affine differential geometry

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English

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Etymology

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The term reflects the categorisation developed by German mathematician Felix Klein for his Erlangen programme (1872, Vergleichende Betrachtungen über neuere geometrische Forschungen), in which he found a useful distinction between projective, affine and Euclidean geometry (in order of increasing restrictiveness). (Riemannian geometry was not initially included.)

Noun

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affine differential geometry (uncountable)

  1. (differential geometry) A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations.
    The basic difference between Riemannian and affine differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics.
    • 1999, Alexander I. Bobenko, Wolfgang K. Schief, 5: Discrete Indefinite Affine Spheres, Alexander I. Bobenko, Ruedi Seiler (editors), Discrete Integrable Geometry and Physics, Oxford University Press (Clarendon Press), page 113,
      Tzitzeica's classical papers are believed to have initiated a new area in mathematics, namely affine differential geometry. [] The present paper extends the above-mentioned approach to affine differential geometry.
    • 2002, C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, page 88:
      The Tzitzeica surfaces are the analogues of spheres in affine differential geometry and, indeed, are known as affine spheres or affinsphären [39]. According to Nomizu and Sasaki [227], the origins of affine differential geometry reside in this work of Tzitzeica at the turn of the nineteenth century.
    • 2012, Miguel Brozos-Vázquez, Peter B. Gilkey, Stana Nikcevic, Geometric Realizations of Curvature[1], World Scientific (Imperial College Press), page 89:
      In Chapter 4, we study questions related to real affine differential geometry. The structure group in Riemannian geometry is the orthogonal group . The structure group in affine differential geometry is the affine group.

Translations

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Further reading

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