Geometrical methods for the elasticity theory of membranes

The elasticity theory of curved membranes is developed using geometrical methods. The strain tensor is shown to be the Lie derivative of the metric tensor with respect to a flow. This formulation is coordinate-free, and can be expressed in any convenient coordinate system. A useful fixed coordinate system for the neighborhood of any given membrane surface is constructed. The formalism is used to expand the curvature free energy of a membrane about its minimum and to find the induced flow which minimizes shear dissipation in a curved membrane when the membrane is deformed at constant density. The shear free energy is expanded about its minimum for deformations of the shape at constant density. The curvature free energy is explicitly expanded about its minimum for shape changes of a closed membrane which conserve area and interior volume.