Various concepts of differential geometry, in particular aspects related to compatibility, will be applied to the kinematics of first- and second-order geometrically nonlinear continuum mechanics. Thereby distinction will be made between the straightforward conditions for the distortion and the double-distortion being integrable and the more involved conditions for the strain (metric) and the double-strain (connection) being integrable. The latter leads to geometrically nonlinear and extended versions of the celebrated St.-Venant compatibility conditions.
