Projective Spaces

  • Igor R. Shafarevich
  • Alexey O. Remizov


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.


Projective Space Projective Plane Projective Line Projective Transformation Cross Ratio 
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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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