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ON THE GAUSS MAP COMING FROM A FRAMING OF THE TANGENT BUNDLE OF A COMPACT MANIFOLD

Byun, Yanghyun;Cheong, Daewoong

  • Received : 2012.04.15
  • Published : 2013.01.31

Abstract

Let W be a parallelizable compact oriented manifold of dimension $n$ with boundary ${\partial}W=M$. We define the so-called Gauss map $f:M{\rightarrow}S^{n-1}$ using a framing of TW and show that the degree of $f$ is equal to Euler-Poincar$\acute{e}$ number ${\chi}(W)$, regardless of the specific framing. As a special case, we get a Hopf theorem.

Keywords

Gauss map;Hopf theorem

References

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  3. T. Sakkalis, On a theorem of H. Hopf, Internat. J. Math. Math. Sci. 13 (1990), no. 4, 813-816. https://doi.org/10.1155/S0161171290001132
  4. V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., 1974.
  5. H. Hopf, Uber die Curvatura integra geschlossener Hyperflachen, Mathematische Annalen 95 (1925-1926), 340-365. https://doi.org/10.1007/BF01206615

Acknowledgement

Supported by : NRF