# Definition of the hyperbolic metric

Let $$\mathbb H$$ be the upper half plane. The hyperbolic metric comes from a Riemannian metric on $$\mathbb H$$: at each point $$z = x+iy \in \mathbb H$$, the tangent space $$T_z(\mathbb H)$$ has a natural identification with $$\mathbb R^2$$, and we define a real inner product on $$T_z(\mathbb H)$$ by the formula

$$\langle v,w \rangle_z = \frac{v \cdot w}{y^2}$$ where $$v \cdot w$$ denotes the standard dot product in $$\mathbb R^2$$. This gives a norm $$||v||_z$$ in $$T_z(\mathbb H)$$. The hyperbolic metric in $$\mathbb H$$ is defined by

$$d(z_1,z_2) = \inf\limits_{\gamma} \int_0^1 ||\gamma'(t)||_{\gamma(t)}dt= \inf\limits_{\gamma} \int_0^1 \frac{\sqrt{x'(t)^2 + y'(t)^2}}{y(t)}dt$$

as $$\gamma(t) = x(t) + i y(t)$$ runs over all smooth curves $$[0,1] \rightarrow \mathbb H$$ satisfying $$\gamma(0) = z_1, \gamma(1) = z_2$$.

Some textbooks I've seen have said "We define the hyperbolic metric by the formula $$ds = \frac{\sqrt{dx^2 + dy^2}}{y}$$." What in the world does this mean? I don't know what $$ds$$ means in this context (it doesn't appear to be a differential form), or what it a priori has to do with a metric. Is this abuse of notation for the definition I gave above, or does it actually mean something?

• In differential geometry, especially Riemannian geometry, a metric is usually given by some equation governing the length of a line element. The inf is implied (locally) due to the properties of geodesics. The reason the metric is a line element is because in order to talk about length, one has to borrow the inner product structure from the tangent space of a manifold and somehow carry it over to connect disparate points (thus disparate tangent spaces). An infinitesimal length fits that intuition. Dec 10, 2019 at 21:31
• Officially, a Riemannian metric is written as a quadratic thing: $ds^2 = A\, dx\otimes dx + B(dx\otimes dy + dy\otimes dx) + C\,dy\otimes dy = A\,dx^2 + 2B\,dxdy + C\,dy^2$ in "old-fashioned" notation. This is a symmetric $(2,0)$-tensor (whereas differential forms are skew-symmetric tensors). Dec 10, 2019 at 22:35

On any Riemannian manifold, here's a description of the formal mathematical relations between $$ds$$ and various other objects, each of which is some kind of function that is defined on the tangent space $$T_z$$ of each point $$z$$.
The Riemannian metric $$\langle \cdot, \cdot \rangle$$ itself is a positive definite inner product denoted $$v,w \mapsto \langle v,w\rangle_z \in \mathbb R$$ for each pair $$v,w \in T_z$$.
Next, $$ds^2$$ is the quadratic form associated to the inner product, defined by $$ds^2_z(v) = \langle v,v \rangle_z \in \mathbb R$$ for each $$v \in T_z$$.
And, finally, $$ds$$ is the line element associated to the quadratic form, defined to be its square root $$ds_z(v) = \sqrt{\langle v, v\rangle }_z$$. The meaning of the line element is what you wrote: it is the thing which you integrate to compute arc lengths. You might compare this to the formula for the length of any single vector $$v$$ in any inner product space, namely $$\sqrt{\langle v, v \rangle}$$.
I added those little $$z$$ subscripts to $$ds^2$$ and to $$ds$$ for clarity and for comparizon to the $$z$$ subscript you wrote in your post for the Riemannian metric itself. But to be honest, I never see subscripts like that used for the length element in Riemannian geometry.