How did Hermann Weyl connect differential geometry and physics?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

Hermann Weyl bridged differential geometry and physics by treating physical fields as geometric properties of space-time. In his influential work Raum, Zeit, Materie (1918), he developed a unified field theory that interpreted Maxwell’s equations and gravitation through a geometric lens.1

A key contribution to differential geometry was his refinement of Tullio Levi-Civita’s concept of parallel displacement. By freeing this concept from its dependence on a Riemann metric, Weyl facilitated the development of projective differential geometry.1

Furthermore, in his attempt to unify electromagnetism and gravitation, Weyl introduced the concept of gauge invariance. This principle, which describes how certain physical quantities remain unchanged despite transformations in the underlying field, became a cornerstone of modern particle physics. Through these efforts, Weyl demonstrated that the fundamental laws of nature could be understood as manifestations of geometric structure, profoundly influencing both the mathematical and physical sciences.1