Understand the Hyperbolic space

Question

I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$. For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ But for $n>2$ I can't seem to find the expression for the metric...

Can anyone help me with this?

Also, on a related question, If I have a scalar field that obeys to the Laplace equation in $\mathbb H^2$, I get that the solution is the same as if I've written the Lapace equation in $\mathbb R^2$...that came as a surprise for me...Why does that happen? (I'm trying to have some understanding of the Hyperbolic space.)

Answer 1

Notes by Manfred Stoll look like a perfect match for your question: both the hyperbolic space (2.3) and harmonic functions on it (Chapters 3-4) are introduced from scratch. For the record, the metric on $\mathbb H^n$ is $ds^2 = \frac{dx^2}{x_n^2}$ where $dx^2$ is the Euclidean metric.

In dimension $2$ (only) the hyperbolic and Euclidean harmonic functions indeed coincide: see formulas (3.2) and the (yet unfinished) section 3.4. One might wonder if this is a feature or a bug. One of possible explanations is that hyperbolic isometries are conformal maps, and conformality is related to the Laplace equation only in dimension $2$. (In dimensions $n\ge 3$ conformal maps are related to the non-linear $n$-Laplace equation.)