Torus and Flat 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

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