The geometrical diffraction theory, in the sense of Keller, is here reconsi
dered as an obstacle problem in the Riemannian geometry. The first result i
s 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 prob
lem for geodesics in a Riemannian manifold with boundary. Then, the axial c
austic is here regarded as a conjugate locus, in the sense of the Riemannia
n 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 di
ffracted rays. These geometrical results are related to the asymptotic appr
oximations of a solution of a boundary value problem for the reduced wave e
quation. In particular, we connect the results of the Morse theory to the M
aslov construction, which is used to obtain the uniformization of the asymp
totic approximations. Then, the border of the diffracting body is the envel
ope 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 th
e damping factors associated with the rays which propagate along the bounda
ry. Finally, the amplitude of the diffracted rays when the diffracting body
is an opaque sphere is explicitly calculated.