2021-28.md

course	course_year	question_number	tags	title	year
Geometry	IB	28	IB 2021 Geometry	Paper 4, Section II, F	2021

Define an abstract smooth surface and explain what it means for the surface to be orientable. Given two smooth surfaces $S_{1}$ and $S_{2}$ and a map $f: S_{1} \rightarrow S_{2}$, explain what it means for $f$ to be smooth

For the cylinder

$$C=\left{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}=1\right},$$

let $a: C \rightarrow C$ be the orientation reversing diffeomorphism $a(x, y, z)=(-x,-y,-z)$. Let $S$ be the quotient of $C$ by the equivalence relation $p \sim a(p)$ and let $\pi: C \rightarrow S$ be the canonical projection map. Show that $S$ can be made into an abstract smooth surface so that $\pi$ is smooth. Is $S$ orientable? Justify your answer.