Science Done Right

Hyperbolic Space

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Def Hyperbolic Geometry The parallel postulate of Euclidean geometry is replaced with: For any given line and point not on , in the plane containing both line and point there are at least two distinct lines through that do not intersect .

Def Hyperbolic 2-Space The hyperbolic 2-space is defined to be the set of points in the upper half plane: equipped with the metric whose first fundamental form is given by Note that a point in can either be thought of as a complex number or as a point . Both perspectives are useful: leads more easily to coordinates and calculations, and C works seamlessly with our definition of isometries below.

Prop Arc length and Volume Suppose is a differentiable curve in for , then we obtain a tangent vector in , called the velocity vector. The arc length of for is then

Missing \begin{aligned} or extra \end{aligned}\gamma|=\int_{a}^{b}\sqrt{\left<\gamma^{\prime}(s),\gamma^{\prime}(s)\right>}\dd s=\int_{a}^{b}\sqrt{(\gamma_{x}'(s))^{2}+(\gamma_{y}'(s))^{2}}\frac1{\gamma_{y}(s)}\dd s \end{aligned}$$ where $\gamma(t)=(\gamma_x(t),\gamma_y(t))$ and $\gamma^{\prime}(t)=\tr{(\gamma_x^{\prime}(t),\gamma_y^{\prime}(t))}$. In the most general setting, if $R ⊂ M$ is contained in a coordinate neighborhood of the Riemannian manifold $M$, with coordinates $(x_{1},\dots,x_{n})$ and metric given by the matrix $g_{ij}$ in these coordinates, then we can compute the volume(area) of $R$ in $\mathbb{H}^2$ to be $$\begin{aligned}\text{vol}(R)=\int_Rd\:\text{vol}=\int_R\sqrt{\det(g_{ij})}\dd x_{1}\dots \dd x_{n}=\int_{R}\frac1{y^{2}}\dd x\dd y\end{aligned}$$ **Def** <i><u>Boundary at Infinity</u></i> We call $\R∪\{\infty\}$ the boundary at infinity for $\mathbb{H}^2$. Note it is homeomorphic to a circle $S^1$, and hence is sometimes called the circle at infinity. It is denoted by $S_{\infty}^{1}$ , $\partial\mathbb{H}^2$, and sometimes $\partial_\infty\mathbb{H}^2$. **Def** <i><u>Geodesic</u></i> and <i><u>Infinite Geodesic</u></i> A geodesic between points $p$ and $q$ is a length minimizing curve between those points. An infinite geodesic is a curve $γ$ from $\R$ to a Riemannian manifold such that for any $s < t ∈ \R$, the curve $γ([s,t])$ minimizes the distance between $γ(s)$ and $γ(t)$. **Thrm** The infinite geodesics in $\mathbb{H}^2$ consist of vertical straight lines and semi-circles with center on the real line. **Def** Isometry

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