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Differential Geometry: A Summary

• Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

• It uses the techniques of differential calculus, integral calculus, linear algebra, and multilinear algebra.

• The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for the development of modern differential geometry during the 18th and 19th centuries.

• Differential geometry finds applications throughout mathematics and the natural sciences, including physics, chemistry, economics, engineering, control theory, computer graphics and computer vision, and machine learning.

• The history and development of differential geometry as a subject begins at least as far back as classical antiquity, where principles that form the foundation of differential geometry and calculus were used in geodesy, although in a much simplified form.

• The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton.

• In the 1800s, Carl Friedrich Gauss and Bernhard Riemann made important contributions to the field, introducing the Gauss map, Gaussian curvature, first and second fundamental forms, the Riemannian metric, and the Riemannian curvature tensor.

• Riemann began the systematic study of differential geometry in higher dimensions, introducing the systematic use of linear algebra and multilinear algebra into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature.

• The development of intrinsic differential geometry in the language of Gauss was spurred on by Riemann, and the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program.

• The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s.

• Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds which do not rely on any additional geometric structure.

• Differential geometry finds applications in physics, including Albert Einstein's theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.

• Differential geometry also finds applications in chemistry, economics, engineering, control theory, computer graphics and computer vision, and recently in machine learning.A Brief History of Differential Geometry

• Differential geometry on curved spaces was studied by Elwin Christoffel who introduced the Christoffel symbols in 1868, and by Eugenio Beltrami who studied many analytic questions on manifolds.

• Luigi Bianchi produced his "Lectures on differential geometry" in 1899, studying differential geometry from Riemann's perspective.

• Differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry.

• Modern differential geometry emerged in the early 1900s, in response to the foundational contributions of many mathematicians, including the work of Henri Poincaré on the foundations of topology.

• The notion of a topological space was distilled by Felix Hausdorff in 1914.

• Interest in the subject grew with the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations.

• Following early development, many mathematicians contributed to the development of the modern theory, including Hermann Weyl.

• Differential geometry expanded in scope and developed links to other areas of mathematics and physics in the middle and late 20th century.

• Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric.

• Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.

• Symplectic geometry is the study of symplectic manifolds.

• Complex differential geometry is the study of complex manifolds.Differential Geometry: A Summary

• Conformal geometry studies angle-preserving transformations on a space.

• Differential topology studies global geometric invariants without a metric or symplectic form.

• Lie groups are groups in the category of smooth manifolds with algebraic and differential geometric properties.

• Geometric analysis uses tools from differential equations to establish new results in differential geometry and differential topology.

• Gauge theory studies connections on vector and principal bundles and arises from problems in mathematical physics.

• Vector bundles, principal bundles, and connections on bundles are essential tools for modern differential geometry.

• The intrinsic point of view of differential geometry considers geometric objects as free-standing, while the extrinsic point of view considers them as lying in a Euclidean space of higher dimension.

• Differential geometry has applications in various fields of science and mathematics, including physics, computer vision, and topology.