I sat through a talk given by Jared Wunsch about a week ago in which he mentioned a certain real valued function (defined for a fixed compact Riemannian manifold) being smooth on all of except for those points where there is a closed geodesic of that length on the manifold. So at the end I asked the question ‘How many lengths can there be?’ as I am curious about whether ‘being smooth outside those lengths’ is a strong statement for all manifolds (or for generic manifolds).
Later on I found this question is quite cool so I went on and thought a bit more about it.
Turns out this ‘set of lengths of closed geodesics’ is called the length spectrum of the manifold.
Without much difficulty, I constructed surfaces with length spectrum containing a sequence of accumulating points or a (measure zero) Cantor set. (by taking the surface of revolutions of graphs of real valued functions with certain properties)
Note that the length spectrum itself need not be closed as one can easily construct examples where there is a sequence of closed geodesics accumulating to a parametrized curve that goes along a closed geodesic twice. However, since there can’t be a sequence of lengths approaching to (because, for example, the injectivity radius is bounded below from by compactness) we may throw in all integer multiples of the lengths of closed geodesics, in each finite interval this is merely taking a union of finitely many copies of the geodesics (hence essentially does not change the size of the set). This resulting set of ‘generalized lengths of closed geodesics’ is closed.
I wish to show that the set of generalized lengths of closed geodesics is both measure and nowhere dense (hence meager in the Baire category sense) by applying Sard’s theorem to a appropriately defined setting.
I’ll try to do this sometime soon, to be continued~