Group structure of the membrane shape equation.

The purpose of this paper is to study the geometry in the plane of the membrane equation or a section equation of a general membrane shape, where the invariance under the group of contact transformations is required. The discussion is mainly based on Cartan's theory of the Lie group. One may find that the relative invariance does not vanish, it is also possible to define a generalized geometry in the plane with the elements of contact of the second order x, y, y', y'' as the elements of the space and with a certain five-parameter group as its fundamental group. In the example of axisymmetric membrane shape equation, one may find that the membrane shape is a five-parameter group and characterized by twelve group structure parameters which are functions of pressure difference, tensile stress and asymmetry effect of the membrane or its environment. When these varieties of membrane or environment change, the structure constants vary; then one can obtain directly the change of symmetric group and the information on the membrane shape variation.