### Conformally Euclidean geometry of the upper half-plane

In Deriving invariant hyperbolic Riemannian metric on the half-plane we have calculated the line element for the SL(2,R) invariant hyperbolic geometry on the upper half-plane:

(1) Using this formula we can easily calculate the length of any segment of any curve. If is a parametric equation of the curve, then and so

(2) (3) In this last formula we have used the fact that in the upper half-plane we have From the studies of geometry of surfaces and its generalization developed mainly by Riemann we know that the formula for the line element allows us not only to find the length, but also the angles.

From Wikipedia:
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” (“On the Hypotheses on which Geometry is Based”). It is a very broad and abstract generalization of the differential geometry of surfaces in . Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.

It allows us to calculate scalar products of tangent vectors. Let us rewrite Eq. (1) in the form: using the symmetric (i.e. ) “Riemannian metric” matrix  Thus, from Eq. (1), we have  and . Or

(4) In general, if and are two tangent vectors at then their scalar product is given by a general formula

(5) which in our case specializes to

(6) The numerator is exactly the same as for the Euclidean scalar product. It has the consequence that the angles in the hyperbolic upper half-plane geometry are the same as in the Euclidean geometry. That it is so follow from the definitions. Angles in any Riemannian geometry are defined the same way as in the Euclidean geometry. If are two tangent vectors at then the angle between them is defined through the formula:

(7) where and are their norms. If we use the scalar product given by Eq. (6), then the denominator in cancels out with the product of denominators is the norms. The end result is

(8) .
Therefore, when we look at the picture like the one obtained in the last post:

Whenever we have two metrics, and that differ a scalar factor we say that they are conformally related, or conformal to each other. In our case the metric in Eq. (4) is proportional to the identity matrix, our geometry is conformally Euclidean.