Bogdan D. Suceavă | Searcharxiv

arXiv2026

A Natural Representation of Volumes Yields a Remarkable Affine Consequence

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.

math.DG↗

arXiv2025

Extrinsic Characterisations of Immersions

We outline the history of the idea of deformation of space, which lead to the concept of curvature invariants, as we understand them today, including contributions of E. Bacaloglu and F. Casorati, among others. We pursue the following question: what is the best way to quantify the deformation of space? This important question could be viewed in a new paradigm after 1956, when John F. Nash, Jr. proved that a Riemannian manifold can be immersed isometrically into an Euclidean ambient space of dimension sufficiently large. This important theorem allowed to view the representation of space from its exterior, from an outside perspective. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic quantities characterising immersions. And such relations seem to be rather few, at least few enough to present us with a technical challenge in applying Nash's Theorem. One technical difficulty is presented by the passing through the narrow gateway provided by Gauss' equation, and that's why in order to obtain some new results it might be useful to include additional natural conditions. A turning point in the history of the question we pursue was an enlightening paper written by B.-Y. Chen in 1993, which paved the way for a deeper understanding of the meaning of the Riemannian inequalities between intrinsic and extrinsic quantities. Our present discussion invites a reflection on whether we could hope to characterise submanifolds by using mainly extrinsic quantities.

math.DG↗