# Smooth transition from Euclidean plane to hyperbolic plane

differential-geometrygeometrymetric-spacesrandom-graphs

If I have a Poisson point process $$\mathcal{X}$$ of density $$\lambda$$ on the Euclidean plane $$\mathbb{R}^2$$, with the Euclidean metric taking pairs of points to the Euclidean distance,

$$\operatorname{dist} (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \sqrt{(x_1 -x_2)^2-(y_1-y_2)^2}$$

can I allow the metric to depend on time, in such a way that it smoothly (or otherwise) approaches the hyperbolic metric as time traverses the closed unit interval $$[0,1]$$?

So, initially, $$\mathcal{X}$$ sees Euclidean space, and finally hyperbolic space,

$$\operatorname{dist} (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname{arcosh} \left( \cosh y_1 \cosh (x_2 – x_1) \cosh y_2 – \sinh y_1 \sinh y_2 \right)$$
with a number (perhaps a finite set) of "negatively curved" spaces in between. Only a small region would need this property, not the whole of $$\mathbb{R}^2$$.

The reason for this is it would affect the structure of a random graph or simplical complex built on the points (e.g. the degree distribution). Is this possible?

Note: I understand Ricci flow is similar to this, but where the metric satisfies a partial differential equation.

As $$t \in [0,1]$$ varies, just use the family of metrics $$e^{2ty} dx^2 + dy^2$$ When $$t=1$$ one gets the metric $$e^{2y}dx^2 + dy^2$$ which is isometric to the upper half plane metric $$\frac{dx^2 + dz^2}{z^2}$$ using the transformation $$z=e^y$$ (or perhaps $$z=e^{-y}$$).
As $$t$$ decreases from $$1$$ to $$0$$ the curvature $$\kappa(t)$$ varies from $$\kappa(1)=-1$$ to $$\kappa(0)=0$$.
Notice that one gets an infinite family of hyperbolic metric spaces between $$t=0$$ and $$t=1$$. In fact its not reasonable to expect only a finite family, if one wants the metrics to vary smoothly, because the curvature will then be a smooth function of $$t$$ and it must therefore decrease smoothly from $$\kappa(1)=-1$$ to $$\kappa(0)=0$$.