I'm currently attempting to come up with an explicit parametrization of the $k$-torus into $\mathbb{R}^{k+1}$. I've been following the answer here but I'm struggling to prove injectivity.
Let $e_1,\ldots, e_{k+1}$ be the standard basis in $\mathbb{R}^{k+1}$ and $\varepsilon<1$. So according to the link, I should start with an element $v_1$ of length 1 in span$(e_1,e_2)$, i.e.
$$
v_1(\theta_1)=\cos(\theta_1)e_1+\sin(\theta_2)e_2
$$
Then from there I should choose an element $v_2$ of length $\varepsilon$ in span$(v_1,e_3)$, i.e.
$$
v_2(\theta_1,\theta_2)=\varepsilon\cos(\theta_2)v_1(\theta_1)+\varepsilon\sin(\theta_2)e_3
$$
and in general (for $1\leq j\leq k$), set
$$
v_j(\theta_1,\ldots,\theta_j)=\varepsilon^{j-1}\cos(\theta_j)v_{j-1}(\theta_1,\ldots ,\theta_j)+\varepsilon^{j-1}\sin(\theta_j)e_{j+1}.
$$
From this, the map
$$
\Phi:(e^{i\theta_1},\ldots,e^{i\theta_{k}})\mapsto\sum_{j=1}^{k}v_j(\theta_1,\ldots,\theta_j)
$$
is supposed to be the necessary embedding. This makes sense intuitively, but I can't seem to prove injectivity.
Things Tried
The only case I can make any progress is in the case $k=3$. I've tried doing induction and immitating this argument in the inductive step, but it doesn't seem to work.
If $k=3$, then $$\Phi(e^{i\theta_1},e^{i\theta_1})=\Phi(e^{i\theta_1'},e^{i\theta_1'})$$
implies
$$
(1+\varepsilon\cos(\theta_2))v_1(\theta_1)+\varepsilon\sin(\theta_2)e_3=(1+\varepsilon\cos(\theta_2'))v_1(\theta_1')+\varepsilon\sin(\theta_2')e_3.
$$
In this case the vectors being added together are orthogonal, so taking the norm-square of both sides gives
$$
(1+\varepsilon\cos(\theta_2))^2+\varepsilon^2\sin(\theta_2)=(1+\varepsilon\cos(\theta_2'))^2+\varepsilon^2\sin(\theta_2'),
$$
which after simplifying turns into
$$
2\varepsilon\cos(\theta_2)=2\varepsilon\cos(\theta_2),
$$
so $\cos(\theta_2)=\cos(\theta_2')$. This combined with the immediate fact that $\sin(\theta_2)=\sin(\theta_2')$ shows that $\theta_2=\theta_2'$ (in $[0,2\pi)$). The rest follows from the fact that $v_1$ is injective.
Any help is greatly appreciated. Thanks in advance.
Best Answer
Here's a description of an embedding, which can be elaborated into an explicit formula if you like.
Take an embedding of $T^n$ into $\Bbb R^{n+1}$. We want to use this as a basis for one of $T^{n+1}=T^n\times S^1$ into $\Bbb R^{n+2}$.
Translate your $T^n$ so that it lies within the half-plane defined by $x_{n+1}>0$. Then let the embedding of $T^n$ be given by functions $p\mapsto (\phi(p),\psi(p))$ where $\phi(p)\in\Bbb R^n$ and $\psi(p)\in(0,\infty)$. Now the embedding of $T^{n+1}$ into $\Bbb R^{n+2}$ by $$(p,e^{it})\mapsto(\phi(p),\psi(p)\cos t,\psi(p)\sin t).$$ As $\psi(p)>0$ the last two coordinates determine $\psi(p)$ and $e^{it}$.