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On Immersion of Manifolds

Published online by Cambridge University Press:  20 November 2018

Hans Samelson
Hans Samelson*
Affiliation:
University of Michigan
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In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'

It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 529 - 534
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Chern, S.S. and Lashof, R.K., On the total curvature of immersed manifolds, Amer. J. Math., 79 (1957), 306–318.Google Scholar
2. Chern, S.S. and Lashof, R.K., On the total curvature of immersed manifolds, II, Mich. Math. J., 5 (1958), 5–12.Google Scholar
3. Lashof, R. and Smale, S., On the immersion of manifolds in Euclidean space, Ann. Math., 68 (1958), 562–583.Google Scholar
4. Morse, M., Calculus of variations in the large, Amer. Math. Soc. Coll. Pub., 18 (1934).Google Scholar