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In this problem set, we will explore the geometry of the hyperbolic plane. This homework set is out of 40 points. Illustration credit: M.C. Escher, Regular Division of The Plane VI, 1957.

Let \(\mathbb{H}\) be the set \(\{ (x,y) \in \mathbb{R}^2 \mid y > 0\}\) equipped with the Riemannian metric: \[ \langle \mathbf{u} ,\mathbf{v} \rangle_{(x,y)}^{\mathbb{H}} := \frac{1}{y^2} \mathbf{u} \cdot \mathbf{v} \] This can also be expressed as \(\frac{1}{y^2} dx^2 + \frac{1}{y^2} dy^2\). This is Poincaré's half-plane model of the hyperbolic plane. A curve in \(\mathbb{H}\) a hyperbolic geodesic if it is either a vertical line segment or else an arc on a circle whose center is on the \(x\)-axis.

1. (10 points) Show that the following transformations are isometries of \(\mathbb{H}\) for any \(a,b \in \mathbb{R}\) with \(a > 0\): \[ T(x,y) = (ax + b,ay) \qquad I(x,y) = \big(\frac{x}{x^2 + y^2},\frac{y}{x^2+y^2}\big) \] Remark: The transformation \(I\) an example of an inversion: it fixes lines through the origin but inverts the distance of a point to the origin. Inversions transform hyperbolic geodesics to hyperbolic geodesics.
2. (10 points) Show that any hyperbolic geodesic can be transformed into a subset of the \(y\)-axis using a sequence of the isometries of \(\mathbb{H}\) from problem 1.
3. (5+5 points) Derive the length of the hyperbolic geodesic which connects \((0,y_0)\) to \((0,y_1)\). Derive the length of the hyperbolic geodesic which connects \((0,1)\) to \((1,1)\).
4. (10 points) Show that hyperbolic geodesics are the shortest curves connecting their endpoints.
   Hint: First prove this when both endpoints are on the \(y\)-axis using a similar argument to the one for Euclidean space. Then use problems 1 and 2 to reduce the general case to this one.