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Projective Spaces

  • Igor R. Shafarevich
  • Alexey O. Remizov

Abstract

The theory of projective spaces and their transformations is presented. The notions of a projective space, projective subspace, homogeneous and inhomogeneous coordinates, projective algebraic variety, projective transformation, cross ratio, etc., are introduced and discussed. The principle of projective duality is presented. At the end of the chapter, the topological properties of real and complex projective spaces are investigated. In particular, compactness of projective spaces and projective algebraic varieties is established. Also, it is proved that a real projective space of arbitrary odd dimension has two orientations, and a real projective space of arbitrary even dimension has one orientation.

Keywords

Projective Space Projective Plane Projective Line Projective Transformation Cross Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  • Alexey O. Remizov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseau CedexFrance

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