We work through the first and second variations of arclength and discuss the rise of index forms and Jacobi fields along with their application in finding conjugate or focal points on Riemannian or Lorentzian manifolds. We then prove two theorems on the maximal distances of two conjugate or focal points along geodesics for manifolds that satisfy certain boundedness conditions for the Ricci tensor. These are Myers’s theorem in Riemannian geometry and Hawking’s theorem in Lorentzian geometry.Peer reviewe
