Abstract:
In this paper, we prove that for every Finsler metric on the 2-dimensional sphere there exist at least two distinct prime closed geodesics. For the case of the two-sphere, this solves an open problem posed by D. V. Anosov in 1974.

Abstract:
Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for "thin" spheres, when the area $A$ is much less than $d^2$, where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when ${A\over d^2}\longrightarrow 0$.

Abstract:
The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.

Abstract:
We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.

Abstract:
The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.

Abstract:
We study 2D quantum gravity on spherical topologies employing the Regge calculus approach with the dl/l measure. Instead of the normally used fixed non-regular triangulation we study random triangulations which are generated by the standard Voronoi-Delaunay procedure. For each system size we average the results over four different realizations of the random lattices. We compare both types of triangulations quantitatively and investigate how the difference in the expectation value of the squared curvature, $R^2$, for fixed and random triangulations depends on the lattice size and the surface area A. We try to measure the string susceptibility exponents through finite-size scaling analyses of the expectation value of an added $R^2$-interaction term, using two conceptually quite different procedures. The approach, where an ultraviolet cut-off is held fixed in the scaling limit, is found to be plagued with inconsistencies, as has already previously been pointed out by us. In a conceptually different approach, where the area A is held fixed, these problems are not present. We find the string susceptibility exponent $\gamma_{str}'$ in rough agreement with theoretical predictions for the sphere, whereas the estimate for $\gamma_{str}$ appears to be too negative. However, our results are hampered by the presence of severe finite-size corrections to scaling, which lead to systematic uncertainties well above our statistical errors. We feel that the present methods of estimating the string susceptibilities by finite-size scaling studies are not accurate enough to serve as testing grounds to decide about a success or failure of quantum Regge calculus.

Abstract:
In this paper we prove that for every bumpy Finsler metric $F$ on every rationally homological $n$-dimensional sphere $S^n$ with $n\ge 2$, there exist always at least two distinct prime closed geodesics.

Abstract:
We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.

Abstract:
We study geodetic lines on a surface generated by a small deformation of the standard 2D-sphere. We construct an auxiliary hamiltonian system with the view of describing geodetic coils and almost closed geodesics, by using the fact that loops of the coil can be well approximated by great circles of the sphere. The phase space of the auxiliary system is determined by the graph generated by separatrixes of its solutions, the vertices of the graph corresponding to almost closed geodesics and the edges to the geodetic coils joining them. Topological types of the graph depend on the parameters determining the deformation. Using the method of averaging in conjunction with the computer modelling of the auxiliary system, we obtain a fairly detailed visualization of geodesics on the deformed sphere.