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Annali di Matematica Pura ed Applicata

, Volume 49, Issue 1, pp 117–128 | Cite as

The existence of non-degenerate functions on a compact differentiablem-manifoldM

  • Marston Morse
Article

Summary

Let M be a compact differentiable m-manifold of class Cm in En, n=2m+1. Let x=(x1, ..., xn) represent a point in En. The union of the direction c on the direction sphere Sn−1 in En such that the scalar product c · x defines a non-degenerate fonction on M is an open subset of Sn−1 whose complement θ has a Lebesgue measure zero on Sn−1. When M is non-compact θ can be everywhere dense on Sn−1, but still has Lebesgue measure zero.

Keywords

Scalar Product Open Subset Lebesgue Measure Measure Zero Lebesgue Measure Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Swets & Zeitlinger B. V. 1960

Authors and Affiliations

  • Marston Morse
    • 1
  1. 1.Princeton

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