In-plane dynamics of membranes having constant curvature

We have studied the dynamics of in-plane displacement of a class of two-dimensional homogeneous and isotropic elastic membranes having constant curvature. The equation of motion of a general in-plane displacement field has been derived in covariant form using the variational formulation. The contribution of curvature on the dynamics of the field variable is clearly observed in the field equation. We observe that for certain membranes with constant curvature, the curvature term introduces a quadratic potential with the sign of the coupling constant being decided by the Ricci curvature of the membrane. Using the Helmholtz theorem, we reduce the field equation to two decoupled scalar equations which exhibit an interesting structure. We consider some examples of membranes with positive, and negative curvatures and comment on the nature of solutions expected. We analyze the deformation of the membranes in terms of the dilatation, shear and vorticity of the displacement field.