Compatibility of Rotations with the Change-of-Metric Measures in a Deformation of a Material Surface

Considering a deformation of a thin shell, modelled by a flexible material surface, one is confronted with diversity of shapes that can be attained by stretching and bending. Yet, not all conceivable shapes are accessible in reality. There is a reasonable limit to this richness, be it due to the fact that any material can withstand stretching only up to a certain point and then falls. Once the stretches are restricted, some deeper reaching consequences of this may be detected. For instance, a flat plate cannot be deformed into a strongly curved cap without generating large stretches. Apparently, the nature of this restraint involves something more than just mere restriction on the elongation of the material fibres of the shell. Curvature must be liable to some restrictions of a similar type and, consequently, so must be the rotations.