Wave Equations in Riemannian Spaces

With regard to applications in quantum theory, we consider the classical wave equation involving the scalar curvature with an arbitrary coefficient $\xi$. General properties of this equation and its solutions are studied based on modern results in group analysis with the aim to fix a physically justified value of $\xi$. These properties depend essentially not only on the values of $\xi$ and the mass parameter but also on the type and dimension of the space. Form invariance and conformal invariance must be distinguished in general. A class of Lorentz spaces in which the massless equation satisfies the Huygens principle and its Green's function is free of a logarithmic singularity exists only for the conformal value of $\xi$. The same value of $\xi$ follows from other arguments and the relation to the known WKB transformation method that we establish.