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The famous theorems of Cartan, related to the axiom of $r$-planes, andLeung-Nomizu about the axiom of $r$-spheres were extended to K\"ahler geometryby 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 specialisometric immersions such that theorems of type "axioms of $r$-specialsubmanifolds" could hold. We verify also that there are plenty of specialsubmanifolds in real and complex space forms and, in the codimension one case,in Einstein manifolds.

Generalizing axioms of $r$-planes and $r$-spheres on Riemannian and K\"ahler manifolds