Can you prove that $d \Omega_1=0$, $d \Omega_2=0$?
Can you write down the expression for the general $n$? Or just prove the general case?
Consider the following 2-cell in $\R^3$ (it parametrized the unit sphere), $$\gamma: [0,1]^2 \to \R^3, \quad (s,t) \mapsto (\sin (\pi s) \cos (2\pi t), \sin (\pi s) \cos (2\pi t), \cos \pi s) $$ What is $\int_\gamma \Omega_2$?
Suppose we use a different parametrization of $S^2$, the stereographic projection $$ \gamma: \R^2 \mapsto \R^3, \quad (a,b) \mapsto (\frac{2a}{1+a^2+b^2}, \frac{2b}{1+a^2+b^2}, \frac{-1+a^2+b^2}{1+a^2+b^2}) $$ Can you explain why $\int_{\gamma} \Omega_2$ is the same as the previous one?
(Optional) Let $\gamma_1, \gamma_2, [0,1] \to \R^3$ be two smooth loops, i.e. $\gamma_i(0)=\gamma_i(1)$ and $\gamma_i'(0) = \gamma_i'(1)$. Suppose they have disjoint images. Define a 2-cell $\phi: [0,1]^2 \to \R^3$ by $\phi(s,t) = \gamma_1(s) - \gamma_2(t)$. Prove that $(4\pi)^{-1} \int_\phi \Omega_2$ is an integer (hence insensitive to small perturbation of $\gamma_1, \gamma_2$). This is called the linking number of two knots, and is a topological invariant of links. Can you compute some examples? Can you see its topological meaning?
math105-s22/hw/hw11.1649873110.txt.gz · Last modified: 2022/04/13 11:05 by pzhou