We introduce a neural implicit framework that exploits the differentiable properties of networks and discrete geometry point-sampled surfaces to approximate them as level sets functions . To train function, we propose loss functional approximates signed distance allows terms with high-order derivatives, such alignment between principal directions curvature, learn more geometric details. During ...

This paper is intended as an introduction to noncommutative geometry for readers with some knowledge of abstract algebra and differential geometry. We show how to extend the theory of differential forms to the “noncommutative spaces” studied in noncommutative geometry. We formulate and prove the Hochschild-Kostant-Rosenberg theorem and an extension of this result involving the Connes differential.

In this paper we describe an approach to differential topology and geometry rooted in supersymmetric quantum theory. We show how the basic concepts and notions of differential geometry emerge from concepts and notions of the quantum theory of non-relativistic particles with spin, and how the classification of different types of differential geometry follows the classification of supersymmetries...

A variety of questions in combinatorics lead one to the task of analyzing the topology of a simplicial complex, or a more general cell complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifo...

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature |K| ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machine Te, e ∈ ω, there is a sequence...