Metric tensor - Wiki slovník

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold $M$ (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point $p$ of $M$ is a bilinear form defined on the tangent space at $p$ (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on $M$ consists of a metric tensor at each point $p$ of $M$ that varies smoothly with $p$ .

A metric tensor $g$ is positive-definite if $g (v, v) > 0$ for every nonzero vector $v$ . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold $M$ , the length of a smooth curve between two points $p$ and $q$ can be defined by integration, and the distance between $p$ and $q$ can be defined as the infimum of the lengths of all such curves; this makes $M$ a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).^{[citation needed]}

While the notion of a metric tensor was known in some sense to mathematicians such as 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 field 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

{\vec {r}}(u,\,v)={\bigl (}x(u,\,v),\,y(u,\,v),\,z(u,\,v){\bigr )}

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's 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.

The metric tensor is ${\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}$ in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.

Arc length

If the variables $u$ and $v$ are taken to depend on a third variable, $t$ , taking values in an interval , then $r \to (u (t), v (t))$ will trace out a parametric curve in parametric surface $M$ . The arc length of that curve is given by the integral

{\begin{aligned}s&=\int _{a}^{b}\left\|{\frac {d}{dt}}{\vec {r}}(u(t),v(t))\right\|\,dt\\&=\int _{a}^{b}{\sqrt {u'(t)^{2}\,{\vec {r}}_{u}\cdot {\vec {r}}_{u}+2u'(t)v'(t)\,{\vec {r}}_{u}\cdot {\vec {r}}_{v}+v'(t)^{2}\,{\vec {r}}_{v}\cdot {\vec {r}}_{v}}}\,dt\,,\end{aligned}}

where $\left\|\cdot \right\|$ represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives:

{\vec {r}}_{u}={\frac {\partial {\vec {r}}}{\partial u}}\,,\quad {\vec {r}}_{v}={\frac {\partial {\vec {r}}}{\partial v}}\,.

The integrand is the restriction^[1] to the curve of the square root of the (quadratic) differential

(ds)^{2}=E\,(du)^{2}+2F\,du\,dv+G\,(dv)^{2},

(1)

where

E={\vec {r}}_{u}\cdot {\vec {r}}_{u},\quad F={\vec {r}}_{u}\cdot {\vec {r}}_{v},\quad G={\vec {r}}_{v}\cdot {\vec {r}}_{v}.

(2)

The quantity $ds$ in (1) is called the line element, while $ds 2$ is called the first fundamental form of $M$ . Intuitively, it represents the principal part of the square of the displacement undergone by $r \to (u, v)$ when $u$ is increased by $du$ units, and $v$ is increased by $dv$ units.

Using matrix notation, the first fundamental form becomes

ds^{2}={\begin{bmatrix}du&dv\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}du\\dv\end{bmatrix}}

[1]