  The famous theorems of Cartan, related to the axiom of $r$-planes, and
Leung-Nomizu about the axiom of $r$-spheres were extended to K\"ahler geometry
by several authors.
  In this paper we replace the strong notions of totally geodesic submanifolds
($r$-planes) and extrinsic spheres ($r$-spheres) by a wider class of special
isometric immersions such that theorems of type "axioms of $r$-special
submanifolds" could hold. We verify also that there are plenty of special
submanifolds in real and complex space forms and, in the codimension one case,
in Einstein manifolds.
