Find a isometric immersion of the Torus $T^{n}$ on $\mathbb{R}^{2n}$.

Question

Find a isometric immersion of the "plane" Torus $T^{n}$ on $\mathbb{R}^{2n}$.

Isometric Immersion, let $f:M^{n}\to N^{n+k}$ a immersion, i.e., $f$ is differentiable and $df_{p}:T_{p}M\to T_{f(p)}N$ is injective for all $p\in M$. If $N$ has a Riemannian structure, $f$ induces a Riemannian structure in $M$ by $\langle u,v\rangle_{p}=\langle df_{p}(u),df_{p}\rangle_{f(p)}$, $u,v\in T_{p}M$, then f is called isometric immersion. For the problem, I think is enough take $i:T^{n}\hookrightarrow \mathbb{R}^{2n}$, but I don't think this was the focus of problem. Any hints, thanks!

Hint: can you do it for n=1? – Steve D Apr 06 '16 at 03:02 — Steve D, Apr 06 '16 at 03:02

score 5 · Answer 1 · answered Apr 06 '16 at 03:13

5

The torus is $$\mathbb T^n=\underbrace{S^1\times...\times S^1}$$ $$\quad\;\;\;\; n\text{ times}.$$ We parametrize the $i$th copy of $S^1$ by the angle $\theta_i$. Let $$f(\theta_1,...,\theta_n)=(\cos(\theta_1),\sin(\theta_1),...,\cos(\theta_n),\sin(\theta_n)).$$ This map should work for you.

answered Apr 06 '16 at 03:13

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That is what I meant with $i:T^{n}\to\mathbb{R}^{2n}$, I thought the problem was more difficult. Thanks! – user123456789 Apr 06 '16 at 03:22

Find a isometric immersion of the Torus $T^{n}$ on $\mathbb{R}^{2n}$.

1 Answers1