Geodesic Research Articles

Spacetime symmetries and geometric diffusion

We examine relativistic diffusion through the frame and observer bundles associated with a Lorentzian manifold (M, g). Our focus is on spacetimes with a non-trivial isometry group, and we detail the conditions required to find symmetric solutions of the relativistic diffusion equation. Additionally, we analyze the conservation laws associated with the presence of Killing vector fields on (M, g) and their implications for the expressions of the geodesic spray and the vertical Laplacian on both the frame and the observer bundles. Finally, we present several relevant examples of symmetric spacetimes.

Visually encoding orthogonal planar graph drawings as graph mazes: An eye tracking study

There are various styles to visually represent relational data in the form of node-link diagrams. In particular, for planar graphs we can find orthogonal node-link diagrams consisting of links bending only at ninety degrees a successful and prominent variant. One of the benefits of such drawings is the tracking of longer paths through a network with the eyes due to their limited number of link orientations, changes, and variations, but on the negative side the links can have arbitrary bending shapes. In this article we developed a novel way to visualize such orthogonal planar drawings by making use of mazes that look more natural to the human eye due to the street-like visual metaphor that many people are familiar with. Tracking paths is one of the major tasks in such graph visualizations, similar to orthogonal node-link diagrams, however, we argue that mazes are a more natural way to find paths. To get insights in the visual scanning behavior when reading graph mazes we conducted a comparative eye tracking study with 26 male versus female participants of different experience levels while also alternating between orthogonal node-link drawings and graph mazes as well as different graph size levels. The major result of this comparative study is that the participants can track paths in both representation styles, including a geodesic path tendency in their visual search behavior, but typically have a longer fixation duration at branching nodes and locations in the mazes that lead to opposite directions to the geodesic path tendency, maybe the viewers had to start a reorientation phase in their visual scanning behavior. We also found out that the size, that is the number of graph vertices has an impact on the visual scanning behavior for both orthogonal node-link diagrams as well as street-like maze representations, but for the mazes we found this impact to be less strong (in terms of the eye movement data metrics fixation durations and saccade lengths) compared to the node-link diagrams. To conclude the article, we discuss limitations and scalability issues of our approach. Moreover, we give an outlook and future work for possible extensions.

Horospherical dynamics in invariant subvarieties

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of C-valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstädt for principal strata. We describe all the leaf closures for the horospherical foliation.

Wide-Band Unambiguous Quantum Sensing via Geodesic Evolution.

We present a quantum sensing technique that utilizes a sequence of π pulses to cyclically drive the qubit dynamics along a geodesic path of adiabatic evolution. This approach effectively suppresses the effects of both decoherence noise and control errors while simultaneously removing unwanted resonance terms, such as higher harmonics and spurious responses commonly encountered in dynamical decoupling control. As a result, our technique offers robust, wide-band, unambiguous, and high-resolution quantum sensing capabilities for signal detection and individual addressing of quantum systems, including spins. To demonstrate its versatility, we showcase successful applications of our method in both low-frequency and high-frequency sensing scenarios. The significance of this quantum sensing technique extends to the detection of complex signals and the control of intricate quantum environments. By enhancing detection accuracy and enabling precise manipulation of quantum systems, our method holds considerable promise for a variety of practical applications.

Gap Between Lyapunov Exponents for Hitchin Representations

Abstract We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two consecutive Lyapunov exponents. Moreover we show that the minimal possible gap for any two consecutive Lyapunov exponents is achieved if and only if the representation is the one uniformizing the hyperbolic structure of the surface. We give two proofs of the previous fact. In the first one, we relate the Lyapunov exponents to a transverse Lyapunov exponent associated to a deformation of the unstable foliation of the geodesic flow, and we establish a general bound for this quantity. In the second one, we relate Lyapunov exponents to the renormalized intersection product in the setting of the thermodynamic formalism developed by Bridgeman, Canary, Labourie, and Sambarino and we use the already existing bound for such a quantity.

MSO-GP: 3-D segmentation of large and complex conjoined tree structures

Robust segmentation of large and complex conjoined tree structures in 3-D is a major challenge in computer vision. This is particularly true in computational biology, where we often encounter large data structures in size, but few in number, which poses a hard problem for learning algorithms. We show that merging multiscale opening with geodesic path propagation, can shed new light on this classic machine vision challenge, while circumventing the learning issue by developing an unsupervised visual geometry approach (digital topology/morphometry). The novelty of the proposed MSO-GP method comes from the geodesic path propagation being guided by a skeletonization of the conjoined structure that helps to achieve robust segmentation results in a particularly challenging task in this area, that of artery-vein separation from non-contrast pulmonary computed tomography angiograms. This is an important first step in measuring vascular geometry to then diagnose pulmonary diseases and to develop image-based phenotypes. We first present proof-of-concept results on synthetic data, and then verify the performance on pig lung and human lung data with less segmentation time and user intervention needs than those of the competing methods.

Spectral correspondences for finite graphs without dead ends

We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non-backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.

Free entropy minimizing persuasion in a predictor–corrector dynamic

Persuasion is the process of changing an agent’s belief distribution from a given (or estimated) prior to a desired posterior. A common assumption in the acceptance of information or misinformation as fact is that the (mis)information must be consistent with or familiar to the individual who accepts it. We model the process as a control problem in which the state is given by a (time-varying) belief distribution following a predictor–corrector dynamic. Persuasion is modeled as the corrector control signal with the performance index defined using the Fisher–Rao information metric, reflecting a fundamental cost associated to altering the agent’s belief distribution. To compensate for the fact that information production arises naturally from the predictor dynamic (i.e., expected beliefs change) we modify the Fisher–Rao metric to account just for information generated by the control signal. The resulting optimal control problem produces non-geodesic paths through distribution space that are compared to the geodesic paths found using the standard free entropy minimizing Fisher metric in several example belief models: a Kalman Filter, a Boltzmann distribution and a joint Kalman/Boltzmann belief system.

Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces

We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu’s bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian H will give rise to the non-vanishing Gromov–Witten invariant.

Stability estimates for the holonomy inverse problem

On a Riemannian manifold (M, g) with Anosov geodesic flow, the problem of recovering a connection from the knowledge of traces of its holonomies along primitive closed geodesics is known as the holonomy inverse problem. In this paper, we prove Hölder type stability estimates for this inverse problem: locally, near generic connections; globally, for line bundles, and for vector bundles satisfying a certain low-rank assumption over negatively curved base (M, g). The proofs are based on a combination of microlocal analysis along with a new non-Abelian approximate Livšic Theorem in hyperbolic dynamics.

Geometric conditions to obtain Anosov geodesic flow for non-compact manifolds

Let ( M , g ) be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold with non-positive curvature, without compact quotient and geodesic flow of Anosov type.

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

Towards geodesic ridge curve for region-wise linear representation of geodesic distance field

This paper addresses the challenge of representing geodesic distance fields on triangular meshes in a piecewise linear manner. Unlike general scalar fields, which often assume piecewise linear changes within each triangle, geodesic distance fields pose a unique difficulty due to their non-differentiability at ridge points, where multiple shortest paths may exist. An interesting observation is that the geodesic distance field exhibits an approximately linear change if each triangle is further decomposed into sub-regions by the ridge curve. However, computing the geodesic ridge curve is notoriously difficult. Even when using exact algorithms to infer the ridge curve, desirable results may not be achieved, akin to the well-known medial-axis problem. In this paper, we propose a two-stage algorithm. In the first stage, we employ Dijkstra's algorithm to cut the surface open along the dual structure of the shortest path tree. This operation allows us to extend the surface outward (resembling a double cover but with distinctions), enabling the discovery of longer geodesic paths in the extended surface. In the second stage, any mature geodesic solver, whether exact or approximate, can be employed to predict the real ridge curve. Assuming the fast marching method is used as the solver, despite its limitation of having a single marching direction in a triangle, our extended surface contains multiple copies of each triangle, allowing various geodesic paths to enter the triangle and facilitating ridge curve computation. We further introduce a simple yet effective filtering mechanism to rigorously ensure the connectivity of the output ridge curve. Due to its merits, including robustness and compatibility with any geodesic solver, our algorithm holds great potential for a wide range of applications. We demonstrate its utility in accurate geodesic distance querying and high-fidelity visualization of geodesic iso-lines.

A Relation between Krylov and Nielsen Complexity.

Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.

Invariant distributions and the transport twistor space of closed surfaces

AbstractWe study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on are determined by their zeroth and first Fourier modes.

The weak mixing property on negatively curved manifolds

Given a complete manifold of negative curvature, we show that weak mixing is a generic property in the set of all probability measures invariant by the geodesic flow, as soon as the flow is topologically weakly mixing in restriction to its non-wandering set.

Datascape: exploring heterogeneous dataspace

Data science is a powerful field for gaining insights, comparing, and predicting behaviors from datasets. However, the diversity of methods and hypotheses needed to abstract a dataset exhibits a lack of genericity. Moreover, the shape of a dataset, which structures its contained information and uncertainties, is rarely considered. Inspired by state-of-the-art manifold learning and hull estimations algorithms, we propose a novel framework, the datascape, that leverages topology and graph theory to abstract heterogeneous datasets. Built upon the combination of a nearest neighbor graph, a set of convex hulls, and a metric distance that respects the shape of the data, the datascape allows exploration of the dataset’s underlying space. We show that the datascape can uncover underlying functions from simulated datasets, build predictive algorithms with performance close to state-of-the-art algorithms, and reveal insightful geodesic paths between points. It demonstrates versatility through ecological, medical, and simulated data use cases.

Geodesic flows of compact higher genus surfaces without conjugate points have expansive factors

In this paper we show that a geodesic flow of a compact surface without conjugate points of genus greater than one is time-preserving semi-conjugate to a continuous expansive flow which is topologically mixing and has a local product structure. As an application we show that the geodesic flow of a compact surface without conjugate points of genus greater than one has a unique measure of maximal entropy. This gives an alternative proof of Climenhaga–Knieper–War Theorem.

Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs

Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs