  Affine deformations serve as basic examples in the continuum mechanics of
deformable 3-dimensional bodies (referred as homogeneous deformations). They
preserve parallelism and are often used as an approximation to general
deformations. However, when the deformable body is a membrane, a shell or an
interface modeled by a surface, the parallelism is defined by the affine
connection of this surface. In this work we study the infinitesimally affine
time - dependent deformations (motions) after establishing formulas for the
variation of the connection, but in the more general context of hypersurfaces
of a Riemannian manifold. We prove certain equivalent formulas expressing the
variation of the connection in terms of geometrical quantities related to the
variation of the metric, as expected, or in terms of mechanical quantities
related to the kinematics of the moving continuum. The latter is achieved using
an adapted version of polar decomposition theorem, frequently used in continuum
mechanics to analyze the motion. Also, we apply our results to special motions
like tangential and normal motions. Further, we find necessary and sufficient
conditions for this variation to be zero (infinitesimal affine motions), giving
insight on the form of the motions and the kind of hypersurfaces that allow
such motions. Finally, we give some specific examples of mechanical interest
which demonstrate motions that are infinitesimally affine but not
infinitesimally isometric.
