Invariant Solutions Of Some Geometric Equations On Jet Bundle Over R～2

In this paper, firstly, using invariant group theory we give the invariant groups of geometric equation k_t = k²(k_θθ + k) and equation S_t = 1/(S_θθ+S) Then we give some invariant solutions of wave equation u_tt= u_xx under some scaling group. At the same time, we consider the interested equation u_xx + u_yy+ λu^p = 0 and we find out the invariant groups and some group-invariant solutions. Finally, we investigate two important equations in Geometry: Gauss curvature equation and mean curvature equation. We give their one-parameter groups and their group-invariant solutions on jet bundle over R² when Gauss curvature and mean curvature are both constants. Noticing the symmetry group's characteristics, we get some invariances of constant Gauss curvature surfaces.