Geodetic heights and holonomity

It is sometimes stated in the geodetic literature that the normal height system, so important in both geodetic theory and practice, is nonholonomic, i.e. the normal height of a point in reality depends also on the integration path of a certain differential. On the contrary, this paper proves that the normal height system is holonomic also identifying the critical point on which the nonholonomic statement is based. Besides that, the general concepts related to the definition of the height system are revised and an overview of the current heights is given. Indeed, given the theorical and practical importance of the subject, this is a key item in Geodesy that must be clearly stated by using definitions and results well known in mathematics.