## Torus and Flat Torus

Torus

$\displaystyle T(u,v)=((R+r\cos u)\cos v,(R+r\cos u)\sin v,r\sin u)$

One can calculate the Gauss curvature by the following formula

$\displaystyle E=T_u\cdot T_u, F=T_u\cdot T_v=0, G=T_v\cdot T_v$

$\displaystyle K=-\frac{1}{2\sqrt{EG}}\left(\frac{\partial }{\partial v}\left(\frac{E_v}{\sqrt{EG}}\right)+\frac{\partial }{\partial u}\left(\frac{G_u}{\sqrt{EG}}\right)\right)$

In fact

$\displaystyle K=\frac{\cos u}{r(R+r\cos u)}$

So one can see that some places ${K}$ is positive and some places ${K}$ is negative. There is another torus homeomorphic to this one, which is also ${S^1\times S^1}$ with different metric,

$\displaystyle T=(\cos u,\sin u,\cos v,\sin v)$

lies in the space ${\mathbb{R}^4}$. Using the formula above, one can get

$\displaystyle E=1,F=0,G=1,\text{ then }K=0$

So this torus has a name as flat torus. Flat torus can not be put in ${R^3}$ because the following therem:

Theorem: On every compact surface ${M\subset\mathbb{R}^3}$ there is some point ${p}$ with ${K(p)>0}$.

Remark:Oprea, John. Differential geometry and its applications. p129