Namely, as far back as Euclid's Elements it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the Earth leads to the conclusion that great circles, which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy, although in a much simplified form. Famously, Eratosthenes calculated the circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for the purposes of mapping the shape of the Earth. In particular, much was known about the geometry of the Earth, a spherical geometry, in the time of the ancient Greek mathematicians. The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to classical antiquity. In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces, and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields.Ĭlassical antiquity until the Renaissance (300 BC – 1600 AD) It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology, especially the study of manifolds. The history and development of differential geometry as a subject begins at least as far back as classical antiquity. Outside of physics, differential geometry finds applications in chemistry, economics, engineering, control theory, computer graphics and computer vision, and recently in machine learning. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics. Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.ĭifferential geometry finds applications throughout mathematics and the natural sciences. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. 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 development of modern differential geometry during the 18th and 19th centuries. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The field has its origins in the study of spherical geometry as far back as antiquity. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
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