The geometry of manifolds and the perception of space

TL;DR

This paper explores the historical development of manifold geometry in the 19th and early 20th centuries, highlighting its foundational role in modern physics and the perception of space.

Contribution

It traces the evolution of the abstract manifold concept and its impact on physics, especially Einstein's theory of space-time.

Findings

Development of manifold geometry in the 19th century

Role of geometric ideas in Einstein's space-time theory

Influence of manifold concepts on modern physics

Abstract

This essay discusses the development of key geometric ideas in the 19th century which led to the formulation of the concept of an abstract manifold (which was not necessarily tied to an ambient Euclidean space) by Hermann Weyl in 1913. This notion of manifold and the geometric ideas which could be formulated and utilized in such a setting (measuring a distance between points, curvature and other geometric concepts) was an essential ingredient in Einstein's gravitational theory of space-time from 1916 and has played important roles in numerous other theories of nature ever since.