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In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors ''v'' and ''w'' at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. A metric tensor is called ''positive-definite'' if it assigns a positive value to every nonzero vector. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function whose value at a pair of points ''p'' and ''q'' is the distance from ''p'' to ''q''. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the ''infinitesimal'' distance on the manifold. While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point. ==Introduction== Carl Friedrich Gauss in his 1827 ''Disquisitiones generales circa superficies curvas'' (''General investigations of curved surfaces'') considered a surface parametrically, with the Cartesian coordinates ''x'', ''y'', and ''z'' of points on the surface depending on two auxiliary variables ''u'' and ''v''. Thus a parametric surface is (in today's terms) a vector valued function : depending on an ordered pair of real variables (''u'',''v''), and defined in an open set ''D'' in the ''uv''-plane. One of the chief aims of Gauss' investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface. One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor. 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「Metric tensor」の詳細全文を読む スポンサード リンク
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