C2 densely the 2-sphere has an elliptic closed geodesic

In the paper we give some necessary conditions for a mapping to be a $\kappa$-geodesic in non-convex complex ellipsoids. Using these results we calculate explicitly the Kobayashi metric in the ellipsoids $\{|z_1|^2+|z_2|^{2m}<1\}\subset\bold C^2$, where $m<\frac12$.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give an exposition of a formula of Daskalopoulos, Hamilton and Sesum for solutions to the Ricci flow on the 2-sphere. This is one of several estimates used by them to classify ancient solutions on the 2-sphere.

We prove the existence of multiple closed geodesics on non-compact cylindrica manifolds.

We show that for an open and dense set non-reversible Finsler metrics on a sphere of odd dimension $n=2m-1 \ge 3$ there is a second closed geodesic with Morse index $\le 4(m+2)(m-1)+2.$

This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit geodesic computation for a Riemannian hypersurface.

In this paper we study some generic properties of the geodesic flows on a convex sphere. We prove that, $C^r$ generically ($2\le r\le\infty$), every hyperbolic closed geodesic admits some transversal homoclinic orbits.

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 the density of algebraically closed geodesics on n-dimensional ellipsoids with respe...

We introduce a flow of $G_2$-structures defining the same underlying Riemannian metric, whose stationary points are those structures with divergence-free torsion. We show short-time existence and uniqueness of the solution.