Related papers: Geodesic Webs and PDE Systems of Euler Equations

We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat…

Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $\Delta(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the…

Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi$. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson,…

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…

In this paper a functional definition of geodesics is introduced which allows to generalize the notion of a geodesic from smooth to topological manifolds. It is shown that in the smooth case the new definition coincides with the classical…

Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of…

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of…

The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting)…

The problem of finding the general classification of geodetic graphs is still open. We believe that one of the obstacles to attain this goal is that geodetic graphs lack a structural description. In other words, their fundamental properties…

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface…

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients…

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of…

Let $G$ be a Lie group, $\Gamma\subset G$ a discrete subgroup, $X=G/\Gamma$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories…

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary…

We study the general properties of fluid spheres satisfying the heuristic assumption that their areas and proper radius are equal (the Euclidean condition). Dissipative and non-dissipative models are considered. In the latter case, all…

In the present study, we derive the problem of constructing a hypersurface family from a given isogeodesic curve in the 4D Galilean space $\mathbf{G}_{4}.$ We obtain the hypersurface as a linear combination of the Frenet frame in…

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a…

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm…

A key task in the study of networked systems is to derive local and global properties that impact connectivity, synchronizability, and robustness; computing shortest paths or geodesics yields measures of network connectivity that can…

This paper investigates flat webs on the projective plane. We present two methods for constructing such webs: the first involves taking the product of finitely many convex reduced foliations and invariant lines, while the second consists of…