| Résume | In a Riemannian manifold, two points along a geodesic are conjugate when there exists a non-trivial Jacobi field vanishing at those two points; equivalently, the geodesic ceases to be a locally length-minimizing path past the second point in the conjugate pair. A celebrated theorem of Burago-Ivanov, first established in dimension two by Hopf, asserts that any Riemannian n-torus without conjugate points must be flat. Inspired by this theorem, I will sketch the proof of an analogous result for the Euclidean space: on the n-dimensional Euclidean space, any Riemannian metric that is asymptotic to an order larger than 2 at infinity to the Euclidean metric and has no conjugate points must be isometric to the Euclidean metric. |
