Task: Question Answering
Input (English): Context: In mathematics, a real number is a value that represents a quantity along a line. The adjective "real" in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials., In planar geometry, an angle is the figure formed by two rays, called the "sides" of the angle, sharing a common endpoint, called the "vertex" of the angle.
Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles., Thierry Aubin ( 6 May 1942 -- 21 March 2009 ) was a French mathematician who worked at the Centre de Mathématiques de Jussieu , and was a leading expert on Riemannian geometry and non-linear partial differential equations . His fundamental contributions to the theory of the Yamabe equation led , in conjunction with results of Trudinger and Schoen , to a proof of the Yamabe Conjecture : every compact Riemannian manifold can be conformally rescaled to produce a manifold of constant scalar curvature . Along with Yau , he also showed that Kähler manifolds with negative first Chern classes always admit Kähler -- Einstein metrics , a result closely related to the Calabi conjecture . The latter result provides the largest class of known examples of compact Einstein manifolds . Aubin was a visiting scholar at the Institute for Advanced Study in 1979 . He was elected to the Académie des sciences in 2003 ., In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions., Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces., In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a "flat" plane, or a curve from being "straight" as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between "extrinsic curvature", which is defined for objects embedded in another space (usually a Euclidean space)  in a way that relates to the radius of curvature of circles that touch the object  and "intrinsic curvature", which is defined in terms of the lengths of curves within a Riemannian manifold. , In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity., Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. , The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The Fields Medal is sometimes viewed as the highest honor a mathematician can receive. The Fields Medal and the Abel Prize have often been described as the mathematician's "Nobel Prize". The Fields Medal differs from the Abel in view of the age restriction mentioned above., In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry., Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a "Riemannian metric", i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions., In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in C, such that the transition maps are holomorphic., In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by and proved by . Yau received the Fields Medal in 1982 in part for this proof., In differential geometry, the Gaussian curvature or Gauss curvature "" of a surface at a point is the product of the principal curvatures, "" and "", at the given point:
For example, a sphere of radius "r" has Gaussian curvature "1/r" everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus., In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other., In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space ("M","g") is a real smooth manifold "M" equipped with an inner product formula_1 on the tangent space formula_2 at each point formula_3 that varies smoothly from point to point in the sense that if "X" and "Y" are vector fields on "M", then formula_4 is a smooth function. The family formula_1 of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry., In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles., Subject: thierry aubin, Relation: field_of_work, Options: (A) algebra (B) differential geometry (C) gas (D) geometry (E) mathematician (F) mathematics
Output (English): differential geometry