Approximate and discrete Euclidean vector bundles

Journal article

We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon> 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples

Authors: Scoccola, L; Perea, JA

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Forum of Mathematics, Sigma
• Volume: 11
• Pages: e20
• Article number: e20
• Publication date: 2023-03-21
• DOI: 10.1017/fms.2023.16
• EISSN: 2050-5094
• ISSN: 2050-5094
• Language: English
• Keywords: Euclidean geometry; Vector-valued differential form; Geometry; Frame bundle; Splitting principle; Euclidean vector; Normal bundle; Pure mathematics; Mathematics; Algebra over a field; Bundle; Vector bundle