Let us define the map $f:\mathbb{R}\to \mathbb{S}^1$ by $x\to e^{2\pi i x}$. I want to prove that this map is an orientation-preserving map and also it's restriction on the unit interval has an orientation-preserving inverse.

What I know:

$f: M\to N$ is an orientation preserving if the determinant of the derivative map on the tangent space level is positive. So here, we need to show $Df(p):T_p\mathbb{R}\to T_{f(p)}\mathbb{S}^1$ has a positive determinant. But here $Df(p)$ is just a real number so we need to show it is a positive number. Consider a curve $\gamma(t)=p+tv$ where $(p,v)\in T_p\mathbb{R}$. $f(\gamma(t))=e^{2\pi i(p+tv)}$. But here I am having difficulty as the derivative will be a $2\times 1$ matrix.

  • $\begingroup$ In the definition of orientation-preserving map I know there is an assumption that source and target manifolds have the same dimension (otherwise you encounter the problem that the derivative is not a square matrix). $\endgroup$ – Paweł Czyż Aug 4 '18 at 22:07
  • $\begingroup$ yes true. Since I am defining orientation as determinant $\endgroup$ – I am pi Aug 11 '18 at 7:57

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