A non-existence theorem for isometric immersions

Let f:M⟶M¯ be an isometric immersion between Riemannian manifolds. For certain conditions on MM and M¯ in terms of curvatures and external diameter, we extend the non-embedding theorem of Chern and Kuiper to the isometric immersions of non-compact manifolds. Also, our results generalize and improve the main results of Jorge and Koutroufiotis [L. Jorge, D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. J. Math. 103 (4) (1981) 711–725] and Veeravalli [A. R. Veeravalli, A sharp lower bound for the Ricci curvature of bounded hypersurfaces in space forms, Bull. Austral. Math. Soc. 62 (1) (2000) 165–170].