Lie algebra and some geodesic properties

Geodesy is a word of Greek origin. Geo=earth, world; dezi=means to divide, measure. Geodesy is the science of measuring and projection of the earth's surface. Therefore; Geodesy teaches how to measure the whole or large parts of the world and how to obtain information about its size, shape, dimensions and gravitational field by evaluating these measurements according to mathematical principles. It also examines the problem of projection of earth points onto a surface. I tried to define Lie groups and Lie algebra based on some definitions about groups and algebra in real numbers. Then, I talked about some Geodesic features and tried to analyze this Geodesy mathematically first. Later, I tried to work on their adaptation to space geometry. For example; I related this to Euler's formula by applying it to the rotational ellipsoid. On the other hand, with the help of Frenet trihedron, I defined tangent and normal unit vectors on the surfaces and obtained curvature and curvature radii accordingly. The reason for all this work was to obtain the Geodesy curve and equation. Therefore; With the help of these studies, I tried to obtain the curvature of a curve and its equation on any surface in space. As a matter of fact, when I adapted this to the earth ellipsoid, I obtained Geodesic equations under the name of Geodesic curvature. Then, using this Geodesic curvature, I obtained the Geodesic torsion and its equation. In conclusion; I have concluded that the shortest curve between any two points on a surface in space is the Geodetic curve. Like this; In this study, I took into consideration the previous studies and tried to find definitions and equations related to different Lie groups and Lie algebras and obtain Geodesic curvature, Geodesic equation and Geodesic torsion.