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

Byun, Yanghyun;Cheong, Daewoong

• Published : 2013.01.31
• 34 11

#### 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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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