Cayley–Klein metric

The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. ... Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight.

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and in his book Vorlesungen über Nicht-Euklidischen Geometrie(1928). The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.

Suppose that Q is a fixed quadric in projective space. If p and q are 2 points then the line through p and q intersects the quadric Q in two further points a and b. The Cayley–Klein distance d(p,q) from p to q is proportional to the logarithm of the cross-ratio:

for some fixed constant C.

Cayley–Klein geometry is the study of the group of motions that leave the Cayley–Klein metric invariant. This group is obtained as the collineations for which the absolute is stable. Indeed, cross-ratio is invariant under any collineation, and the stable absolute enables the metric comparison, which will be equality.

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