Problem
Problem 3.10 Study the orientability of the following $C^{\infty}$ manifolds:
(i) A cylindrical surface of $\mathbb{R}^{3}$, with the atlas given in Problem 1.30.
(ii) The Möbius strip, with the atlas given in Problem 1.31.
(iii) The real projective space $\mathbb{R} P^{2}$, with the atlas given in Problem 1.81.