Real Scalar Fields on Manifolds

A generic theory of a single real scalar field is considered, and a simple method is presented for obtaining a class of solutions to the equation of motion. These solutions are obtained from a simpler equation of motion that is generated by replacing a set of the original coordinates by a set of generalized coordinates, which are harmonic functions in the spacetime. These ansatz solutions solve the original equation of motion on manifolds that are defined by simple constraints. These manifolds, and their dynamics, are independent of the form of the scalar potential. Some scalar field solutions, and manifolds upon which they exist, are presented for Klein-Gordon and quartic potentials as examples. Solutions existing on leaves of a foliated space may allow inferences of the characteristics expected of exact bulk solutions.