Curtiss Lyman

The minimization of distance is an important problem in mathematics, andits solution is well understood in Euclidean space. However, oneencounters difficulties when generalizing this problem to other spaces. Inorder to address this problem, this talk will introduce some of the basicnotions of Riemannian geometry, such as the definitions of a manifold anda metric. This will then lead us to appropriate generalizations of theconcepts of a derivative and a straight line, namely a covariant derivativeand a geodesic, respectively. We will then explore how these concepts areused to determine the minimization of distance on Riemannian manifolds.Lastly, we will see that there can exist points, called conjugate points, pastwhich geodesics no longer minimize distance.