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Classification of Geometries with Projective Metric

Published online by Cambridge University Press:  20 January 2009

D. M. Y. Sommerville
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In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , February 1909 , pp. 25 - 41
Copyright
Copyright © Edinburgh Mathematical Society 1909

References

page 25 note * A distinction is not made here between the “antipodal” and the “polar” forms of elliptic geometry. The antipodal form (as in ordinary spherioal geometry) does not in fact make its appearanoe at all in the projective metric, for two lines determine just one point, as two points determine just one line.

page 28 note * The term “aotual” here is opposed to “ideal,” and is preferred to “real,” which is opposed to “imaginary.”

page 29 note * It may happen that the harmonio conjugate A′ of A is at a real finite distance from A, but points on AA′ in the vicinity of A′ are ideal. In this case A′ is ideal.

page 37 note * The discussion for a ruled quadric is not complete, as in space of n dimensions, Rn, there are ruled quadrica of different ranks, viz., in R2p–1 or R2p ruled quadrics may contain lines, planes, 3-flats, … or (p–l)-flats. At any stage these may become imaginary, so that there are quadrics of rank p – 1 down to 0 (unruled) and – 1 (imaginary). Central quadrics of each rank exist. If the equation of a central quadric in Rn–1 be written in homogeneous coordinates , where k of the coefficients are positive and n–k negative, the rank is ½{n– |2k–n |}–1. For some of the properties of ruled quadrics see Bertini, Introduzione alla geometria proiettiva degli iperspazi, Pisa, 1907.

The discussion in the text is sufficient, however, for the classification. The theorems relating to the tangent (n–l)-flats through an (n–2) flat may be regarded merely as existence-theorems, the circumstances under which they are true not being completely discussed.