  The geometrical diffraction theory, in the sense of Keller,is here
reconsidered as an obstacle problem in the Riemannian geometry. The first
result is the proof of the existence and the analysis of the main properties of
the diffracted rays, which follow from the non-uniqueness of the Cauchy problem
for geodesics in a Riemannian manifold with boundary. Then,the axial caustic is
here regarded as a conjugate locus, in the sense of the Riemannian geometry,
and the results of the Morse theory can be applied.The methods of the algebraic
topology allow us to introduce the homotopy classes of diffracted rays. These
geometrical results are related to the asymptotic approximations of a solution
of a boundary value problem for the reduced wave equation. In particular, we
connect the results of the Morse theory to the Maslov construction, which is
used to obtain the uniformization of the asymptotic approximations. Then, the
border of the diffracting body is the envelope of the diffracted rays and,
instead of the standard saddle point method, use is made of the procedure of
Chester, Friedman and Ursell to derive the damping factors associated with the
rays which propagate along the boundary. Finally, the amplitude of the
diffracted rays when the diffracting body is an opaque sphere is explicitly
calculated.
