Differential geometry gave Einstein the language of curved spacetime
Differential geometry studies smooth shapes and spaces, called manifolds, using vector calculus and linear and multilinear algebra. It grew from ancient measurements of a spherical Earth, and its language later framed Einstein's general relativity before physicists used it for quantum field theory and the particle Standard Model.
Since the late nineteenth century the subject has focused on structures laid over differentiable manifolds that fix some notion of size or shape. Riemannian geometry specifies distances and angles, symplectic geometry lets volumes be computed, conformal geometry keeps only angles, and gauge theory places fields on the space. Close cousins include differential topology, which drops the extra structure, and geometric analysis. Beyond physics the methods reach chemistry, economics, control theory, computer vision and machine learning.
The roots are ancient. Eratosthenes worked out Earth's circumference around 200 BC, and about 150 AD Ptolemy's Geography introduced stereographic projection for mapmaking. Euclid treated tangency to a circle and the straight line as the shortest route, which on a globe makes great circles the shortest paths, and Archimedes found areas and volumes of circles, spheres, cones and cylinders by exhaustion. Little changed until Gerardus Mercator, whose conformal map showed shortest routes as bent curves, evidence that no flat map preserves distance.
Calculus arrived in the 1600s with Newton and Leibniz, building on Descartes' coordinates, and Fermat, Newton and Leibniz studied inflection points and osculating circles to measure curvature. Johann Bernoulli's lectures, compiled by L'Hôpital into the first differential calculus textbook, computed tangents, and the first formula for curvature followed. Alexis Clairaut began studying space curves at 16 and coined curvature and double curvature.
Leonhard Euler, a student of Johann Bernoulli, derived the first geodesic equation and introduced intrinsic coordinates on a surface, and his Mechanica showed that a mass sliding on a surface with no force acting follows a geodesic, an early hint of relativity. Gaspard Monge's French school then took up surfaces of revolution and envelopes.
Source: Differential geometry
