I am familiarizing myself with a new topic and want to calculate an actual example (no homework). So far, I have defined the parameter coordinates, the diffeomorphism and the pullback metrics, but I don't know how to handle the quasi-isometry. It seems to be trivial and I probably could just guess the upper and lower bounds, but I really want to understand how to estimate the bounds with this easy example before I move on to harder ones.
This is a definition that I found for quasi-isometry:
I have the manifold (finite paraboloid with height $h$) $M=\{(x_{1},\, x_{2},\, x_{3}):\, h(x_{1}^{2}+x_{2}^{2})=x_{3},\,0<x_{3}\leq h\}$.
The parameneter equations are:
$x_{1}=\rho cos(\varphi)$
$x_{2}=\rho sin(\varphi)$
$x_{3}=\rho^{2}$, with $\rho\in[0,\sqrt{h}]$ and $\varphi\in]0,2\pi]$
The induced metric on $M$ is $g=(1+4\rho^{2})d\rho^{2}+\rho^{2}d\varphi^{2}$.
The manifold (disk) $N=\{(y_{1},\, y_{2}):\, y_{1}^{2}+y_{2}^{2}\leq1\}$ has the parameter equations:
$y_{1}=R\cdot cos(\phi)$
$y_{2}=R\cdot sin(\phi)$, with $R\in[0,1]$ and $\phi\in]0,2\pi]$.
The induced metric on $N$ is $d=dR^{2}+R^{2}d\phi^{2}$.
I defined the diffeomorphism $F$ as a projection:
$F:\, M\rightarrow N$
$(\rho,\,\varphi)\longmapsto(\frac{1}{\sqrt{h}}\rho,\,\varphi)=(R,\phi)$ .
Then I pulled back the metric $d$ on $N$ and got the pulled-back metric $d*$ on $M$:
$d*=\frac{1}{\sqrt{h}}d\rho+1d\varphi$.
Now I am ready to show that $M$ and $N$ are quasi-isometric, but I am totally stuck:
Question 1) I don't know how to handle the Riemannian metrics and how to actually calculate with them. Do I need to use the matrix representation? Or how do I multiply, divide etc with line elements metrics?
Question 2) I don't know how to start to estimate the lower and upper bound for the Riemannian case. I have worked with bounds in calculus a while ago and there was always a trick how to estimate certain terms.
I would highly appreciate if someone could show me with a full calculation how to deal with Riemannian metrics to show Quasi-Isometry.
THANKS!
First of all, if you consider the pullback metric then you have actually an isometry, which is in fact a quasi-isometry. Thus, in $N$ you should consider the Euclidean metric.
Since $M$ and $N$ are compact and $F$ is continuous there exist $D>0$ such that $$\max_{x,y\in M} d_M(x,y)\le D$$ and $$\max_{x,y\in M} d_N(F(x),F(y))\le D.$$ Thus, it is
$$d_N(F(x),F(y))-D\le 0 \le d_M(x,y) \le D\le d_N(F(x),F(y))+D.$$
What if we consider $$M=\{(x_{1},\, x_{2},\, x_{3}):\, h(x_{1}^{2}+x_{2}^{2})=x_{3},\,0<x_{3}< h\}$$ and $$N=\{(y_{1},\, y_{2}):\, y_{1}^{2}+y_{2}^{2}< 1\}?$$ In this case we can follow the same argument as in the compact case, because there exist $D>0$ such that $$\max_{x,y\in M} d_M(x,y)\le D$$ and $$\max_{x,y\in M} d_N(F(x),F(y))\le D.$$ So, in this case you don't need to perform any computation with the metrics.