Geodesic chords for closed geodesics in a Riemannian manifold.

A geodesic chord for a closed geodesic C in a manifold M is a geodesic segment G : {0,1] –> M with endpoints in C and image not entirely in C. The chord G is said to be proper when its endpoints are distinct. I’ll discuss some existence results for (proper) chords for closed geodesics and a conjecture aiming to characterize real projective spaces in terms of geodesic chords.