----- > [!proposition] Proposition. ([[condition to be an extended surface of revolution]]) > [[parameterized curve|Parameterize]] via $\gamma:[0,1] \to \mathbb{R}^{3}$, $\gamma([0,1])=C$. WLOG the $r$-axis is the $z$-axis (a convention set in the book). > First note that automatically $p$ and $q$ will need to be the endpoints of $C$ or else the [[surface of revolution]] $S$ will not actually be a [[surface]] (suppose $\gamma(\ell)=p$ and $\ell \in (0,1)$, then any [[neighborhood]] $V \ni p$ in $S$ about $p$ can't be [[homeomorphism]] to $\mathbb{R}^{2}$ because $\mathbb{R}^{2}$ is [[path-connected]] while $V$ is not). So we assume hereon that $\gamma(0)=p$ and $\gamma(1)=q$. > One condition ensuring that $S$ is a [[differentiable Euclidean submanifold|regular surface]] would be that the closed [[parameterized curve|parameterized curve]] $\gamma * \hat{\gamma}:[0,1] \to \mathbb{R}^{3}$ obtained as the [[fundamental groupoid|concatenation]] of $C$ with its [[reflection]] $\hat{C}$ across the $z$-axis,$(\gamma * \hat{\gamma})(t)= \begin{cases} \big( \gamma_{1}(2t), \gamma_{2}(2t), \gamma_{3}(2t) \big) & t \in \left[ 0, \frac{1}{2} \right] ;\\ \big(- {\gamma}_{1}(2-2t), -{\gamma}_{2}(2-2t), {\gamma}_{3}(2-2t) \big) & t \in \left[ \frac{1}{2 }, 1 \right], \end{cases}$ is [[regular curve|regular]]. (It is [[well-defined]] since $\begin{align} \gamma\left( 2 \cdot \frac{1}{2} \right)= & \gamma(1)=(q_{1},q_{2},q_{3}) \\ = & (-q_{1},-q_{2},q_{3}) \text{ since } q \text{ lives on } z\text{-axis} \\ = & - {\gamma}_{1}\left( 2-2\cdot \frac{1}{2} \right), -{\gamma}_{2}(2-2\cdot \frac{1}{2}), {\gamma}_{3}(2-2\cdot \frac{1}{2}). \end{align}$ . It is closed since $\begin{align} (\gamma * \hat{\gamma})(0)= & \gamma(0) \\ = & (p_{1},p_{2},p_{3}) \\ = & (-p_{1},-p_{2},p_{3}) \text{ since }p \text{ lives on }z\text{-axis} \\ = & - {\gamma}_{1}(2-2\cdot0), -{\gamma}_{2}(2-2 \cdot 0), {\gamma}_{3}(2-2 \cdot 0) \\ = & (\gamma * \hat{\gamma})(1). \end{align}$. Note that it is automatically regular away from $p$ and $q$, so we actually just need to check whether $(\gamma * \hat{\gamma})(0) \neq \b 0$ (wrap around to make differentiation make sense here) and $(\gamma * \hat{\gamma})\left( \frac{1}{2} \right) \neq \b 0$.) > This condition is sufficient. Because the image of $\gamma * \hat{\gamma}$ will be symmetric about the $z$-axis, *remark 4* of Do Carmo section 2-3 applies to guarantee that $S$ is an extended regular [[surface of revolution]]. > This condition is necessary. Indeed, suppose that the [[surface of revolution]] induced by $\gamma * \hat{\gamma}$ is not [[differentiable Euclidean submanifold|regular]] at $p$ or $q$. WLOG $p$. The book tells us that one [[coordinate patch]] about $p$ is $\alpha(u,v)=(\gamma_{1}(v) \cos u, \gamma_{1}(v) \sin u, \gamma_{3}(v)), \alpha(u_{*}, v_{*})=p$ where $v$ belongs to some [[open interval]] $(a,b)$ and $f(v)>0$. Now we see that $D\alpha(u_{*},v_{*})=\begin{bmatrix} -\gamma_{1}(v_{*})\sin u_{*} & \cancel{\gamma_{1}'(v_{*})}^{0} \cos u_{*}) \\ \gamma_{1}(v_{*})\cos u_{*} & \cancel{\gamma_{1}'(v_{*})}^{0} \sin u_{*} \\ 0 & \cancel{\gamma_{3}'(v_{*}) }^{0} \end{bmatrix}$ is obviously not [[injection|injective]]. Hence $S$ must not be an extended regular [[surface of revolution]]. ^b4eaa2 > [!proof]- Proof. ([[condition to be an extended surface of revolution]]) > ~ ----- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag