Download Projective and Cayley-Klein Geometries by Arkadij L. Onishchik, Rolf Sulanke PDF

By Arkadij L. Onishchik, Rolf Sulanke

This publication deals an creation into projective geometry. the 1st half provides n-dimensional projective geometry over an arbitrary skew box; the true, the advanced, and the quaternionic geometries are the vital issues, finite geometries enjoying just a minor half. the second one offers with classical linear and projective teams and the linked geometries. the ultimate part summarizes chosen effects and difficulties from the geometry of transformation teams.

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Example text

The base points of a homogeneous coordinate system for P " form the coordinate simplex. Denoting by Bj the opposite face of aj the points in the hyperplane spanned by Bj are characterized by the equation x^ = 0. The j-th projection Pj of a point x onto the face Bj (from the point aj) is defined by Pj : a; G P " I—> Xj := Bj A {aj V x) e Bj, (3) thus Pj{aj) = o. T h e unit point in the face Bj is the image of the unit point under the corresponding projection, Cj :=pj{e). Fig. 7 illustrates the situation for the plane.

Since 030 7^ ^0, this is only possible for y = o. Thus the hnes ZVZQ, XVXQ are skew, and hence L := (zV ZQ) V (a; V a;o) is a three-dimensional subspace of H™^^. Since f f ^ + i = I, V A, this implies Dim L AA = 2. Because of q\A = i d ^ we arrive at the contradiction AAL C q{L) = q{x) V q{xo) V q{z) V q{zo) = XQV ZQ. The converse is trivial. Now we can easily deduce the foUowing. • Corollary 8. Let f : P'^^ -^ Q"^ he collinear with D i m / ( P ^ ) > 1. Then f can be represented as a composition, f = g o p, of a central projection p and an injective collinear map g in the following way: For B = /^^(o) and any subspace A complementary to B in P " , let p be the central projection p : P " -^ A with center B.

C o r o l l a r y 5. Let f : P " -^ Q™ be coUinear. Then for any two projective subspaces A, B C P " the restriction, f\A : A -^ Q™, is also coUinear, and D i m / ( A ) < Dim A . Moreover, f{A\/B) = f{A)\/f{B). (14) P r o o f . Both claims follow from (13) by considering A , B as generated by projectively independent points. • T h e following proposition characterizes injective coUinear maps: P r o p o s i t i o n 6. Let f : P " -^ Q™ be coUinear. Then the foUowing ments are equivalent: a) / is injective; b) D i m / ( P : ) = n ; c) D i m / ( A ) = Dim A for every projective subspace A.

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