  For a pair of reductive groups $G \supset G'$, we prove a geometric criterion
for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional
in the Archimedean case.
  This criterion leads us to a complete classification of the symmetric pairs
$(G,G')$ having finite-dimensional Shintani spaces.
  A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is
also obtained.
  Furthermore, we prove that symmetry breaking operators of the restriction of
smooth admissible representations yield Shintani functions of moderate growth,
of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.
