Riemannian Manifolds and Geodesics

We refer to differential geometry of manifolds equipped with the Riemannian metric, in particular we deal with various aspects of smooth surfaces in order to optimize connections among points on them. To succeed in this, one can investigate the Gaussian curvature of surfaces or embed them in an Euclidean space. Gauss curvature is defined in terms of the curvatures of certain plane curves related to the surface, for instance in dimension 3, it is the product of the principal curvatures at a point, being the mean curvature their average. The smooth surfaces of constant Gaussian curvature play an important role because the uniformization theorem applies to them, thus theyare conformally equivalent to one of the three Riemann surfaces and admit a Riemannian metric of constant curvature 1,0 or−1. For any surface embedded in an Euclidean space such as minimal surfaces and ruled surfaces, we are able to measure lengths of curves lying on it, angles between two curves and areas of regions on it. Using the variational method, this can be well illustrated by functions that maximize or minimize functionals through the nonlinear Euler-Lagrange equation. Typical models of calculus of variations in differential geometry are the geodesics. Mathematically speaking, they are described using PDEs, and geometry of surfaces revolves around the study of geodesics. The geodesic curvature at a point of a curve on a surface is the measure of how far the curve is from being a geodesic: it is an intrinsic invariant depending only on the metric near the point. We are interested to establish links between Riemannian geometry and geodetic datums: the latter is a set of reference points in order to locate places on Earth. In mathematical terms, a reference datum is a constant surface used to describe location of unknown points on Earth. Reference datums can have different radii or center points,so it is necessary to combine their better-approximated coordinates with specific points. With the geometric methods above described, we are able to derive functions that allow us to give in a geodetic system measurements closer to reality and to minimize them.