The Fundamental Theorem of Differential Geodesy

Abstract

In this chapter we take the first step towards investigating the global, i.e. large-scale, structure of the geopotential field. We first state the Pizzetti Theorem, which we call the Fundamental Theorem of Differential Geodesy, since it essentially specifies the natural domain of validity of differential geodesy. Intuitively, one is accustomed to assuming that the nearby equipotential surfaces of a uniformly rotating Earth are closed surfaces which are locally isometrically imbedded in an Euclidean 3-space E₃, and Pizzetti’s result indicates when this is a reasonable assumption. We then examine the Extension Problem which, in effect, considers to what extent Gaussian differential geometry can in practice be applied to the study of these equipotential surfaces.