On the Gauss map coming from a framing of the tangent bundle of a compact manifold

Yanghyun Byun; Daewoong Cheong

doi:10.4134/CKMS.2013.28.1.183

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2013; 28(1): 183-189

Printed January 31, 2013

https://doi.org/10.4134/CKMS.2013.28.1.183

On the Gauss map coming from a framing of the tangent bundle of a compact manifold

Yanghyun Byun and Daewoong Cheong

Hanyang University, Seoul National University

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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\'{e} number $\chi{(W)}$, regardless of the specific framing. As a special case, we get a Hopf theorem.

Keywords: Gauss map, Hopf theorem

MSC numbers: 57R70, 58E05