Study on the Space Geodesy

Abstract

Geodesic is a kind of shortest path among all curves in a metric space, which originally appeared in the Gaussian period. Since then, it was extensively applied in various branches of mathematics and physics, such as Riemannian geometry, digital geometry, Einstein’s relativity, etc. This thesis mainly discusses geodesic definitions in Euclidean space and those in smooth manifolds after introducing the basic theory of smooth manifolds. At first, the thesis applies three ways to define geodesics on a surface, including the geodesic curvature method, the shortest distance method, and the relation of the principal normal of a curve and the normal vector of a surface. Then, the paper introduces some basic concepts in Riemannian manifolds. Finally, it gives a general definition of geodesics in a smooth Riemannian manifold.