Existence and nonexistence of half-geodesics on \(S^2\) (Q2802125)

It is a classical and fundamental problem in differential geometry to prove existence of closed geodesics, especially when the fundamental group of the Riemannian manifold is trivial. The most basic example is the sphere. In this paper, the author studies half-geodesics on the sphere \(S^2\), where a closed geodesic of total length \(L\) is called a half-geodesic if it is distance minimizing on every subinterval of length \(L/2\). For each nonnegative integer \(n\), the author constructs a Riemannian metric on \(S^2\) which admits exactly \(n\) half-geodesics, and also constructs a sequence of Riemannian metrics on \(S^2\) which does not admit any half-geodesic, but converges in the sense of Gromov-Hausdorff to a limit space which admits infinitely many half-geodesic.