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I'm having a difficulty understanding how to go on proving a certain map is an isometry. It should be really basic and simple, but for some reason I can't understand how to do this..


The situation is this: I have 2 manifolds, $D,M$:

  • $D$ is the Poincare disk $\{x\in\mathbb{R}^{n+1}:x_0=0,\sum_{i=1}^n{x_i}<1\}$, and
  • $M$ is the positive-half-space, $\{x\in\mathbb{R}^n:x_n>0\}$).

Each has its own metric:

  • for $D$: $g_{ij}^{(1)} = \frac{4\delta_{ij}}{1-\sum_\alpha u_\alpha^2}$.
  • for $M$: $g_{ij}^{(2)} = \frac{\delta_{ij}}{x_n^2}$.

And I have a map, $f:D\to M$, given by $f(p) = 2\frac{p-p_0}{|p-p_0|^2}+p_0$ where $p_0=(0,\ldots,0,-1)\in\mathbb{R}^n$.


By my understanding of the definition of isometry, I should take 2 vectors, $p_1,p_2\in D$, and show that $g^{(1)}(p_1,p_2)=g^{(2)}(f(p_1),f(p_2))$.

From here I'm really confused.. I have $g_{ij}$ (not $g$), which is defined for $T_pD$, and we get $g$ from $g_{ij}$ (though how exactly I'm not sure). What I thought is that I need to find $df$, and show that for a basis $\partial_i$ of $T_pD$, $g_{ij}^{(1)}(\partial_i,\partial_j)=g_{ij}^{(2)}(df_p(\partial_i),df_p(\partial_j))$, but I'm really not sure on how to do this, and I'll be glad if you could give me a direction..

Thanks

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    What is $\delta_{ij}$? Is it the discrete metric? I.e. the distance between distinct points is 1.2012-07-08
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    @Arthur $\delta_{ij}$ in differential geometry is the standard notation for the euclidean metric, i.e. the usual dot product on $\mathbb R^n$.2012-07-12

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Using your notation, we can get $g$ from $g_{ij}$ by integrating along paths. Given $\gamma:[0,1] \to M$ with $\gamma(p_1)=0, \gamma(p_2)=1$, its arclength can be computed as

$$L(\gamma)=\int_0^1 g_{ij}(\gamma'(t),\gamma'(t)) \, dt \, .$$

Then $g(p_1,p_2)$ is defined to be $\inf_{\gamma(0)=p_1,\gamma(1)=p_2} L(\gamma)$.

So your last sentence is correct; if you can show that $df$ preserves the inner product on $TD$, that will imply that $f$ preserves arclengths of curves and thus also the metric-space structure on $D$. In fact, "$df$ preserves the inner product" is generally taken as the definition of an isometry between Riemannian manifolds, since the actual metric-space structure can be difficult to work with directly (being an infimum over an infinite-dimensional space of curves).

So you just need to be able to compute $df_p$ for all $p$; to do this, remember that, once you've chosen local coordinates for your source and target manifolds, $df$ becomes the Jacobian of $f$. The problem statement hands you a set of coordinates (and in fact gives you your two metrics in those coordinates), so...

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    I understand what you mean, but get really confused with the notation (coordinates in the space and tangent space), and how to parse $g_{ij}$ accordingly.. for example, when I have $g_{ij}=4\delta_{ij}/(1-\sum{u_i^2})$, then $<\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_i}>_p=4/(1-\sum{u_i})$ where $u_i$ are the coordinates of p? Even more confusing is calculating $$ - how should I treat $v_i$? also $df$ is dependent on $x_i$, so what is $x_i$ there?2012-07-08
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    The answer to your first question is "yes", that's exactly what you should do (except that I'm not sure where the $u$s come from since everywhere except in that line your coordinates are $x$s). The $v_i$ are vectors; if you have $df$ in coordinates as a matrix, $dfv_i$ is just ordinary matrix-by-vector multiplication, giving a vector that you should think of as an element of $T_{f(p)}M$; if you compute $df$ on basis vectors as your OP suggests, there's much less to worry about here. The $x_i$ are again the coordinates of the point you're computing at (which in this case is $f(p)$).2012-07-08
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    Ok. I'm still really struggling. I tried differentiating $f$, then calculating the inner products, but it's not working - I'm not able to get to the equality $<\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}>_p=_{f(p)}$.. I believe I mix up the "general" coordinates of the tangent space and the coordinates of $p,f(p)$, but can't find where is my mistake.. Could you help me organize everything - I'm not even sure if I differentiated f correctly...2012-07-08
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    I'm not sure how much I can help remotely without seeing your algebra. One problem you might be having is that the metric you've written down for $D$ appears to be wrong (see this article: http://en.wikipedia.org/wiki/Poincaré_disc_model); it should be $g_{ij}=\frac{\delta_{ij}}{(1-\sum_\alpha u_\alpha^2)^2}$, so you're missing a square in the denominator.2012-07-08
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    correct, a square was missing, but that's not the problem (yet).. I'm not able to get to a "good" expression for $$. I should get that if $i\neq j$ then the product is 0, but I don't get it.. I got that $(df)_{ij}=\frac{1}{r^4}\cdot(-2x_ix_j)$, tried to multiply by $\frac{\partial}{\partial x_i}$, getting the i-th column of df, and then activating $<,>$, but it just doesn't work out.. I know I'm missing something, but don't know what..2012-07-09