Generalizing axioms of r-planes and r-spheres on Riemannian and Kähler manifolds

Original · EN

The famous theorems of Cartan, related to the axiom of r-planes, and Leung-Nomizu about the axiom of r-spheres were extended to Kähler 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.