Supplementary mathematics/Differential geometry

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Differential geometry is a branch of mathematics that deals with the geometry of smooth shapes and smooth spaces, otherwise called smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multi-linear algebra. This field has its roots in the study of spherical geometry since ancient times. It is also related to astronomy, geodesy of the earth and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of flat spaces are curves and plane surfaces in Euclidean three-dimensional space, and the study of these forms was the basis for the development of modern differential geometry during the 18th and 19th centuries. A triangle immersed in a saddle-shaped plane (a hyperbolic parabola), as well as two divergent hyperparallel lines. Since the late 19th century, differential geometry has evolved into a field generally concerned with geometric constructions on differentiable manifolds. A geometric structure is a structure that defines a concept of size, distance, shape, volume, or other rigid structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be calculated, in isomorphic geometry only angles are specified, and in gauge theory certain fields are given on space. Differential geometry is closely related to, and sometimes includes, differential topology, which concerns properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for further discussion of the distinction between the two ). Differential geometry is also related to the geometric aspects of the theory of differential equations, which is also called geometric analysis. Differential geometry is used throughout mathematics and the natural sciences. The language of differential geometry was further 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. Outside of physics, differential geometry is used in chemistry, economics, engineering, control theory, computer graphics and computer vision, and more recently in machine learning.