A Natural Representation of Volumes Yields a Remarkable Affine Consequence
Wladimir G. Boskoff, Bogdan D. Suceavă
TL;DR
This work connects curvature and affine invariants by expressing the ratio $\dfrac{K_f(p)}{d_f^4(p)}$ as a function of parallelepiped volumes determined by the surface patch $f(x,y)$. The authors derive a volume-based formula $\dfrac{K_f(p)}{d_f^4(p)}=\dfrac{V_xV_y-(V_{xy})^2}{V^4}$ and show its transformation under centro-affine maps: $\dfrac{K_{ar f}}{d_{ar f}^4}=\dfrac{1}{(\det \mathbb{A})^2}\dfrac{K_f}{d_f^4}$, leading to the notion of a Ţiţeica surface as one with constant ratio $R_f$. They illustrate that geometry alone does not fix this invariant by comparing Euclidean examples (sphere, pseudosphere) with Minkowski settings, where the invariant can be preserved or altered depending on the ambient space. The results underscore how the Erlangen Program guides finding geometric invariants through affine-volume considerations and extend to Minkowski spaces, advancing affine differential geometry in a Kleinian framework.
Abstract
At the beginning of the 20th Century there was a growing interest for the investigation of the action of linear groups on the geometry of surfaces. In that context of ideas, the quest for a connection between curvature and the behaviour of linear groups rose naturally. Pursuing the original thought, we investigate how the geometric meaning of this idea is intimately related to the concept of volume of parallelepiped boxes. We show how the ratio of the Gaussian curvature divided by the fourth power of a certain distance of interest in the geometry of surfaces can be represented as a function of volumes. This geometric description explores the profound meaning of a quantity considered by {Ţ}i{ţ}eica in 1907, in a work that sparked a growing interest in affine differential geometry, as an illustration of Felix Klein's Erlangen Program, in which the quest for geometric invariants was the main point of inquiry.