  We discuss infinitesimal isometries of the middle surfaces and present some
characteristic conditions for a function to be the normal component of an
infinitesimal isometry. Our results show that those characteristic conditions
depend on the Gaussian curvature of the middle surfaces: Normal components of
infinitesimal isometries satisfy an elliptic problem, or a parabolic one, or a
hyperbolic one according to the middle surface being elliptic, or parabolic, or
hyperbolic, respectively. In those cases, a problem of determining an
infinitesimal isometry is changed into that of 1-dimension. Then we apply those
results to the energy functionals of bending of shells which has been obtained
as two-dimensional problems by the limit theory of Gamma-convergence from the
three-dimensional nonlinear elasticity. Therefore the limit theory of
Gamma-convergence reduces to be a one-dimensional problem in the those cases.
