affine differential geometry

English

Etymology

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

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 ${\mathcal {O}}$. The structure group in affine differential geometry is the affine group.

Translations

type of differential geometry

Further reading

• Affine geometry on Wikipedia.Wikipedia
• Erlangen program on Wikipedia.Wikipedia
• Differential invariant on Wikipedia.Wikipedia
• Differential form on Wikipedia.Wikipedia
• Volume form on Wikipedia.Wikipedia