Differential geometry
Study of smooth shapes and spaces using calculus and algebra.
Differential geometry is the branch of mathematics concerned with the geometry of smooth shapes and smooth spaces, which are called smooth manifolds. Its tools come from vector calculus, linear algebra, and multilinear algebra. The field traces back to ancient studies of spherical geometry, and it has long been tied to astronomy, the measurement of the Earth (geodesy), and later to Lobachevsky's work on hyperbolic geometry. The simplest examples of smooth spaces are curves and surfaces in three-dimensional Euclidean space, and studying these shapes laid the groundwork for modern differential geometry in the 18th and 19th centuries. Since the late 1800s, the field has broadened to focus on geometric structures on differentiable manifolds. A geometric structure defines some idea of size, distance, shape, volume, or other rigid property. For instance, Riemannian geometry specifies distances and angles; symplectic geometry allows volumes to be computed; conformal geometry specifies only angles; and gauge theory assigns certain fields over the space. Differential geometry is closely related to differential topology, which studies properties of differentiable manifolds without any extra geometric structure (the distinction between the two is discussed in that article). It also connects to the geometric aspects of differential equations, an area known as geometric analysis. The subject has applications throughout mathematics and the natural sciences. Albert Einstein used the language of differential geometry in his general theory of relativity, and physicists later applied it in quantum field theory and the Standard Model of particle physics. Beyond physics, differential geometry is used in chemistry, economics, engineering, control theory, computer graphics, computer vision, and recently in machine learning.
**History and development**
The history of differential geometry begins at least in classical antiquity. It is closely linked to the broader development of geometry, the concepts of space and shape, and topology—especially the study of manifolds.
- Field
- Mathematics
- Known for
- Study of smooth manifolds, geometric structures, and applications in physics
- Origins
- Classical antiquity (spherical geometry, geodesy)
- Key developments
- 18th–19th century: plane and space curves, surfaces; late 19th century: differentiable manifolds
Lore & Background
Since the late 19th century, differential geometry has focused on geometric structures on differentiable manifolds, including Riemannian geometry (distances and angles), symplectic geometry (volumes), conformal geometry (angles only), and gauge theory. It is closely related to differential topology and geometric analysis. The language of differential geometry was used by Albert Einstein in general relativity and later in quantum field theory and the Standard Model. Applications also appear in chemistry, economics, engineering, control theory, computer graphics, computer vision, and machine learning.
Reader's Guide
Differential geometry is significant as the mathematical framework for understanding smooth shapes and spaces, from ancient geodesy to modern physics. Its development from classical antiquity through the calculus era to contemporary manifold theory has provided essential tools for describing curvature, geodesics, and geometric structures. The field's application in Einstein's general relativity revolutionized physics, and its continued use in quantum field theory, the Standard Model, and diverse fields like computer vision and machine learning underscores its broad impact. The study of intrinsic geometry, initiated by Euler and formalized by Gauss's Theorema Egregium, remains foundational. Differential geometry bridges pure mathematics and applied science, enabling precise modeling of curved spaces and physical phenomena.
Did You Know?
- The Mercator projection, developed by Gerardus Mercator, is conformal but does not preserve distances, reflecting Gauss's later Theorema Egregium.
- Leonhard Euler derived the first analytical geodesic equation and began the theory of intrinsic geometry.
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