The development of modern geometry: From Gauss and Riemann to the present - Lecture 1

Bernhard Riemann's seminal work that founded Riemannian geometry was inspired by Gauss' insights on the geometry of surfaces. Riemann introduced what is now called a Riemannian manifold and showed that the sectional curvatures derived from the Riemann curvature tensor yield a complete set of local invariants of a Riemannian metric. Riemannian geometry became fundamental for Einstein's theory of general relativity theory as well as modern quantum field theory. Currently, it also provides the basis for novel methods in machine learning. From a philosophical perspective, it offers new insight into the nature of space.