CayleyKleinMetric

A Cayley-Klein metric is a type of metric defined on a pseudo-Riemannian manifold that is derived from the Cayley-Klein parameters. This name stems from the works of mathematicians Arthur Cayley and Felix Klein, who used these parameters in the context of projective geometry.

The Cayley-Klein metrics arise from the so-called Cayley-Klein geometries, which are specific models of the classical geometries, such as Euclidean, hyperbolic, elliptic, and Galilean. These geometries can be unified by a common framework using Cayley-Klein metrics, which allows for a deeper understanding of their interconnections and general properties.

In a Cayley-Klein geometry, distances between points are calculated using a bilinear form associated with the metric, which can incorporate different degrees and types of curvature depending on the specific parameters used. For instance, with certain parameter values, the metric can model the geometry of the hyperbolic plane, spherical geometry, or Euclidean geometry, among others.

Spaces that carry a Cayley-Klein metric are often referred to as Cayley-Klein spaces. However, the exact terminology might vary according to the context and specifics of the model or theory under consideration.

Remember that as of my knowledge cutoff in September 2021, the concept of a Cayley-Klein metric is a rather specialized one, mainly found in the literature of differential geometry, algebraic geometry, and mathematical physics. For an accurate and up-to-date understanding, consulting resources or experts in these fields is recommended.

More Detail

The Cayley-Klein geometries, from which the Cayley-Klein metrics derive, are constructed through projective geometry with the use of a process known as "quadrance".

A Cayley-Klein metric can be given by a function of the cross-ratio of four points, which is a fundamental invariant in projective geometry. For a set of four points in a projective line ${0, 1, x, \infty}$, the cross-ratio is defined as

$$[ [0, 1, x, \infty] = \frac{(0 - x)}{(0 - \infty)} \cdot \frac{(1 - \infty)}{(1 - x)} ]$$

Now, define a function of this cross-ratio to be the quadrance $Q(x)$ between the points $0$ and $x$. For example, in Euclidean geometry, $Q(x) = x^2$. The Cayley-Klein geometries are then defined by setting

$$[Q(x) = \frac{(ax + b)^2}{(cx + d)^2}]$$

for some parameters $a$, $b$, $c$, and $d$, with the condition

$$ad - bc = 1$$

for projective invariance.

From the quadrance, one can define the spread $s$ between two lines in a plane, which is analogous to the sine of the angle between the lines in Euclidean geometry. In the Cayley-Klein geometries, the spread can be defined using a function of the cross-ratio of four lines, in a way that parallels the definition of quadrance.

Given the quadrance and spread, a Cayley-Klein metric can be induced on higher-dimensional projective spaces by specifying that the quadrance and spread between points and lines, respectively, are invariant under projective transformations.

As you can see, the specifics of the Cayley-Klein metrics are quite involved, and a complete understanding would require a solid background in projective and differential geometry. I hope this gives you a starting point for further study.