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I would appreciate some help/clarification on the following:

Suppose I have a torus $T=(0,2\pi)\times(0,2\pi)$ and a map $f:T\to \mathbb R^3$ where $f$ is a function of $\alpha, \beta$, what does it mean to find a Riemannian metric on $T$ induced by $f$? 

Perhaps take a concrete example, say $f(\alpha,\beta)=(c(\beta)\cos\alpha, c(\beta)\sin\alpha, d\sin\beta)$ where $c(\beta)=a+b\cos\beta$ for some $a>b$ and $a,b,d$ are constants. Incidentally, what is the image of such a map? My guess is the image is symmetric in the $x-y$ plane and the top half looks something like "a jelly pudding with a hole on top containing a tornado shaped hollow"? I don't know how to describe it, and I may be totally wrong. Please correct me if I am!

Also does this kind of map have a name? Something like embedding or ...?

By the way, can Wolfram Alpha plot graphs like these?


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up vote 3 down vote accepted

If $f:M\to N$ is a smooth immersion of a smooth manifold $M$ into a Riemannian manifold $N$ with metric $g$, on each tangent space $T_pM$, one can define the pullback metric $f^*g$ via $f^*g(v,w) = g(dfv,dfw).$

For your example, explicitly, you want to define a metric at each tangent space of $T$. You are given some map $f(\alpha,\beta)$. You can push tangent vectors forward via $df$, and then take the inner product in $\mathbb{R}^3$:

$$(dfv)^T(dfw) = v^T(df^Tdf)w.$$

So the pullback metric on the torus is given by the matrix $df^Tdf$.

Edit: For your explicit example, if $f(\alpha,\beta) = (c(\beta)\cos\alpha, c(\beta)\sin(\alpha),d\sin\beta)$, then

$$df = \begin{pmatrix} -c(\beta)\sin\alpha && c'(\beta)\cos\alpha \\ c(\beta)\cos\alpha && c'(\beta)\sin\alpha \\ 0 && d\cos\beta \end{pmatrix}$$


$$\begin{eqnarray} f^*g = df^Tdf &= \begin{pmatrix} -c(\beta)\sin\alpha && c(\beta)\cos\alpha && 0 \\ c'(\beta)\cos\alpha && c'(\beta)\sin\alpha && d\cos\beta \end{pmatrix} \begin{pmatrix} -c(\beta)\sin\alpha && c'(\beta)\cos\alpha \\ c(\beta)\cos\alpha && c'(\beta)\sin\alpha \\ 0 && d\cos\beta \end{pmatrix} \\ &= \begin{pmatrix}c^2(\beta) && 0 \\ 0 && c'^2(\beta) + d^2\cos^2\beta \end{pmatrix}. \end{eqnarray}$$

Edit 2: Playing around with Wolfram Alpha, try the command "parametric plot 3d f(x,y) = blah blah blah." The standard torus embedding is $$f(\alpha,\beta) = \bigg(\big(R - r\cos(\beta)\big)\sin(\alpha),\big(R - r\cos(\beta)\big)\cos(\alpha), r\sin(\beta)\bigg).$$ Here $R$ is the major radius of the torus, $r$ is the minor radius of the torus, and $\alpha$ and $\beta$ are, of course, surface parameters. The intuition is twofold: a rotation about the $z$-axis, along the major radius with coordinate $\alpha$, and then a rotation (coordinate $\beta$) about that radius for every fixed $\alpha$.

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Thank you, @Neal! – Greg Dawson Feb 22 '12 at 16:57

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