Math Seminars
Wing Kai Ho (PSU)
Manifolds without 1/k-geodesics
This is an elementary talk, definitely accessible to graduate
students. We answer a question of C. Sormani asking if, for every k,
there is a metric on S^2 without 1/k geodesics.
Consider constant speed geodesic \gamma: S --> M parameterized by a
circle of unit length. We say that \gamma is a 1/k geodesic if its restriction
to every segment of length 1/k is a shortest path.
If M is not simply connected, the shortest non-contractible geodesic
is a 1/2-geodesic (its length is a very important characteristic of M
called 1-systole). It is also known that there is a closed geodesic on
every (simply-connected) manifold. Every Riemannian metric on S^2
has infinitely many closed geodesics. However, for every given k, we
will construct a Riemannian metric on S^2 without 1/k-geodesics.