The Space of Spheres

  • Marcel Berger
  • Pierre Pansu
  • Jean-Pic Berry
  • Xavier Saint-Raymond
Part of the Problem Books in Mathematics book series (PBM)

Abstract

We fix an n-dimensional Euclidean space E, and denote by Q ( E ) the space of affine quadratic forms q over E; \(\overrightarrow q \), an element of Q (\(\overrightarrow E \)), will be the symbol of q. A sphere is given by a form q, written as
$$q = k{\left\| {\left. \cdot \right\|} \right.^2} + (\alpha \left| \cdot \right.) + h, where \alpha \in \overrightarrow E and k, h \in R;$$
it is an actual sphere if k ≠ 0 and ‖α2 > 4kh; if k ≠ 0 and ‖α2 = 4kh the image is a single point (sphere of zero radius), and if ‖α2 < 4kh the image is empty, and we say that q represents a sphere “of imaginary radius”. For k = 0, h ≠ 0, we obtain a hyperplane (if α ≠ 0); the case k = 0 and α = 0, h ≠ 0 represents the point at infinity of E.

Keywords

Conjugate Point Hyperbolic Geometry Conformal Group Exceptional Point Tangent Hyperplane 
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.

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Copyright information

© Marcel Berger 1984

Authors and Affiliations

  • Marcel Berger
    • 1
  • Pierre Pansu
    • 2
  • Jean-Pic Berry
    • 3
  • Xavier Saint-Raymond
    • 3
  1. 1.U.E.R. de Mathematique et InformatiqueUniversité Paris VIIParis, Cedex 05France
  2. 2.Centre National de la Recherche ScientifiqueParisFrance
  3. 3.P.U.K.GrenobleFrance

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