affine differential geometry

Definitions

A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations.

uncountable

Examples

“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.”

“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.”

“In Chapter 4, we study⟳ questions related to real affine differential geometry. The structure⟳ group in Riemannian geometry is the orthogonal group #92;mathcal#123;O#125;. The structure⟳ group in affine differential geometry is the affine group.”

CEFR level