Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T04:47:59.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-immersion theorems for differentiable manifolds

Published online by Cambridge University Press:  24 October 2008

B. J. Sanderson  and
R. L. E. Schwarzenberger
B. J. Sanderson
Affiliation:
Department of Pure Mathematics, University of Liverpool
R. L. E. Schwarzenberger
Affiliation:
Department of Pure Mathematics, University of Liverpool
Get access Rights & Permissions [Opens in a new window]

Extract

Let X be a compact differentiable n-manifold of class C∞ and let f be a C∞ map of X into euclidean (n + k)-space. Then f is an immersion if its jacobian has rank n at each point of X. It is an embedding if it is also one-one. We write X ⊆ n + k to denote the existence of an immersion, X ⊂ n + k to denote the existence of an embedding. By a theorem of Whitney, we have X ⊆ 2n − 1, X ⊂ 2n.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 319 - 322
Copyright
Copyright © Cambridge Philosophical Society 1963

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Atiyah, M. F., and Hirzebruch, F., Quelques theéorèmes de non-plongement pour les variétés différentiables. Bull. Soc. Math. France, 87 (1959), 383–396.Google Scholar
(2)Hirsch, M. W.Immersions of manifolds. Trans. American Math. Soc. 93 (1959), 242–276.CrossRefGoogle Scholar
(3)James, I. M., Some embeddings of projective spaces. Proc. Cambridge Philos. Soc. 55 (1959),294–298.CrossRefGoogle Scholar
(4)Sanderson, B. J., Immersions and embeddings of projective spaces. Proc. London Math. Soc. (In the Press.)Google Scholar
(5)Schwarzenberger, R. L. E., Embeddings in euclidean space. Proc. Cambridge Philos. Soc. 59 (1963) 505–507.CrossRefGoogle Scholar