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The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups. The term "manifold" comes from German ''Mannigfaltigkeit,'' by Riemann. In Romance languages, this is translated as "variety" – such spaces with a differentiable structure are called "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. ==Background== Ancestral to the modern concept of a manifold were several important results of 18th and 19th century mathematics. The oldest of these was Non-Euclidean geometry, which considers spaces where Euclid's parallel postulate fails. Saccheri first studied this geometry in 1733. Lobachevsky, Bolyai, and Riemann developed the subject further 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these are called hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature, respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. In modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more topological example of an intrinsic property of a manifold is the Euler characteristic. For a non-intersecting graph in the Euclidean plane, with ''V'' vertices (or corners), ''E'' edges and ''F'' faces (counting the exterior) Euler showed that ''V''-''E''+''F''= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of the torus is 0, since the complete graph on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a useful topological invariant, which has been extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes or dimensions (known as generalized coordinates in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth. In the early 19th century the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi, the answer was formulated: the resulting integral would involve functions of two complex variables, having four independent ''periods'' (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the ''Jacobian of a hyperelliptic curve of genus 2''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「History of manifolds and varieties」の詳細全文を読む スポンサード リンク
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