Geodesic Nets Research Articles

Generic density of geodesic nets

Generic density of geodesic nets

Stable Geodesic Nets in Convex Hypersurfaces

Stable Geodesic Nets in Convex Hypersurfaces

Bumpy metrics theorem for geodesic nets

Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a Baire-generic Riemannian metric on a smooth manifold [Formula: see text], all connected embedded stationary geodesic nets are nondegenerate.

On the equidistribution of closed geodesics and geodesic nets

We show that given a closed n n -manifold M M , for a Baire-generic set of Riemannian metrics g g on M M there exists a sequence of closed geodesics that are equidistributed in M M if n = 2 n=2 ; and an equidistributed sequence of embedded stationary geodesic nets if n = 3 n=3 . One of the main tools that we use is the Weyl law for the volume spectrum for 1 1 -cycles, proved by Liokumovich, Marques, and Neves [Ann. of Math. (2) 187 (2018), pp. 933–961] for n = 2 n=2 and by Guth and Liokumovich [Preprint, arXiv:2202.11805, 2022] for n = 3 n=3 . We show that our proof of the equidistribution of stationary geodesic nets can be generalized for any dimension n ≥ 2 n\geq 2 provided the Weyl Law for 1 1 -cycles in n n -manifolds holds.

Geodesic nets on non-compact Riemannian manifolds

Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.

The p-widths of a surface

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by lfloor sqrt{p}rfloor great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be sqrt{pi }.En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.

Geodesic nets with three boundary vertices

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.

On critical nets in $${\mathbb {R}}^k$$

Critical nets in \({\mathbb {R}}^k\) (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices have a fixed position. In contrast to what happens on generic manifolds, we show that, if the embedding is bounded and n is the number of 1-valent vertices, the total length of the edges not incident with a 1-valent vertex is bounded by rn (where r is the outer radius), the degree of any vertex is bounded by n and that the number of edges (and hence the number of vertices) is bounded by \(n\ell \) (where \(\ell \) is related to the combinatorial diameter of the graph).

Geodesic Nets: Some Examples and Open Problems

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. In the first part of this paper, we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets with edges of multiplicity one on the Euclidean plane. Our main focus is on relationships between the number of boundary vertices, the number of inner (or balanced) vertices, and some basic geometric characteristic of geodesic nets (such as the length or the imbalances at boundary vertices). The second part contains a new construction providing a partial answer for one of these questions: We describe an infinite family of geodesic nets with edges of multiplicity one on the Euclidean plane with a constant number (namely, 14) of boundary vertices and arbitrarily many inner (or balanced) vertices of degree The fact that all edges of the constructed geodesic nets have multiplicity one is not proven but strongly supported by numerical evidence obtained from experimentation.

Minimizing geodesic nets and critical points of distance

We establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n-sphere. This result is used to give conditions under which a minimizing geodesic flower degenerates into a simple closed geodesic.

Adjusting ADJUST: Optimizing the ADJUST algorithm for pediatric data using geodesic nets.

A major challenge for electroencephalograph (EEG) studies on pediatric populations is that large amounts of data are lost due to artifacts (e.g., movement and blinks). Independent component analysis (ICA) can separate artifactual and neural activity, allowing researchers to remove such artifactual activity and retain a greater percentage of EEG data for analyses. However, manual identification of artifactual components is time-consuming and requires subjective judgment. Automated algorithms, like ADJUST and ICLabel, have been validated on adults, but to our knowledge, no such algorithms have been optimized for pediatric data. Therefore, in an attempt to automate artifact selection for pediatric data collected with geodesic nets, we modified ADJUST's algorithm. Our "adjusted-ADJUST" algorithm was compared to the "original-ADJUST" algorithm and ICLabel in adults, children, and infants on three different performance measures: respective classification agreement with expert coders, the number of trials retained following artifact removal, and the reliability of the EEG signal after preprocessing with each algorithm. Overall, the adjusted-ADJUST algorithm performed better than the original-ADJUST algorithm and no ICA correction with adult and pediatric data. Moreover, in some measures,it performed better than ICLabel for pediatric data. These results indicate that optimizing existing algorithms improves artifact classification and retains more trials, potentially facilitating EEG studies with pediatric populations. Adjusted-ADJUST is freely available under the terms of the GNU General Public License at: https://github.com/ChildDevLab/MADE-EEG-preprocessing-pipeline/tree/master/adjusted_adjust_scripts.

The shape space of discrete orthogonal geodesic nets

Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold's tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors by solving a linear system, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We further show how to extend the shape space of DOGs by supporting creases and curved folds and apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.

Flowers on Riemannian manifolds

In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let Mn be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n − 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of Mn. We also show that there exists a minimal geodesic net that consists of one vertex and at most \({3^{(n+1)^2}}\) loops of total length \({\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}\), where Fill Rad Mn denotes the filling radius and vol(Mn) denotes the volume of Mn.

Shapes of geodesic nets

Let M n be a closed Riemannian manifold of dimension n. In this paper we will show that either the length of a shortest periodic geodesic on M n does not exceed .nC1/d , where d is the diameter of M n or there exist infinitely many geometrically distinct stationary closed geodesic nets on this manifold. We will also show that either the length of a shortest periodic geodesic is, similarly, bounded in terms of the volume of a manifold M n , or there exist infinitely many geometrically distinct stationary closed geodesic nets on M n .

The length of a shortest geodesic net on a closed Riemannian manifold

In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold M n in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p , q ∈ M n of total length ≤ m d , where m ∈ { 2 , … , ( n + 1 ) } and d is the diameter of M n . We also show that there exists a minimal geodesic net with at most n + 1 vertices and ( n + 1 ) ( n + 2 ) 2 geodesic segments of total length ≤ ( n + 1 ) ( n + 2 ) FillRad M n ≤ ( n + 1 ) 2 n n ( n + 2 ) ( n + 1 ) ! vol ( M n ) 1 n . These results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234–254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748–790].

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let M n be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on M n of length at most 3 diameter(M n ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(M n ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of M n . If n=2, then this critical θ-graph is not only immersed but embedded.

Special networks in the hypersurfaces of generalised recurrent and birecurrent Weyl spaces

In this paper, we study some properties of the Chebyshev nets and geodesic nets in the hypersurfaces of the generalised recurrent and birecurrent Weyl spaces.

On the nets of equi-conjugate pairs of curves in a Weyl hypersurface

In this paper, equi-conjugate pairs of curves in a hypersurfaceWn(gij,Tk) are defined and the Chebyshev nets of the first kind, the second kind and the geodesic nets which are formed by the tangent vector fields of these pairs of curves are investigated.

On the partition of the 2-sphere by geodesic nets

The main result of the paper is that for every natural number n n there exists a geodesic net with vertices of degree 3 or 4 partitioning the round 2-sphere into n n regions.

Geodesic nets on the 2-sphere

In this paper we introduce the concept of a geodesic net, an idea which plays the role among graphs that geodesics play among simple closed curves. We establish the existence of specific geodesic nets on the 2-sphere in certain cases.