By Barmak J.A., Minian E.G.
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This e-book is a translation of an authoritative introductory textual content in keeping with a lecture sequence added through the popular differential geometer, Professor S S Chern in Beijing college in 1980. the unique chinese language textual content, authored by way of Professor Chern and Professor Wei-Huan Chen, was once a different contribution to the maths literature, combining simplicity and financial system of process with intensity of contents.
A research of linear order and continuity to offer a origin for traditional genuine or complicated geometry.
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Additional resources for 2-Dimension from the Topological Viewpoint
11). In the former view the points of the plane are arranged in triangles with edges on the points fi. One or three of the edges must be produced. Exercise 2 - If we take two such triangles x, y. z and x.. Y.. z. then x - x.. y - Y•• z - z. are parallel. The corresponding edges y - z and Y. - z. meet at f •. Consider the next case, 8 18 2 8 s8 4 =1. Here 8 1 8 2 and 8 a8 4 are reciprocal stretches with the same fixed point f, the intersection of fl - f2 and fa - f4. Taking any point x, the 28 THE EUCLIDEAN GROUP points x, xSl> XS l S 2, XS l S 2S S form a closed cycle.
If the circles are external, the ring includes the point 00. We consider all the arcs orthogonal to both. These terminate at the image pair II' 12' Any arc meets Cl say at Xl and C 2 say at X 2• Consider the cross-ratio (Xl (Xl - Il)(x 2 - 12) I2)(x 2 -II) First, it is real because the points are on a circle. Second, it is the same for all arcs. For inversion in any of the arcs will not alter it, but will send an arc I l x l x 2I2 into another, say, II Yl yd2' Third, it is positive, for it is manifestly positive for the segment II -12' Of this positive number, which is the same for all arcs, we take the logarithm, A.
The stretch 8 1 sends a circle C with centre y into a circle C8 1 with centre z. The stretch 8 2 sends this latter into a circle with centre x. Thus this circle is C81 8 2 • The stretch 8 s sends this into the original circle, since 8 18 28 s is to be I. The product of ratios PIP2PS is then 1, (1) But there is a further relation. We obtain it by placing y and therefore z at A. Then x8 s is 11 and 1182 is x. We have then 11 - and Is= Ps(x -Is) x - 12=P2(X - 12) whence, eliminating x, (2) This is one of three equivalent forms.