Rigidity of spheres in Riemannian manifolds and a non-embedding theorem

Letf:M →\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar M$$ \end{document} be an isometric immersion between Riemannian manifolds. The purpose of this paper is to find the minimum possible conditions onM and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar M$$ \end{document} (in the terms of curvatures and external diameter) in order to the image off be contained in a sphere. Our results generalize the other authors work in three major steps, domain, range and the codimension of immersions. As a byproduct, we obtain the non-embedding theorems Chern-Kuiper, Moore and Jacobowitz. The proofs are based on the maximum (comparison) principle.