area of $𝐧(U)$ equals $|∫_U K\rmd A|$

hbghlyj

a2 geometry Richard Earl 1998

第b问就是Gauss-Bonnet Formula对于torus的$\chi=0$吧.

第a问呢

Czhang271828

Use $K(p)=\det \left(\mathrm dN(p)\right)$.

hbghlyj

Czhang271828 发表于 2023-8-21 16:42
Use $K(p)=\det \left(\mathrm dN(p)\right)$.

想起来了😯$K$正好是$\bf n$的Jacobian
page 21 Definition 58(a)中$W_p$是这道题的$\bf n$的导数

hbghlyj

(b) if $X$ is a torus of revolution then what is $\mathbf n(X)$? A double cover of the sphere?

Two points on $X$ have the same unit normal, so they are mapped to the same point by $\mathbf n$

Czhang271828

hbghlyj 发表于 2023-8-21 21:23
本帖最后由 hbghlyj 于 2023-8-21 23:09 编辑 (b) if $X$ is a torus of revolution then what is $\mathb ...

$N: \text{torus}\to \text{sphere}$ is almost a double cover, except for bifurcation on the antipodes.

hbghlyj

Czhang271828 发表于 2023-8-22 15:19
$N: \text{torus}\to \text{sphere}$ is almost a double cover, except for bifurcation on the antipodes

What is “bifurcation on the antipodes”
The topmost of the torus is mapped to the north pole and the lowermost circle of the torus is mapped to the south pole