Integration of Differential Forms on Manifolds With Locally-Finite Variations

Abstract

Given the fundamental role of integration in measuring distances, there has always been a need to develop this concept, especially due to the diversity and complexity of shapes in the real world. Geometers have historically sought to extend integration to more complex geometric objects. The first major obstacle arose when Riemann and Lebesgue integrals failed to apply on non-flat surfaces. This prompted the intervention of differential geometers, most notably George Stokes, who formulated the famous Stokes's theorema partial resolution connecting integration over a manifold with its boundary. However, this did not fully solve the issue. Therefore, Alexey V. Potepun approached the problem differently by modifying the foundations of integration on manifolds, adapting it to the framework of locally-finite variation. This was a major challenge, but it allowed for powerful new generalizations of the integral. Let us now explore what exactly happened.