# The 3 Geometries Geometry is the study space and shapes. It focus on objects like points, line and surfaces and concepts like distance and angles. It defines axioms that describe fundamental relationships between objects and ask what theorem can be proven and answer concrete questions about space. Until the 19th century, mathematicians studied a single geometry - the Euclidean geometry. In 1830's Lobachevsky and Bolyai published independently their work on hyperbolic geometry - the first non-Euclidean geometry. In 1854, Riemann published a work on a new geometry type of geometry called spherical geometry. A natural question at this point - are there more geometries out there? If we restrict our attention to an interesting class of surfaces which are compact, closed, orientable [[Topological manifold (manifold)|2D-manifold]] without boundary then the answer is no. Furthermore, each such surface has exactly one natural geometry from the list above. This can be seen from the following claims. ## Claim 1 - The surfaces can be classified by genus A well known theorem in Topology is the "Classification theorem of closed surfaces" that state that [[Topological manifold (manifold)|2D-manifold]] which is compact, closed, orientable and without boundary can be classified according to its genus. If its genus $g$ is 0 then the 2D-manifold is homeomorphic to the Sphere, if $g=1$ it is homeomorphic to a Torus and for $g>1$ it is homeomorphic to $g$ copies of Torus which are glued together (see image below). ![[Pasted image 20260603174442.png|282]] ## Claim 2 - The surfaces has a unique constant-curvature metric A 2D-manifolds can be given a complex structure (not trivial fact) and so can be treated as [[Riemann surface]]. This Riemann surface (not necessarily simply-connected) has a covering space which is simply connected Riemann surface and according to the [[The Uniformization theorem]] must be one of $\widehat{\mathbb C}, \mathbb C, \mathbb D$. There is a local-isometry from the covering space and the Riemann manifold (not a trivial fact) and so the existing canonical constant-curvature metric on the covering space is induced locally to the Riemann manifold. So each Riemann manifold has unique constant-curvature metric (canonical metric). ## Claim 3 - The constant curvature on surface determines the surface's class and genus The Gauss-Bonnet theorem state that for a closed, compact, orientable 2D-manifold $M$ without boundary we have $\int_M KdA=4\pi(1-g)$ where $g$ is the genus, $K$ is Gauss curvature , $dA$ area element. It is interesting theorem that connect two branches of mathematics: geometry on the left and topology on the right. A conclusion from this theorem is that $K=0\rightarrow g=1$ , $K=1\rightarrow g=0$ and for $K=-1\rightarrow g>1$. Combining the claims we get a remarkable result: the 3 geometries above are the only geometries for those surfaces. There are no more. The sphere has spherical geometry, the torus has Euclidean geometry and all the rest have hyperbolic geometry. Another interesting fact to note is that the unfamiliar hyperbolic geometry is much more common on surfaces (all surfaces with genus >1) than the popular and most familiar Euclidean geometry which exist only on the torus (genus=1). ## Created 2026-06-05 10:04