We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid Q that are cut out by algebraic surfaces in R. Such geodesics are either connected components of spatial elliptic curves or rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves and the addition...

Let G4 be the space of all simple closed geodesics on the punctured sphere Σ4. We construct an explicit homeomorphism of the completion of G4 onto a circle by using geometric intersection numbers. Also, we relate these geometric intersection numbers to trace polynomials of transformations corresponding to geodesics in G4 in a representation of π1(Σ4) into PSL(2,C).

We study “flat knot types” of geodesics on compact surfaces M2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.

Causal radial geodesics with a positive interval in the Schwarzschild metric include a subset of trajectories completely confined under a horizon, which compose a thermal statistical ensemble with the Hawking-Gibbons temperature. The Bekenstein–Hawking entropy is given by an action at corresponding geodesics of particles with a summed mass equal to that of black hole in the limit of large mass.

We introduce a novel method for the approximation of shortest geodesics on smooth triangulated surfaces. The method relies on the theory of normal curves on triangulated surfaces and their relations with geodesics. We also relate in this work to normal surfaces and comment on the possible extension of the method for finding minimal surfaces inside 3-manifolds.

Hedlund [He] constructed Riemannian metrics on n-tori, n ≥ 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert’s existence results of minimal geodesics [Ba2] are optimal for nilpotent fundamental groups.

We exhibit an example of a homogeneous Lorentzian manifold G/H whose homogeneous geodesics form just the light-cone and a hyperplane in the tangent space. For all light-like homogeneous geodesics, the natural parameter of the orbit is not the affine parameter of the geodesic.

Thirteen simple closed geodesics are found in the lemniscate. Among these are nine “mirrors”—geodesics of reflection symmetry—which generate the full octahedral group and determine a triangulation of the lemniscate as a disdyakis dodecahedron. New visualizations of the lemniscate are presented.

We shall prove a Poincaré-Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.

We prove that the language of all geodesics of any Garside group, with respect to the generating set of divisors of the Garside element, forms a regular language. In particular, the braid groups admit generating sets where the associated language of geodesics is regular.