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

TL;DR

This paper extends classical theorems about $r$-planes and $r$-spheres to broader classes of submanifolds in Riemannian and K"ahler manifolds, introducing special isometric immersions.

Contribution

It generalizes the axioms of $r$-planes and $r$-spheres by replacing strong conditions with special isometric immersions, broadening applicability in various geometric contexts.

Findings

Existence of many special submanifolds in space forms

Extension of classical theorems to K"ahler geometry

Applicability in Einstein manifolds for codimension one cases

Abstract

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.