On the derivation of several second order partial differential equations from a generalization of the Einstein equation

A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The roles of spatial variance are studied based on energy estimates,and several dissipative or antidissipative properties are remarked.