Topological quantum numbers and curvature -- examples and applications

TL;DR

This paper explores topological quantum numbers through the degree of mappings between manifolds, providing classifications, geometric proofs, and applications to vector fields, monopoles, and instantons.

Contribution

It offers a comprehensive classification of mappings between spheres and other structures, along with original elementary proofs of key geometric theorems.

Findings

Classification of mappings between spheres of the same dimension

Elementary proof of the Gauss-Bonnet theorem

Elementary proof of the Poincaré-Hopf theorem

Abstract

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss-Bonnet theorem and the Poincar\'{e}-Hopf theorem.