Geometry & Topology 2024, 29: Spherical & Hyperbolic Geometry
Spherical and hyperbolic geometry are 2D geometries where distance, lines and angles are defined and invariant under motions, which act transitively on points and directions, and can be treated as Euclidean locally. They also include parallel and non-intersecting lines, have intrinsic curvature properties and contain a definition of a unit of length intrinsic to the geometry. A sphere with radius r and centered at the origin is defined by x² + y² + z² = r². Define the spherical line/great circle as the intersection of the sphere with any plane containing the origin, and "antipodal" points as those that have opposite position vectors. For any 2 points on the sphere, there exists a unique spherical line containing them, and a spherical distance, the length of the shortest arc of said great circle.
Assume for simplicity that r = 1. A spherical triangle consists of the the 3 vertices between 3 points on the sphere, and the 3 arcs of the great circle joining them, though they are not necessarily the shortest arcs. Two of the points may be antipodal. Define the spherical angle between the two lines in the triangle as the angle between two lines cut out by the planes in an auxiliary plane orthogonal to the line connecting the common point with the one forming the shared planes. The tangent plane and the tangent vectors to the curves are the lines cut out by the auxiliary planes, the orthogonality of which defines an axis, with an angle between the curves equal to the dihedral angle. If the sides of a triangle are shorter arcs with angles below π, then α ≤ β + γ iff the vertices are collinear with a point on the shorter arc of the other two. This is the Triangle Inequality.
An isometry is in essence a spherical motion, which in turn is a map T: S² → S² preserving spherical distance. It takes pairs of antipodal points to pairs of antipodal points and spherical lines to spherical lines. Any motion given in coordinates by x → Ax, where A is a 3*3 orthogonal matrix. Given a space with the metric with t > 0, define the Hyperbolic space H²
A line of hyperbolic geometry is the hyperbola obtained as the intersection with a 2-dimensional vector subspace. This is a Lorentz plane, meaning it contains time-like vectors with the signature (-1, +1). There is a unique line through any two distinct points in the hyperbolic space. These are analogues to the great circles on the sphere. The hyperbolic distance is defined through
Equality holds iff P = Q. If the points are different, there exists a Lorentz basis of ℝ³, that can set P = (1, 0, 0) and Q = (cosh α, sinh α, 0), α = d(P, Q) > 0. A hyperbolic triangle can thus be set so that P = (1, 0, 0), and PQ is on the hyperbolic line {y = 0} with Q' = (0, 1, 0). The last point R can be placed on R' = (0, cos a, sin a). The side QR is thus determined by the sides PQ, PR and the dihedral angle.
Equality holds iff P is on the interval [Q, R] (basically not a real triangle). Hyperbolic motions are defined to maps preserving hyperbolic distance. It's given by coordinates x → Ax where A is a Lorentz matrix, preserving the two halves of the cone.
Take a line V in hyperbolic space. It's either space-like, q(v) > 0, lines on the equatorial plane and those with (sin a)x = (cos a)y are disjoint. When V is time-like, q(v) < 0, they intersect, and when V is on the light-cone, q(v) = 0, the lines are disjoint, but asymptotic. However, all lines approach one another arbitrarily closely in infinity.
If L is a hyperbolic line and P a point external to L, then there is a unique perpendicular line PQ to L through P. If the plane M is orthogonal to PQ in P, then L diverges from M. If there is an angle below 90 degrees, with L' going through P, then L' and L meet iff the angle of L' and PQ is below that angle. In a hyperbolic triangle,
