Question Number 226638 by Lara2440 last updated on 08/Dec/25
Commented by Lara2440 last updated on 08/Dec/25
![Boy′s Surface can be parametrized in several ways Given complex number w whose ∣∣w∣∣<1, Let g_μ = { ((g_1 =−(3/2)Im[ ((w(1−w^4 ))/(w^6 +(√5)w^3 −1)) ])),((g_2 =−(3/2)Re[ ((w(1+w^4 ))/(w^6 +(√5)w^3 −1)) ])),((g_3 =Im[ ((w^6 +1)/(w^6 +(√5)w^3 −1)) ]−(1/2))) :} and then Set Boy′s Surface X_μ =(g_μ /(∣∣g∣∣)) 1) Prove Boy′s Surface is an Immerion of the Real Projective plane RP^2 into the Euclidean Space.](https://www.tinkutara.com/question/Q226639.png)
$$\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:\mathrm{can}\:\mathrm{be}\:\mathrm{parametrized}\:\mathrm{in}\:\mathrm{several}\:\mathrm{ways} \\ $$$$\mathrm{Given}\:\mathrm{complex}\:\mathrm{number}\:{w}\:\mathrm{whose}\:\mid\mid{w}\mid\mid<\mathrm{1}, \\ $$$$\mathrm{Let}\:{g}_{\mu} =\begin{cases}{{g}_{\mathrm{1}} =−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Im}\left[\:\frac{{w}\left(\mathrm{1}−{w}^{\mathrm{4}} \right)}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]}\\{{g}_{\mathrm{2}} =−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Re}\left[\:\frac{{w}\left(\mathrm{1}+{w}^{\mathrm{4}} \right)}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]}\\{{g}_{\mathrm{3}} =\mathrm{Im}\left[\:\frac{{w}^{\mathrm{6}} +\mathrm{1}}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]−\frac{\mathrm{1}}{\mathrm{2}}}\end{cases}\: \\ $$$$\mathrm{and}\:\mathrm{then}\:\mathrm{Set}\:\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:{X}_{\mu} =\frac{{g}_{\mu} }{\mid\mid{g}\mid\mid}\: \\ $$$$\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Prove}\:\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:\mathrm{is}\:\mathrm{an}\:\mathrm{Immerion}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{Real}\:\mathrm{Projective}\:\mathrm{plane}\:\mathbb{RP}^{\mathrm{2}} \:\mathrm{into}\:\mathrm{the}\:\mathrm{Euclidean}\:\mathrm{Space}. \\ $$
Answered by MrAjder last updated on 09/Dec/25
![D={w∈C:∣w∣<1},Δ(w)=w^6 +(√5)w^3 −1 ∀w∈D:Δ(w)≠0⇒g_μ ∈C^∞ (D,R^3 ) g_1 (w)=−(3/2)Im((w(1−w^4 ))/(Δ(w))),g_2 (w)=−(3/2)Re((w(1+w^4 ))/(Δw)),g_3 (w)=Im((w^6 +1)/(Δ(w)))−(1/2) X_μ (w)=((g_μ (w))/(∣∣g_μ (w)∣∣))∈C^∞ (D,S^2 ) ∀θ∈[0,2π):lim_(r→1^− ) X_μ (re^(iθ) )=lim_(r→1^− ) X_μ (re^(i(θ+π)) ) ⇒∃X^∼ :RP→R^3 ,X^∼ ([w])=X_μ (w) w=u+iv,dX_μ (w)=(∂_u X_μ ∂_v X_μ ) det(dX_μ (w)^T dX_μ (w))=((9∣w^6 +1∣^4 )/(4∣Δ(w)∣^8 ))>0,∀w∈D rank dX_μ (w)=2 ∴X^∼ ∈Imm(RP^2 ,R^3 )](https://www.tinkutara.com/question/Q226669.png)
$${D}=\left\{{w}\in\mathbb{C}:\mid{w}\mid<\mathrm{1}\right\},\Delta\left({w}\right)={w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1} \\ $$$$\forall{w}\in{D}:\Delta\left({w}\right)\neq\mathrm{0}\Rightarrow{g}_{\mu} \in{C}^{\infty} \left({D},\mathbb{R}^{\mathrm{3}} \right) \\ $$$${g}_{\mathrm{1}} \left({w}\right)=−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Im}\frac{{w}\left(\mathrm{1}−{w}^{\mathrm{4}} \right)}{\Delta\left({w}\right)},{g}_{\mathrm{2}} \left({w}\right)=−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Re}\frac{{w}\left(\mathrm{1}+{w}^{\mathrm{4}} \right)}{\Delta{w}},{g}_{\mathrm{3}} \left({w}\right)=\mathrm{Im}\frac{{w}^{\mathrm{6}} +\mathrm{1}}{\Delta\left({w}\right)}−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${X}_{\mu} \left({w}\right)=\frac{{g}_{\mu} \left({w}\right)}{\mid\mid{g}_{\mu} \left({w}\right)\mid\mid}\in{C}^{\infty} \left({D},{S}^{\mathrm{2}} \right) \\ $$$$\forall\theta\in\left[\mathrm{0},\mathrm{2}\pi\right):\underset{{r}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}{X}_{\mu} \left({re}^{{i}\theta} \right)=\underset{{r}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}{X}_{\mu} \left({re}^{{i}\left(\theta+\pi\right)} \right) \\ $$$$\Rightarrow\exists\overset{\sim} {{X}}:\mathbb{RP}\rightarrow\mathbb{R}^{\mathrm{3}} ,\overset{\sim} {{X}}\left(\left[{w}\right]\right)={X}_{\mu} \left({w}\right) \\ $$$${w}={u}+{iv},{dX}_{\mu} \left({w}\right)=\left(\partial_{{u}} {X}_{\mu} \partial_{{v}} {X}_{\mu} \right) \\ $$$$\mathrm{det}\left({dX}_{\mu} \left({w}\right)^{{T}} {dX}_{\mu} \left({w}\right)\right)=\frac{\mathrm{9}\mid{w}^{\mathrm{6}} +\mathrm{1}\mid^{\mathrm{4}} }{\mathrm{4}\mid\Delta\left({w}\right)\mid^{\mathrm{8}} }>\mathrm{0},\forall{w}\in{D} \\ $$$$\mathrm{rank}\:{dX}_{\mu} \left({w}\right)=\mathrm{2} \\ $$$$\therefore\overset{\sim} {{X}}\in\mathrm{Imm}\left(\mathbb{RP}^{\mathrm{2}} ,\mathbb{R}^{\mathrm{3}} \right) \\ $$
Commented by Lara2440 last updated on 10/Dec/25
$$\mathrm{Wow}….\mathrm{thx} \\ $$