Isometric immersions with singularities between space forms of the same positive curvature

TL;DR

This paper introduces a generalized concept of space forms called coherent tangent bundles, classifies their realizations as wave fronts in spheres, and extends classical results on isometric immersions of spheres.

Contribution

It defines coherent tangent bundles of space form type and classifies their realizations, generalizing classical theorems about isometric immersions.

Findings

Coherent tangent bundles of space form type are introduced.

Classifications of realizations as wave fronts are provided.

Any isometric immersion of an n-sphere into an (n+1)-sphere of the same curvature is totally geodesic.

Abstract

In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere of the same sectional curvature is totally geodesic.