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)
$$\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:\mathrm{can}\:\mathrm{be}\:\mathrm{parametrized}\:\mathrm{in}\:\mathrm{several}\:\mathrm{ways} \\ $$$$\mathrm{Given}\:\mathrm{complex}\:\mathrm{number}\:{w}\:\mathrm{whose}\:\mid\mid{w}\mid\mid<\mathrm{1}, \\ $$$$\mathrm{Let}\:{g}_{\mu} =\begin{cases}{{g}_{\mathrm{1}} =−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Im}\left[\:\frac{{w}\left(\mathrm{1}−{w}^{\mathrm{4}} \right)}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]}\\{{g}_{\mathrm{2}} =−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{Re}\left[\:\frac{{w}\left(\mathrm{1}+{w}^{\mathrm{4}} \right)}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]}\\{{g}_{\mathrm{3}} =\mathrm{Im}\left[\:\frac{{w}^{\mathrm{6}} +\mathrm{1}}{{w}^{\mathrm{6}} +\sqrt{\mathrm{5}}{w}^{\mathrm{3}} −\mathrm{1}}\:\right]−\frac{\mathrm{1}}{\mathrm{2}}}\end{cases}\: \\ $$$$\mathrm{and}\:\mathrm{then}\:\mathrm{Set}\:\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:{X}_{\mu} =\frac{{g}_{\mu} }{\mid\mid{g}\mid\mid}\: \\ $$$$\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Prove}\:\mathrm{Boy}'\mathrm{s}\:\mathrm{Surface}\:\mathrm{is}\:\mathrm{an}\:\mathrm{Immerion}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{Real}\:\mathrm{Projective}\:\mathrm{plane}\:\mathbb{RP}^{\mathrm{2}} \:\mathrm{into}\:\mathrm{the}\:\mathrm{Euclidean}\:\mathrm{Space}. \\ $$