Differential geometry and curvatures of equipotential surfaces in the realization of the World Height System

The notion of an equipotential surface of the Earth’s gravity potential is of key importance for vertical datum definition. The aim of this contribution is to focus on differential geometry properties of equipotential surfaces and their relation to parameters of Earth’s gravity field models. The discussion mainly rests on the use of Weingarten’s theorem that has an important role in the theory of surfaces and in parallel an essential tie to Brun’s equation (for gravity gradient) well known in physical geodesy. Also Christoffel’s theorem and its use will be mentioned. These considerations are of constructive nature and their content will be demonstrated for high degree and order gravity field models. The results will be interpreted globally and also in merging segments expressing regional and local features of the gravity field of the Earth. They may contribute to the knowledge important for the realization of the World Height System.