Le theoreme de conservation de la courbure et de la torsion

Abstract

Based on exterior calculus, the G. Frobenius integration theorem, holonomic and anholonomic Riemannian geometry, the typical geodetic problems are summarized in a unified manner. The E. Cartan pseudotorsion of natural orthogonal coordinates causes the misclosure of a closed three dimensional traverse. Natural coordinate differences are path dependent, anholonomic, nonintegrable, nonunique, therefore. The geodetic pseudotorsion form depends only on the components of the A. Marussi tensor of gravity gradients. A physically defined coordinate system can be found which is pseudotorsion free, whose coordinates are holonomic, integrable, unique. The G. Frobenius transformation matrix is of rank three, explaining the number of three dimensions of an intrinsic surface geometry. The matrix elements depend on either the second derivatives of the real gravity potential and the Euclidean norm of its gravity vector or the second derivatives of the standard gravity potential, the Euclidean norm of its standard gravity vector and the vertical deflections. Incomplete information of the earth's gravity field leads to the concept of boundary value problems and satellite geodesy.