of a quasi‐objective Lagrangian metric derived directly from drifter trajectories.

Mechanics of geodesics in information geometry and black hole thermodynamics

In this paper, we shall discuss the theory of geodesics in information geometry, and an application in astrophysics. We will study how gradient flows in information geometry describe geodesics, explore the related mechanics by introducing a constraint, and apply our theory to Gaussian model and black hole thermodynamics. Thus, we demonstrate how deformation of gradient flows leads to more general Randers–Finsler metrics, describe Hamiltonian mechanics that derive from a constraint, and prove duality via canonical transformation. We also verified our theories for a deformation of the Gaussian model, and described dynamical evolution of flat metrics for Kerr and Reissner–Nordström black holes.

Geodesic theory of long association fibers arrangement in the human fetal cortex.

Association fibers connect different areas of the cerebral cortex over long distances and integrate information to achieve higher brain functions, particularly in humans. Prototyped association fibers are developed to the respective tangential direction throughout the cerebral hemispheres along the deepest border of the subplate during the fetal period. However, how guidance to remote areas is achieved is not known. Because the subplate is located below the cortical surface, the tangential direction of the fibers may be biased by the curved surface geometry due to Sylvian fissure and cortical poles. The fiber length can be minimized if the tracts follow the shortest paths (geodesics) of the curved surface. Here, we propose and examine a theory that geodesics guide the tangential direction of long association fibers by analyzing how geodesics are spatially distributed on the fetal human brains. We found that the geodesics were dense on the saddle-shaped surface of the perisylvian region and sparse on the dome-shaped cortical poles. The geodesics corresponded with the arrangement of five typical association fibers, supporting the theory. Thus, the geodesic theory provides directional guidance information for wiring remote areas and suggests that long association fibers emerge from minimizing their tangential length in fetal brains.

Lagrangian coherent structures in flow past a backward-facing step

This paper investigates flow past a backward-facing step (BFS) in a duct at Reynolds number $Re = 5080$ based on step height, mean inflow velocity and fluid kinematic viscosity. The flow configuration matches a combustion experiment conducted by Pitz and Daily in 1983. High-resolution velocity fields are obtained in OpenFOAM by direct numerical simulation, and the flow field analysed by Lagrangian approaches. Trajectories of fluid particles in areas of interest are obtained by high-order numerical integration, and used to compute finite-time Lyapunov exponents (FTLEs) and polar rotation angles. Lagrangian coherent structures (LCSs) are extracted using geodesic theory, including hyperbolic LCSs and elliptic LCSs. We use complementary qualitative and quantitative LCS analyses to uncover the underlying flow structures. Notably, we find that a flow pathway in which fluid particles rarely diverge from adjacent particles is opened and closed by FTLE ridges determined by the periodic shedding of vortices from the BFS. Two dominant vortices with significant Lagrangian coherence, generated respectively by the separated boundary layer and shear layer, are self-sustaining and of comparable strength. Hyperbolic repelling LCSs act as transport barriers between the pathway and cores of the coherent vortices, thus playing a major part in the fluid entrainment process. Interactions between these different geometric regions partitioned by LCSs lead to intrinsic complexity in the BFS flow.

Information geometry of the space of probability measures and barycenter maps

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical Laplacian comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical Laplacian comparison theorems for the Riemannian distances approximations. As a corollary we prove that, under suitable curvature conditions, sub-Riemannian Sasakian spaces are actually limits of Riemannian spaces satisfying a uniform measure contraction property.

On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in $\mathbb{S}^1\times \mathbb{S}^2$, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in $\mathbb{T}^3$ for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of `integrable' and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

Surface mixing and biological activity in the North-West African upwelling.

Near-shore water along the North-West African margin is one of the world's major upwelling regions. It is associated with physical structures of oceanic fronts which influence the biological productivity. The study of these coherent structures in connection with chlorophyll concentration data is of fundamental importance for understanding the spatial distributions of the plankton. In this work, we study the horizontal stirring and mixing in different upwelling areas using Lagrangian coherent structures (LCSs). These LCSs are calculated using the recent geodesic theory of LCSs. We use these LCSs to study the link between the chlorophyll fronts concentrations and surface mixing, based on 10 years of satellite data. These LCSs move with the flow as material lines, thus the horizontal mixing is calculated from the intersection of these LCSs with the finite time Lyapunov exponent maps. We compare our results with those of a recent study conducted over the same area, but based on finite size Lyapunov exponents (FSLEs), whose output is a plot of scalar distributions. We discuss the differences between FSLE and geodesic theory of LCS. The latter yields analytical solutions of LCSs, while FSLEs can only provide LCSs for sharp enough ridges of nearly constant height.

Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles

We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an N-disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert (Math Z 51:197–216, 1948).

Comparison theorems in pseudo-Hermitian geometry and applications

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.

Shear and shearless Lagrangian structures in compound channels

Abstract Transport processes in a physical model of a natural stream with a composite cross-section (compound channel) are investigated by means of a Lagrangian analysis based on nonlinear dynamical system theory. Two-dimensional free surface Eulerian experimental velocity fields of a uniform flow in a compound channel form the basis for the identification of the so-called Lagrangian Coherent Structures. Lagrangian structures are recognized as the key features that govern particle trajectories. We seek for two particular class of Lagrangian structures: Shear and shearless structures. The former are generated whenever the shear dominates the flow whereas the latter behave as jet-cores. These two type of structures are detected as ridges and trenches of the Finite-Time Lyapunov Exponents fields, respectively. Besides, shearlines computed applying the geodesic theory of transport barriers mark Shear Lagrangian Coherent Structures. So far, the detection of these structures in real experimental flows has not been deeply investigated. Indeed, the present results obtained in a wide range of the controlling parameters clearly show a different behaviour depending on the shallowness of the flow. Shear and Shearless Lagrangian Structures detected from laboratory experiments clearly appear as the flow develops in shallow conditions. The presence of these Lagrangian Structures tends to fade in deep flow conditions.

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

LCS Tool: A computational platform for Lagrangian coherent structures

We give an algorithmic introduction to Lagrangian coherent structures (LCSs) using a newly developed computational engine, LCS Tool. LCSs are most repelling, attracting and shearing material lines that form the centrepieces of observed tracer patterns in two-dimensional unsteady dynamical systems. LCS Tool implements the latest geodesic theory of LCSs for two-dimensional flows, uncovering key transport barriers in unsteady flow velocity data as explicit solutions of differential equations. LCS Tool makes theoretical results accessible to the fluid mechanics community since implementing these results directly could be time consuming. After a review of the underlying theory, we explain the steps and numerical methods used by LCS Tool, and illustrate its capabilities on three unsteady fluid flow examples.

Analysis of water quality and circulation of four recreational Miami beaches through the use of Lagrangian Coherent Structures

Four popular, recreational beaches in Miami, FL are Hobie Beach, Virginia Key Beach, Crandon Park Beach, and Bill Baggs Cape Florida State Park. While all of the beaches are within a few miles of each other in Biscayne Bay, they have greatly differing water qualities, as determined by the testing for fecal indicator bacteria performed by the Florida Department of Health. Using the geodesic theory of transport barriers, we identify Lagrangian Coherent Structures (LCSs) in each area. We show how these material curves, which shape circulation and mixing patterns, can be used to explain the incongruous states of the water at beaches that should be comparable. The LCSs are computed using a hydrodynamic model and verified through field experimentation at each beach.

Differential Geometry Perspective of Shape Coherence and Curvature Evolution by Finite-Time Nonhyperbolic Splitting

Mixing and coherence are essential topics for understanding and describing transport in fluid dynamics and other nonautonomous dynamical systems. Only recently has the idea of coherence gained more serious footing, particularly with the recent advances of finite-time studies of nonautonomous dynamical systems. Here we define shape coherent sets as a means to emphasize the intuitive notion of ensembles which “hold together" for some period of time, and we contrast this notion to other recent perspectives of coherence, notably “coherent pairs," and likewise also to the geodesic theory of material lines. We will relate shape coherence to the differential geometry concept of curve congruence through matching curvatures. We show that points in phase space where there is a zero-splitting between stable and unstable foliations locally correspond to points where curvature will evolve only slowly in time. Then we develop curves of points with zero-angle, meaning nonhyperbolic splitting, by continuation methods in terms of the implicit function theorem. From this follows a simple ODE description of the boundaries of shape coherent sets. We will illustrate our method with popular benchmark examples and further investigate the intricate structure of foliation geometry.

Kajian Sketsa Digital dalam Proses Reka-Struktur Bentuk-Bentuk Surface

Free form surface has a multiple complexity curve intrinsically, so that special approach is needed to form the structure. Geodesic shell is well-known as the shortest path between two points on a curved surface. This method is used to generate the structure of form, which can be easily determined by its basic shape or the complex form. This paper explains how Geodesic theory can assist to generate structures in the early stage of digital sketching. A workshop that has initially started becomes a topic to this discussion. The digital sketching process to apply the theory is the aim of this paper.

Addendum to “Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric” [Ann. I. H. Poincaré – AN 27 (3) (2010) 857–876

We give the details of the proof of equality (29) in Caponio et al. (2010) [3].

EQUATIONS OF MOTION FOR SPACE-TIME-MATTER THEORY

The purpose of this paper is to present, in a covariant form and in their full generality, the equations of motion for space-time-matter (STM) theory. The whole study is based on the new approach of STM theory developed in our first paper [1] of this series. We show that the theory of geodesics in a general Kaluza–Klein space is best presented and explained by splitting the set of all geodesics into horizontal and non-horizontal geodesics. It is noteworthy that the horizontal geodesics (respectively, non-horizontal geodesics) project on the base manifold on motions which generalize the motions from general relativity theory (respectively, motions from Lorentz force equations).

Using Lagrangian Coherent Structures to understand coastal water quality

The accumulation of pollutants near the shoreline can result in low quality coastal water with negative effects on human health. To understand the role of mixing by tidal flows in coastal water quality we study the nearshore Lagrangian circulation. Specifically, we reveal Lagrangian Coherent Structures (LCSs), i.e., distinguished material curves which shape global mixing patterns and thus act as skeletons of the Lagrangian circulation. This is done using the recently developed geodesic theory of transport barriers. Particular focus is placed on Hobie Beach, a recreational subtropical marine beach located in Virginia Key, Miami, Florida. According to studies of water quality, Hobie Beach is characterized by high microbial levels. Possible sources of pollution in Hobie Beach include human bather shedding, dog fecal matter, runoff, and sand efflux at high tides. Consistent with the patterns formed by satellite-tracked drifter trajectories, the LCSs extracted from simulated currents reveal a Lagrangian circulation favoring the retention near the shoreline of pollutants released along the shoreline, which can help explain the low quality water registered at Hobie Beach.