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The metric ds^2=(dx^2+dy^2)/((1-x^2-y^2)^2) for the Poincaré hyperbolic disk, which is a model for hyperbolic geometry. The hyperbolic metric is invariant under conformal ...

If ABB^' and AC^'C are straight lines with BC and B^'C^' intersecting at D and AB+BD=AC^'+C^'D, then AB^'+B^'D=AC+CD. The origin and some history of this theorem are ...

Given a point P and a line AB, draw the perpendicular through P and call it PC. Let PD be any other line from P which meets CB in D. In a hyperbolic geometry, as D moves off ...

Two metrics g_1 and g_2 defined on a space X are called equivalent if they induce the same metric topology on X. This is the case iff, for every point x_0 of X, every ball ...

1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line ...

In the hyperbolic plane H^2, a pair of lines can be parallel (diverging from one another in one direction and intersecting at an ideal point at infinity in the other), can ...

Linear Algebra

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

A set within which the values of a function lie (as opposed to the range, which is the set of values that the function actually takes).

A map is called "one-to-one" (or "1:1") if it is injective. However, being in "one-to-one correspondence" is synonymous with being a bijection. Unfortunately, this confusing ...