Constrictions on a surface are defined as simple closed curves whose length is locally minimal. In particular, constrictions are periodic geodesics. We use constrictions in order to segment objects. In [4], we proposed an approach based on progressive surface simplification and local geodesic computation. The drawback of this approach is that constrictions are approximated by closed piecewise g...

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern– Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than o...

We analyze here the structure of non-radial nonspacelike geodesics terminating in the past at a naked singularity formed as the end state of inhomogeneous dust collapse. The spectrum of outgoing nonspacelike geodesics is examined analytically. The local and global visibility of the singularity is also examined by integrating numerically the null geodesics equations. The possible implications of...

in this work, different types of chaotic 1-manifolds which lie on the chaotic spheres or on a torus are introduced. some types of retractions of the chaotic spheres affect on the 1-chaotic systems, and other types of retractions occur to the geometric manifold but make the 1-chaotic manifold invariant. the existed retractions are discussed through new proved theorems. also we construct diffe...

In this paper, the equations of motion for geodesics in the neutral rotating Black Ring metric are derived and the separability of these equations is considered. The bulk of the paper is concerned with sets of solutions where the geodesic equations can be examined analytically specifically geodesics confined to the axis of rotation, geodesics restricted to the equatorial plane, and geodesics th...

The closedness condition for real geodesics on n–dimensional ellipsoids is in general transcendental in the parameters (semiaxes of the ellipsoid and constants of motion). We show that it is algebraic in the parameters if and only if both the real and the imaginary geodesics are closed and we characterize such double–periodicity condition via real hyperelliptic tangential coverings. We prove th...

Let S = Sg,n be a connected, orientable surface of genus g ≥ 0 with n ≥ 0 punctures. Let Teich(S) denote the corresponding Teichmüller space, and let Mod(S) denote the mapping class group of S. Understanding the analogy of Teich(S) with symmetric spaces is a wellknown theme. Recall that a complete Riemannian manifold X is symmetric if it is symmetric at each point x ∈ X: the map γ(t) 7→ γ(−t) w...

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.

In this note based on paper [3] we deal with domains D (i.e. connected open subsets) of a Finsler manifold (M, F ). At first we carry out a comparison between different notions of convexity for their boundaries. Then a careful application of variational methods to the geodesic problem yields that the convexity of ∂D is equivalent to the existence of a minimal geodesic for each pair of points of...

In this paper, we prove that on every Finsler n-sphere (S, F ) for n ≥ 6 with reversibility λ and flag curvature K satisfying ( λ λ+1 )2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exist [ 2 ] − 2 closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n − 3 closed geodesics possessing irrational average indice...