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| In the [[mathematics|mathematical]] field of [[differential geometry]], a '''metric tensor''' is a type of function defined on a [[manifold]] (such as a [[surface]] in space) which takes as input a pair of [[tangent vector]]s ''v'' and ''w'' and produces a [[real number]] ([[scalar (mathematics)|scalar]]) ''g''(''v'',''w'') in a way that generalizes many of the familiar properties of the [[dot product]] of [[Vector (geometry)|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.
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| A metric tensor is called ''positive definite'' if every vector has positive length with respect to the metric. A manifold equipped with a positive definite metric tensor is known as a [[Riemannian manifold]]. By [[integral|integration]], the metric tensor allows one to define and compute the length of curves on the 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 [[metric (mathematics)|distance function]] ''d''(''p'',''q'') 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.
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| 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]], meaning that relative to a [[local coordinate system]] on the manifold, a metric tensor takes on the form of a [[symmetric matrix]] whose entries transform [[covariance and contravariance of vectors|covariantly]] under changes to the coordinate system. Thus a metric tensor is a covariant [[symmetric tensor]]. From the [[Coordinate-free|coordinate-independent]] point of view, a metric tensor is defined to be a [[nondegenerate form|nondegenerate]] [[symmetric bilinear form]] on each tangent space that varies [[smooth function|smoothly]] from point to point.
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| ==Introduction==
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| [[Carl Friedrich Gauss]] in his 1827 ''[[#CITEREFGauss1827|Disquisitiones generales circa superficies curvas]]'' (''General investigations of curved surfaces'') considered a surface [[parametric 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]]
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| :<math>\vec{r}(u,v) = ( x(u,v), y(u,v), z(u,v) )</math>
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| 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.
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| One natural such invariant quantity is the [[arclength|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, or [[tangent vector]]s at the same point of the surface. 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.
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| ===Arclength===
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| If the variables ''u'' and ''v'' are taken to depend on a third variable, ''t'', taking values in an [[interval (mathematics)|interval]] [''a'',''b''], then <math>\scriptstyle{\vec{r}(u(t),v(t))}</math> will trace out a [[parametric curve]] in parametric surface ''M''. The [[arclength]] of that curve is given by the [[integral]]
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| : <math> \begin{align}
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| s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\
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| &= \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.
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| \end{align}</math>
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| where <math> \left\| \cdot \right\| </math> represents the [[Norm (mathematics)#Euclidean norm|Euclidean norm]]. Here the [[chain rule]] has been applied, and the subscripts denote [[partial derivative]]s (<math>\scriptstyle \vec{r}_u=\tfrac{\partial \vec{r}}{\partial u}</math>, <math>\scriptstyle \vec{r}_v=\tfrac{\partial \vec{r}}{\partial v}</math>). The integrand is the restriction<ref>More precisely, the integrand is the [[pullback (differential geometry)|pullback]] of this differential to the curve.</ref> to the curve of the square root of the ([[quadratic form|quadratic]]) [[differential (infinitesimal)|differential]]
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| {{NumBlk|:|<math>ds^2 = E \,du^2 + 2F \,du\, dv + G\, dv^2\,</math>|{{EquationRef|1}}}}
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| where
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| {{NumBlk|:|<math> E=\vec r_u\cdot\vec r_u, \quad
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| F=\vec r_u\cdot\vec r_v, \quad
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| G=\vec r_v\cdot \vec r_v.</math>|{{EquationRef|2}}}}
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| The quantity ''ds'' in ({{EquationNote|1}}) is called the '''[[line element]]''', while ''ds''<sup>2</sup> is called the '''[[first fundamental form]]''' of ''M''. Intuitively, it represents the [[principal part]] of the square of the displacement undergone by <math>\scriptstyle{\vec{r}(u,v)}</math> when ''u'' is increased by ''du'' units, and ''v'' is increased by ''dv'' units.
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| Using matrix notation, the first fundamental form becomes
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| :<math>
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| \begin{align}
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| ds^2 &=
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| \begin{bmatrix}
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| du&dv
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| \end{bmatrix}
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| \begin{bmatrix}
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| E&F\\
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| F&G
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| \end{bmatrix}
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| \begin{bmatrix}
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| du\\dv
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| \end{bmatrix}\\
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| \end{align}
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| </math>
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| ===Coordinate transformations===
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| Suppose now that a different parameterization is selected, by allowing ''u'' and ''v'' to depend on another pair of variables ''u''′ and ''v''′. Then the analog of ({{EquationNote|2}}) for the new variables is
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| {{NumBlk|:|<math> E'=\vec r_{u'}\cdot\vec r_{u'}, \quad
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| F'=\vec r_{u'}\cdot\vec r_{v'}, \quad
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| G'=\vec r_{v'}\cdot \vec r_{v'}.</math>|{{EquationRef|2'}}}}
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| The [[chain rule]] relates ''E''′, ''F''′, and ''G''′ to ''E'',''F'', and ''G'' via the [[matrix (mathematics)|matrix]] equation
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| {{NumBlk|:|<math>\begin{bmatrix}
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| E'&F'\\
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| F'&G'
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| \end{bmatrix} =
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| \begin{bmatrix}
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| \frac{\partial u}{\partial u'}&\frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'}&\frac{\partial v}{\partial v'}
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| \end{bmatrix}^\mathrm{T}
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| \begin{bmatrix}
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| E&F\\
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| F&G
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| \end{bmatrix}
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| \begin{bmatrix}
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| \frac{\partial u}{\partial u'}&\frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'}&\frac{\partial v}{\partial v'}
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| \end{bmatrix}
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| </math>|{{EquationRef|3}}}}
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| where the superscript ''T'' denotes the [[matrix transpose]]. The matrix with the coefficients ''E'', ''F'', and ''G'' arranged in this way therefore transforms by the '''[[Jacobian matrix]]''' of the coordinate change
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| :<math>J=\begin{bmatrix}
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| \frac{\partial u}{\partial u'}&\frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'}&\frac{\partial v}{\partial v'}
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| \end{bmatrix}.</math>
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| A matrix which transforms in this way is one kind of what is called a [[tensor]]. The matrix
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| :<math>
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| \begin{bmatrix}
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| E&F\\
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| F&G
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| \end{bmatrix}
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| </math>
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| with the transformation law ({{EquationNote|3}}) is known as the '''metric tensor''' of the surface.
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| ===Invariance of arclength under coordinate transformations===
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| {{harvtxt|Ricci-Curbastro|Levi-Civita|1900}} first observed the significance of a system of coefficients ''E'', ''F'', and ''G'', that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form ({{EquationNote|1}}) is ''invariant'' under changes in the coordinate system, and that this follows exclusively from the transformation properties of ''E'', ''F'', and ''G''. Indeed, by the chain rule,
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| :<math>
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| \begin{bmatrix}
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| du\\dv
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| \end{bmatrix}
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| =\begin{bmatrix}
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| \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
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| \end{bmatrix}
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| \begin{bmatrix}
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| du'\\dv'
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| \end{bmatrix}
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| </math>
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| so that
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| :<math>
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| \begin{align}
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| ds^2 &=
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| \begin{bmatrix}
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| du&dv
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| \end{bmatrix}
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| \begin{bmatrix}
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| E&F\\
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| F&G
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| \end{bmatrix}
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| \begin{bmatrix}
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| du\\dv
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| \end{bmatrix}\\
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| &=\begin{bmatrix}
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| du'&dv'
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| \end{bmatrix}
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| \begin{bmatrix}
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| \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
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| \end{bmatrix}^\mathrm{T}
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| \begin{bmatrix}
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| E&F\\
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| F&G
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| \end{bmatrix}
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| \begin{bmatrix}
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| \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'}\\
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| \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
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| \end{bmatrix}
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| \begin{bmatrix}
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| du'\\dv'
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| \end{bmatrix}\\
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| &=
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| \begin{bmatrix}
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| du'&dv'
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| \end{bmatrix}
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| \begin{bmatrix}
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| E'&F'\\
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| F'&G'
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| \end{bmatrix}
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| \begin{bmatrix}
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| du'\\dv'
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| \end{bmatrix}\\
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| &=(ds')^2.
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| \end{align}
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| </math>
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| ===Length and angle===
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| Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of [[tangent vector]]s to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the [[dot product]] of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface ''M'' can be written in the form
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| :<math> \mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v</math>
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| for suitable real numbers ''p''<sub>1</sub> and ''p''<sub>2</sub>. If two tangent vectors are given
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| :<math>\mathbf{a} = a_1\vec{r}_u + a_2\vec{r}_v </math>
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| :<math> \mathbf{b} = b_1\vec{r}_u + b_2\vec{r}_v</math>
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| then using the [[bilinear transform|bilinearity]] of the dot product,
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| :<math>
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| \begin{align}
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| \mathbf{a} \cdot \mathbf{b} &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + b_1a_2 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v\\
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| &= a_1 b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G \\
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| &=\begin{bmatrix}
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| a_1 & a_2
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| \end{bmatrix}
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| \begin{bmatrix}
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| E&F\\F&G
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| \end{bmatrix}
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| \begin{bmatrix}
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| b_1 \\ b_2
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| \end{bmatrix}
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| \end{align}.
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| </math>
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| This is plainly a function of the four variables ''a''<sub>1</sub>, ''b''<sub>1</sub>, ''a''<sub>2</sub>, and ''b''<sub>2</sub>. It is more profitably viewed, however, as a function that takes a pair of arguments '''a''' = [''a''<sub>1</sub> ''a''<sub>2</sub>] and '''b''' = [''b''<sub>1</sub> ''b''<sub>2</sub>] which are vectors in the ''uv''-plane. That is, put
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| :<math>g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G.</math>
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| This is a [[symmetric function]] in '''a''' and '''b''', meaning that
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| :<math>g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a}).</math>
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| It is also [[bilinear form|bilinear]] meaning that it is [[linear functional|linear]] in each variable '''a''' and '''b''' separately. That is,
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| :<math>g(\lambda\mathbf{a}+\mu\mathbf{a'},\mathbf{b}) = \lambda g(\mathbf{a},\mathbf{b}) + \mu g(\mathbf{a'},\mathbf{b}),\quad\text{and}</math>
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| :<math>g(\mathbf{a}, \lambda\mathbf{b}+\mu\mathbf{b'}) = \lambda g(\mathbf{a},\mathbf{b}) + \mu g(\mathbf{a},\mathbf{b'})</math>
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| for any vectors '''a''', '''a'''′, '''b''', and '''b'''′ in the ''uv'' plane, and any real numbers μ and λ.
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| In particular, the length of a tangent vector '''a''' is given by
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| :<math>\|\mathbf{a}\| = \sqrt{g(\mathbf{a},\mathbf{a})}</math>
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| and the angle θ between two vectors '''a''' and '''b''' is calculated by
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| :<math>\cos\theta = \frac{g(\mathbf{a},\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.</math>
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| ===Area===
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| The [[surface area]] is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface ''M'' is parameterized by the function <math>\vec{r}(u,v)</math> over the domain ''D'' in the ''uv''-plane, then the surface area of ''M'' is given by the integral
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| :<math>\iint_D \left|\vec{r}_u\times\vec{r}_v\right|\,du\,dv</math>
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| where × denotes the [[cross product]], and the absolute value denotes the length of a vector in Euclidean space. By [[Lagrange's identity]] for the cross product, the integral can be written
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| :<math>\begin{align}
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| \iint_D &\sqrt{(\vec{r}_u\cdot\vec{r}_u)(\vec{r}_v\cdot\vec{r}_v)-(\vec{r}_u\cdot\vec{r}_v)^2}\,du\,dv\\
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| &\quad=\iint_D\sqrt{EG-F^2}\,du\,dv\\
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| &\quad=\iint_D\sqrt{\operatorname{det}\begin{bmatrix}E&F\\ F&G\end{bmatrix}}
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| \, du\, dv\end{align}
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| </math>
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| where det is the [[determinant]].
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| ==Definition==
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| Let ''M'' be a smooth manifold of dimension ''n''; for instance a [[surface]] (in the case ''n'' = 2) or [[hypersurface]] in the [[Cartesian space]] '''R'''<sup>''n''+1</sup>. At each point ''p'' ∈ ''M'' there is a [[vector space]] T<sub>''p''</sub>''M'', called the [[tangent space]], consisting of all tangent vectors to the manifold at the point ''p''. A '''metric at ''p''''' is a function ''g''<sub>''p''</sub>(''X''<sub>''p''</sub>,''Y''<sub>''p''</sub>) which takes as inputs a pair of tangent vectors ''X''<sub>''p''</sub> and ''Y''<sub>''p''</sub> at ''p'', and produces as an output a [[real number]] ([[scalar (mathematics)|scalar]]), so that the following conditions are satisfied:
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| * ''g''<sub>''p''</sub> is '''[[bilinear form|bilinear]]'''. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if ''U''<sub>''p''</sub>, ''V''<sub>''p''</sub>, ''Y''<sub>''p''</sub> are three tangent vectors at ''p'' and ''a'' and ''b'' are real numbers, then
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| ::<math>g_p(aU_p+bV_p,Y_p) = ag_p(U_p,Y_p)+bg_p(V_p,Y_p),\ \ \text{and}</math>
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| ::<math>g_p(Y_p,aU_p+bV_p) = ag_p(Y_p,U_p)+bg_p(Y_p,V_p).\,</math>
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| * ''g''<sub>''p''</sub> is '''[[symmetric function|symmetric]]'''.<ref>In several formulations of [[classical unified field theories]], the metric tensor was allowed to be non-symmetric; however, the antisymmetric part of such a tensor plays no role in the contexts described here, so it will not be further considered.</ref> A function of two vector arguments is symmetric provided that for all vectors ''X''<sub>''p''</sub> and ''Y''<sub>''p''</sub>,
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| ::<math>g_p(X_p,Y_p) = g_p(Y_p,X_p).\,</math>
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| * ''g''<sub>''p''</sub> is '''[[nondegenerate]]'''. A bilinear function is nondegenerate provided that, for every tangent vector ''X''<sub>''p''</sub> ≠ 0, the function
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| ::<math>Y_p\mapsto g_p(X_p,Y_p)</math>
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| :obtained by holding ''X''<sub>''p''</sub> constant and allowing ''Y''<sub>''p''</sub> to vary is not [[identically zero]]. That is, for every ''X''<sub>''p''</sub> ≠ 0 there exists a ''Y''<sub>''p''</sub> such that ''g''<sub>''p''</sub>(''X''<sub>''p''</sub>,''Y''<sub>''p''</sub>) ≠ 0.
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| A '''metric tensor''' ''g'' on ''M'' assigns to each point ''p'' of ''M'' a metric ''g''<sub>''p''</sub> in the tangent space at ''p'' in a way that varies [[smooth function|smoothly]] with ''p''. More precisely, given any [[open set|open subset]] ''U'' of manifold ''M'' and any (smooth) [[vector field]]s ''X'' and ''Y'' on ''U'', the real function
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| :<math>g(X,Y)(p) = g_p(X_p,Y_p)\,</math>
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| is a smooth function of ''p''.
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| ==Components of the metric==
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| {{Hatnote|This section assumes some familiarity with [[coordinate vector]]s.}}
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| The components of the metric in any [[basis of a vector space|basis]] of [[vector field]]s, or [[frame bundle|frame]], '''f''' = (''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>) are given by<ref>The notation of using square brackets to denote the basis in terms of which the components are calculated is not universal. The notation employed here is modeled on that of {{harvtxt|Wells|1980}}. Typically, such explicit dependence on the basis is entirely suppressed.</ref>
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| {{NumBlk|:|<math>g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right).</math>|{{EquationRef|4}}}}
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| The ''n''<sup>2</sup> functions ''g''<sub>''ij''</sub>['''f'''] form the entries of an ''n''×''n'' [[symmetric matrix]], ''G''['''f''']. If
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| :<math>v = \sum_{i=1}^n v^iX_i,\quad w = \sum_{i=1}^n w^iX_i</math>
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| are two vectors at ''p'' ∈ ''U'', then the value of the metric applied to ''v'' and ''w'' is determined by the coefficients ({{EquationNote|4}}) by bilinearity:
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| :<math>g(v,w) = \sum_{i,j=1}^n v^iw^jg\left(X_i,X_j\right) = \sum_{i,j=1}^n v^iw^jg_{ij}[\mathbf{f}]</math>
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| Denoting the [[matrix (mathematics)|matrix]] (''g''<sub>''ij''</sub>['''f''']) by ''G''['''f'''] and arranging the components of the vectors ''v'' and ''w'' into [[column vector]]s '''v'''['''f'''] and '''w'''['''f'''],
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| :<math>g(v,w) = \mathbf{v}[\mathbf{f}]^\mathrm{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathrm{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]</math>
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| where '''v'''['''f''']<sup>''T''</sup> and '''w'''['''f''']<sup>''T''</sup> denote the [[matrix transpose|transpose]] of the vectors '''v'''['''f'''] and '''w'''['''f'''], respectively. Under a [[change of basis]] of the form
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| :<math>\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A</math>
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| for some [[invertible matrix|invertible]] ''n''×''n'' matrix ''A'' = (''a''<sub>''ij''</sub>), the matrix of components of the metric changes by ''A'' as well. That is,
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| :<math>G[\mathbf{f}A] = A^\mathrm{T} G[\mathbf{f}]A</math>
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| or, in terms of the entries of this matrix,
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| :<math>g_{ij}[\mathbf{f}A] = \sum_{k,\ell=1}^n a_{ki}g_{k\ell}[\mathbf{f}]a_{\ell j}.</math>
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| For this reason, the system of quantities ''g''<sub>''ij''</sub>['''f'''] is said to '''transform covariantly''' with respect to changes in the frame '''f'''.
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| ===Metric in coordinates===
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| A system of ''n'' real valued functions (''x''<sup>1</sup>, …, ''x''<sup>''n''</sup>), giving a [[local coordinates|local coordinate system]] on an [[open set]] ''U'' in ''M'', determines a basis of vector fields on ''U''
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| :<math>\mathbf{f}=\left(X_1=\frac{\partial}{\partial x^1},\dots,X_n=\frac{\partial}{\partial x^n}\right).</math>
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| The metric ''g'' has components relative to this frame given by
| |
| | |
| :<math>g_{ij}[\mathbf{f}] = g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right).</math>
| |
| | |
| Relative to a new system of local coordinates, say
| |
| :<math>y^i = y^i(x^1,x^2,\dots,x^n),\quad i=1,2,\dots,n</math>
| |
| the metric tensor will determine a different matrix of coefficients,
| |
| :<math>g_{ij}[\mathbf{f}'] = g\left(\frac{\partial}{\partial y^i},\frac{\partial}{\partial y^j}\right).</math>
| |
| This new system of functions is related to the original ''g''<sub>''ij''</sub>('''f''') by means of the [[chain rule]]
| |
| :<math>\frac{\partial}{\partial y^i} = \sum_{k=1}^n\frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}</math>
| |
| so that
| |
| :<math>g_{ij}[\mathbf{f'}]=\sum_{k,\ell=1}^n \frac{\partial x^k}{\partial y^i}g_{k\ell}[\mathbf{f}]\frac{\partial x^\ell}{\partial y^j}.</math>
| |
| Or, in terms of the matrices ''G''['''f'''] = (''g''<sub>''ij''</sub>['''f''']) and ''G''['''f'''′] = (''g''<sub>''ij''</sub>['''f'''′]),
| |
| :<math>G[\mathbf{f}'] = \left((Dy)^{-1}\right)^\mathrm{T} G[\mathbf{f}](Dy)^{-1}\,</math>
| |
| where ''Dy'' denotes the [[Jacobian matrix]] of the coordinate change.
| |
| | |
| ===Signature of a metric===
| |
| {{main|Metric signature}}
| |
| | |
| Associated to any metric tensor is the [[quadratic form]] defined in each tangent space by
| |
| | |
| :<math>q_m(X_m) = g_m(X_m,X_m),\quad X_m\in T_mM.</math>
| |
| | |
| If ''q''<sub>''m''</sub> is positive for all non-zero ''X''<sub>''m''</sub>, then the metric is [[definite bilinear form|positive definite]] at ''m''. If the metric is positive definite at every ''m'' ∈ ''M'', then ''g'' is called a [[Riemannian metric]]. More generally, if the quadratic forms ''q''<sub>''m''</sub> have constant [[signature of a quadratic form|signature]] independent of ''m'', then the '''signature of ''g''''' is this signature, and ''g'' is called a [[pseudo-Riemannian metric]].<ref>{{harvnb|Dodson|Poston|1991|loc=Chapter VII §3.04}}</ref> If ''M'' is [[connected space|connected]], then the signature of ''q''<sub>m</sub> does not depend on ''m''.<ref>{{harvnb|Vaughn|2007|loc=§3.4.3}}</ref>
| |
| | |
| By [[Sylvester's law of inertia]], a basis of tangent vectors ''X''<sub>i</sub> can be chosen locally so that the quadratic form diagonalizes in the following manner
| |
| | |
| :<math>q_m\left(\sum_i\xi^iX_i\right) = (\xi^1)^2+(\xi^2)^2+\cdots+(\xi^p)^2 - (\xi^{p+1})^2-\cdots-(\xi^n)^2</math>
| |
| | |
| for some ''p'' between 1 and ''n''. Any two such expressions of ''q'' (at the same point ''m'' of ''M'') will have the same number ''p'' of positive signs. The signature of ''g'' is the pair of integers (''p'', ''n'' − ''p''), signifying that there are ''p'' positive signs and ''n'' − ''p'' negative signs in any such expression. Equivalently, the metric has signature (''p'',''n'' − ''p'') if the matrix ''g''<sub>''ij''</sub> of the metric has ''p'' positive and ''n'' − ''p'' negative [[eigenvalue]]s.
| |
| | |
| Certain metric signatures which arise frequently in applications are:
| |
| * If ''g'' has signature (''n'', 0), then ''g'' is a Riemannian metric, and ''M'' is called a [[Riemannian manifold]]. Otherwise, ''g'' is a pseudo-Riemannian metric, and ''M'' is called a [[pseudo-Riemannian manifold]] (the term semi-Riemannian is also used).
| |
| * If ''M'' is four-dimensional with signature (1,3) or (3,1), then the metric is called [[Lorentzian metric|Lorentzian]]. More generally, a metric tensor in dimension ''n'' other than 4 of signature (1,''n'' − 1) or (''n'' − 1, 1) is sometimes also called Lorentzian.
| |
| * If ''M'' is 2''n''-dimensional and ''g'' has signature (''n'',''n''), then the metric is called [[ultrahyperbolic metric|ultrahyperbolic]].
| |
| | |
| ===Inverse metric===
| |
| Let '''f''' = (''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>) be a basis of vector fields, and as above let ''G''['''f'''] be the matrix of coeffients
| |
| :<math>g_{ij}[\mathbf{f}] = g(X_i,X_j).\,</math>
| |
| One can consider the [[inverse matrix]] ''G''['''f''']<sup>−1</sup>, which is identified with the '''inverse metric''' (or ''conjugate'' or ''dual metric''). The inverse metric satisfies a transformation law when the frame '''f''' is changed by a matrix ''A'' via
| |
| | |
| {{NumBlk|:|<math>G[\mathbf{f}A]^{-1} = A^{-1}G[\mathbf{f}]^{-1}(A^{-1})^\mathrm{T}.</math>|{{EquationRef|5}}}}
| |
| | |
| The inverse metric transforms ''[[Covariance and contravariance of vectors|contravariantly]]'', or with respect to the inverse of the change of basis matrix ''A''. Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) [[covector]] fields; that is, fields of [[linear functional]]s.
| |
| | |
| To see this, suppose that α is a covector field. To wit, for each point ''p'', α determines a function α<sub>''p''</sub> defined on tangent vectors at ''p'' so that the following [[linear transformation|linearity]] condition holds for all tangent vectors ''X''<sub>''p''</sub> and ''Y''<sub>''p''</sub>, and all real numbers ''a'' and ''b'':
| |
| | |
| :<math>\alpha_p(aX_p+bY_p) = a\alpha_p(X_p)+b\alpha_p(Y_p).\,</math>
| |
| | |
| As ''p'' varies, α is assumed to be a [[smooth function]] in the sense that
| |
| | |
| :<math>p\mapsto \alpha_p(X_p)</math>
| |
| | |
| is a smooth function of ''p'' for any smooth vector field ''X''.
| |
| | |
| Any covector field α has components in the basis of vector fields '''f'''. These are determined by
| |
| | |
| :<math>\alpha_i = \alpha(X_i),\quad i=1,2,\dots,n.</math>
| |
| | |
| Denote the [[row vector]] of these components by
| |
| | |
| :<math>\alpha[\mathbf{f}] = \left[\alpha_1\ \ \alpha_2\ \ \dots\ \ \alpha_n\right].</math>
| |
| | |
| Under a change of '''f''' by a matrix ''A'', α['''f'''] changes by the rule
| |
| | |
| :<math>\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A.</math>
| |
| | |
| That is, the row vector of components α['''f'''] transforms as a ''covariant'' vector.
| |
| | |
| For a pair α and β of covector fields, define the inverse metric applied to these two covectors by
| |
| | |
| {{NumBlk|:|<math>\tilde{g}(\alpha,\beta) = \alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^\mathrm{T}.</math>|{{EquationRef|6}}}}
| |
| | |
| The resulting definition, although it involves the choice of basis '''f''', does not actually depend on '''f''' in an essential way. Indeed, changing basis to '''f'''''A'' gives
| |
| | |
| :<math>\begin{align}
| |
| \alpha[\mathbf{f}A]G[\mathbf{f}A]^{-1}\beta[\mathbf{f}A]^\mathrm{T} &= (\alpha[\mathbf{f}]A)\left(A^{-1}G[\mathbf{f}]^{-1}(A^{-1})^\mathrm{T}\right)A^\mathrm{T}\beta[\mathbf{f}]^\mathrm{T}\\
| |
| &=\alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^\mathrm{T}.
| |
| \end{align}
| |
| </math>
| |
| | |
| So that the right-hand side of equation ({{EquationNote|6}}) is unaffected by changing the basis '''f''' to any other basis '''f'''''A'' whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix ''G''['''f'''] are denoted by ''g''<sup>''ij''</sub>, where the indices ''i'' and ''j'' have been raised to indicate the transformation law ({{EquationNote|5}}).
| |
| | |
| ===Raising and lowering indices===
| |
| {{See also|Raising and lowering indices}}
| |
| In a basis of vector fields '''f''' = (''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>), any smooth tangent vector field ''X'' can be written in the form
| |
| | |
| {{NumBlk|:|<math>X = v^1[\mathbf{f}]X_1+v^2[\mathbf{f}]X_2+\dots+v^n[\mathbf{f}]X_n = \mathbf{f}\begin{bmatrix}v^1[\mathbf{f}]\\v^2[\mathbf{f}]\\ \vdots\\ v^n[\mathbf{f}]\end{bmatrix} = \mathbf{f}v[\mathbf{f}]\,</math>|{{EquationRef|7}}}}
| |
| | |
| for some uniquely determined smooth functions ''v''<sup>1</sup>, …, ''v''<sup>''n''</sup>. Upon changing the basis '''f''' by a nonsingular matrix ''A'', the coefficients ''v''<sup>''i''</sup> change in such a way that equation ({{EquationNote|7}}) remains true. That is,
| |
| | |
| :<math>
| |
| X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}].
| |
| </math>
| |
| | |
| Consequently, ''v''['''f'''''A''] = ''A''<sup>−1</sup>''v''['''f''']. In other words, the components of a vector transform ''contravariantly'' (with respect to the inverse) under a change of basis by the nonsingular matrix ''A''. The contravariance of the components of ''v''['''f'''] is notationally designated by placing the indices of ''v''<sup>''i''</sup>['''f'''] in the upper position.
| |
| | |
| A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields '''f''' = (''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>) define the [[dual basis]] to be the [[linear functional]]s (θ<sup>1</sup>['''f'''], …, θ<sup>''n''</sup>['''f''']) such that
| |
| | |
| :<math>\theta^i[\mathbf{f}](X_j) = \begin{cases} 1 & \mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}</math>
| |
| | |
| That is, θ<sup>''i''</sup>['''f'''](''X''<sub>''j''</sub>) = δ<sub>''j''</sub><sup>''i''</sub>, the [[Kronecker delta]]. Let
| |
| | |
| :<math>\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}]\\\theta^2[\mathbf{f}]\\\vdots\\\theta^n[\mathbf{f}]\end{bmatrix}.</math>
| |
| | |
| Under a change of basis '''f''' → '''f'''''A'' for a nonsingular matrix ''A'', θ['''f'''] transforms via
| |
| | |
| :<math>\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].</math>
| |
| | |
| Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ
| |
| | |
| {{NumBlk|:|<math>\begin{align}
| |
| \alpha &= a_1[\mathbf{f}]\theta^1[\mathbf{f}] + a_2[\mathbf{f}]\theta^2[\mathbf{f}] + \cdots + a_n[\mathbf{f}]\theta^n[\mathbf{f}]\\
| |
| &= \left[\frac{}{}a_1[\mathbf{f}]\ \ a_2[\mathbf{f}]\ \ \dots\ \ a_n[\mathbf{f}]\right]\theta[\mathbf{f}] = a[\mathbf{f}]\theta[\mathbf{f}]
| |
| \end{align}
| |
| </math>|{{EquationRef|8}}}}
| |
| | |
| where ''a''['''f'''] denotes the [[row vector]] [''a''<sub>1</sub>['''f'''] … ''a''<sub>''n''</sub>['''f'''] ]. The components ''a''<sub>''i''</sub> transform when the basis '''f''' is replaced by '''f'''''A'' in such a way that equation ({{EquationNote|8}}) continues to hold. That is,
| |
| | |
| :<math>\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]</math>
| |
| | |
| whence, because θ['''f'''''A''] = ''A''<sup>−1</sup>θ['''f'''], it follows that ''a''['''f'''''A''] = ''a''['''f''']''A''. That is, the components ''a'' transform ''covariantly'' (by the matrix ''A'' rather than its inverse). The covariance of the components of ''a''['''f'''] is notationally designated by placing the indices of ''a''<sub>''i''</sub>['''f'''] in the lower position.
| |
| | |
| Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding ''X''<sub>''p''</sub> fixed, the function
| |
| | |
| :<math>g_p(X_p,-) : Y_p \mapsto g_p(X_p,Y_p)</math>
| |
| | |
| of tangent vector ''Y''<sub>''p''</sub> defines a [[linear functional]] on the tangent space at ''p''. This operation takes a vector ''X''<sub>''p''</sub> at a point ''p'' and produces a covector ''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, −). In a basis of vector fields '''f''', if a vector field ''X'' has components ''v''['''f'''], then the components of the covector field ''g''(''X'', −) in the dual basis are given by the entries of the row vector
| |
| :<math>a[\mathbf{f}] = v[\mathbf{f}]^\mathrm{T} G[\mathbf{f}].</math>
| |
| Under a change of basis '''f'''→'''f'''''A'', the right-hand side of this equation transforms via
| |
| :<math>v[\mathbf{f}A]^\mathrm{T} G[\mathbf{f}A] = v[\mathbf{f}]^\mathrm{T} (A^{-1})^\mathrm{T} A^\mathrm{T} G[\mathbf{f}]A = v[\mathbf{f}]^\mathrm{T} G[\mathbf{f}]A</math>
| |
| so that ''a''['''f'''''A''] = ''a''['''f''']''A'': ''a'' transforms covariantly. The operation of associating to the (contravariant) components of a vector field ''v''['''f'''] = [''v''<sup>1</sup>['''f'''] ''v''<sup>2</sup>['''f'''] … ''v''<sup>''n''</sup>['''f'''] ]<sup>T</sup> the (covariant) components of the covector field ''a''['''f'''] = [''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] … ''a''<sub>''n''</sub>['''f'''] ] where
| |
| :<math>a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]</math>
| |
| is called '''lowering the index'''.
| |
| | |
| To ''raise the index'', one applies the same construction but with the inverse metric instead of the metric. If ''a''['''f'''] = [''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] … ''a''<sub>''n''</sub>['''f'''] ] are the components of a covector in the dual basis θ['''f'''], then the column vector
| |
| {{NumBlk|:|<math>v[\mathbf{f}] = G^{-1}[\mathbf{f}]a[\mathbf{f}]^\mathrm{T}</math>|{{EquationRef|9}}}}
| |
| has components which transform contravariantly:
| |
| :<math>v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].</math>
| |
| Consequently, the quantity ''X'' = '''f'''v['''f'''] does not depend on the choice of basis '''f''' in an essential way, and thus defines a vector field on ''M''. The operation ({{EquationNote|9}}) associating to the (covariant) components of a covector ''a''['''f'''] the (contravariant) components of a vector ''v''['''f'''] given is called '''raising the index'''. In components, ({{EquationNote|9}}) is
| |
| :<math>v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}]a_k[\mathbf{f}].</math>
| |
| | |
| ===Induced metric===
| |
| <!--{{main|Induced metric}} Not currently well-written. -->
| |
| Let ''U'' be an [[open set]] in '''R'''<sup>''n''</sup>, and let φ be a [[continuously differentiable]] function from ''U'' into the [[Euclidean space]] '''R'''<sup>''m''</sup> where ''m'' > ''n''. The mapping φ is called an [[immersion (mathematics)|immersion]] if its differential is injective at every point of ''U''. The image of φ is called an [[immersed submanifold]].
| |
| | |
| Suppose that φ is an immersion onto the submanifold ''M'' ⊂ '''R'''<sup>''m''</sup>. The usual Euclidean [[dot product]] in '''R'''<sup>''m''</sup> is a metric which, when restricted to vectors tangent to ''M'', gives a means for taking the dot product of these tangent vectors. This is called the '''induced metric'''.
| |
| | |
| Suppose that ''v'' is a tangent vector at a point of ''U'', say
| |
| :<math>v = v^1\mathbf{e}_1+\dots+v^n\mathbf{e}_n</math>
| |
| where '''e'''<sub>i</sub> are the standard coordinate vectors in '''R'''<sup>''n''</sup>. When φ is applied to ''U'', the vector ''v'' goes over to the vector tangent to ''M'' given by
| |
| :<math>\phi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \phi^a}{\partial x^i}\mathbf{e}_a.</math>
| |
| (This is called the [[pushforward (differential)|pushforward]] of ''v'' along φ.) Given two such vectors, ''v'' and ''w'', the induced metric is defined by
| |
| | |
| :<math>g(v,w) = \phi_*(v)\cdot \phi_*(w).</math>
| |
| | |
| It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields '''e''' is given by
| |
| | |
| :<math>G(\mathbf{e}) = (D\phi)^\mathrm{T}(D\phi)</math>
| |
| | |
| where Dφ is the Jacobian matrix:
| |
| | |
| :<math>D\phi = \begin{bmatrix}
| |
| \frac{\partial\phi^1}{\partial x^1}&\frac{\partial\phi^1}{\partial x^2}&\dots&\frac{\partial\phi^1}{\partial x^n}\\[1ex]
| |
| \frac{\partial\phi^2}{\partial x^1}&\frac{\partial\phi^2}{\partial x^2}&\dots&\frac{\partial\phi^2}{\partial x^n}\\
| |
| \vdots&\vdots&\ddots&\vdots\\
| |
| \frac{\partial\phi^m}{\partial x^1}&\frac{\partial\phi^m}{\partial x^2}&\dots&\frac{\partial\phi^m}{\partial x^n}\\
| |
| \end{bmatrix}.
| |
| </math>
| |
| | |
| ==Intrinsic definitions of a metric==
| |
| The notion of a metric can be defined intrinsically using the language of [[fiber bundle]]s and [[vector bundle]]s. In these terms, a '''metric tensor''' is a function
| |
| | |
| {{NumBlk|:|<math>g : TM\times_M TM\to \mathbf{R}</math>|{{EquationRef|5}}}}
| |
| | |
| from the [[fiber product]] of the [[tangent bundle]] of ''M'' with itself to '''R''' such that the restriction of ''g'' to each fiber is a nondegenerate bilinear mapping
| |
| | |
| :<math>g_p : T_pM\times T_pM \to \mathbf{R}.</math>
| |
| | |
| The mapping ({{EquationNote|5}}) is required to be [[continuous function|continuous]], and often [[continuously differentiable]], [[smooth function|smooth]], or [[real analytic]], depending on the case of interest, and whether ''M'' can support such a structure.
| |
| | |
| ===Metric as a section of a bundle===
| |
| By the [[Tensor product#Universal property|universal property of the tensor product]], any bilinear mapping ({{EquationNote|5}}) gives rise [[natural transformation|naturally]] to a [[section (fiber bundle)|section]] ''g''<sub>⊗</sub> of the [[dual space|dual]] of the [[tensor product bundle]] of T''M'' with itself
| |
| | |
| :<math>g_\otimes\in \Gamma\left((TM\otimes TM)^*\right).</math>
| |
| | |
| The section ''g''<sub>⊗</sub> is defined on simple elements of T''M''⊗T''M'' by
| |
| | |
| :<math>g_\otimes(v\otimes w) = g(v,w)</math>
| |
| | |
| and is defined on arbitrary elements of T''M''⊗T''M'' by extending linearly to linear combinations of simple elements. The original bilinear form ''g'' is symmetric if and only if
| |
| :<math>g_\otimes\circ\tau = g_\otimes</math>
| |
| where
| |
| :<math>\tau : TM\otimes TM\stackrel{\cong}{\to} TM\otimes TM</math>
| |
| is the [[Tensor product#Tensor powers and braiding|braiding map]].
| |
| | |
| Since ''M'' is finite-dimensional, there is a [[natural isomorphism]]
| |
| | |
| :<math>(TM\otimes TM)^*\cong T^*M\otimes T^*M,</math>
| |
| | |
| so that ''g''<sub>⊗</sub> is regarded also as a section of the bundle T*''M''⊗T*''M'' of the [[cotangent bundle]] T*''M'' with itself. Since ''g'' is symmetric as a bilinear mapping, it follows that ''g''<sub>⊗</sub> is a [[symmetric tensor]].
| |
| | |
| ===Metric in a vector bundle===
| |
| More generally, one may speak of a metric in a [[vector bundle]]. If ''E'' is a vector bundle over a manifold ''M'', then a metric is a mapping
| |
| | |
| :<math>g : E\times_M E\to \mathbf{R}</math>
| |
| | |
| from the [[fiber product]] of ''E'' to '''R''' which is bilinear in each fiber:
| |
| | |
| :<math>g_p : E_p \times E_p\to \mathbf{R}.</math>
| |
| | |
| Using duality as above, a metric is often identified with a [[section (fiber bundle)|section]] of the [[tensor product]] bundle <math>\scriptstyle E^*\otimes E^*</math>, (See [[metric (vector bundle)]].)
| |
| | |
| ===Tangent-cotangent isomorphism===
| |
| {{see also|Musical isomorphism}}
| |
| The metric tensor gives a [[natural isomorphism]] from the [[tangent bundle]] to the [[cotangent bundle]], sometimes called the [[musical isomorphism]].<ref>For the terminology "musical isomorphism", see {{harvtxt|Gallot|Hulin|Lafontaine|2004|p=75}}. See also {{harvtxt|Lee|1997|pp=27–29}}</ref> This isomorphism is obtained by setting, for each tangent vector ''X''<sub>''p''</sub> ∈ T<sub>''p''</sub>''M'',
| |
| | |
| :<math>S_gX_p\, \stackrel{def}{=}\, g(X_p,-),</math>
| |
| | |
| the [[linear functional]] on T<sub>''p''</sub>''M'' which sends a tangent vector ''Y''<sub>''p''</sub> at ''p'' to ''g''<sub>''p''</sub>(''X''<sub>''p''</sub>,''Y''<sub>''p''</sub>). That is, in terms of the pairing [−,−] between T<sub>''p''</sub>''M'' and its [[dual space]] ''T''<sub>''p''</sub>*''M'',
| |
| | |
| :<math>[S_gX_p,Y_p] = g_p(X_p,Y_p)\,</math>
| |
| | |
| for all tangent vectors ''X''<sub>''p''</sub> and ''Y''<sub>''p''</sub>. The mapping ''S''<sub>''g''</sub> is a [[linear transformation]] from T<sub>''p''</sub>''M'' to T<sub>''p''</sub>*''M''. It follows from the definition of non-degeneracy that the [[kernel (set theory)|kernel]] of ''S''<sub>''g''</sub> is reduced to zero, and so by the [[rank-nullity theorem]], ''S''<sub>''g''</sub> is a [[linear isomorphism]]. Furthermore, ''S''<sub>''g''</sub> is a [[symmetric linear transformation]] in the sense that
| |
| | |
| :<math>[S_gX_p,Y_p] = [S_gY_p,X_p] \,</math>
| |
| | |
| for all tangent vectors ''X''<sub>''p''</sub> and ''Y''<sub>''p''</sub>.
| |
| | |
| Conversely, any linear isomorphism ''S'' : T<sub>''p''</sub>''M'' → T<sub>''p''</sub>*''M'' defines a non-degenerate bilinear form on T<sub>''p''</sub>''M'' by means of
| |
| | |
| :<math>g_S(X_p,Y_p) = [SX_p,Y_p].\,</math>
| |
| | |
| This bilinear form is symmetric if and only if ''S'' is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on T<sub>''p''</sub>''M'' and symmetric linear isomorphisms of T<sub>''p''</sub>''M'' to the dual T<sub>''p''</sub>*''M''.
| |
| | |
| As ''p'' varies over ''M'', ''S''<sub>''g''</sub> defines a section of the bundle Hom(T''M'',T*''M'') of [[vector bundle morphism|vector bundle isomorphisms]] of the tangent bundle to the cotangent bundle. This section has the same smoothness as ''g'': it is continuous, differentiable, smooth, or real-analytic according as ''g''. The mapping ''S''<sub>''g''</sub>, which associates to every vector field on ''M'' a covector field on ''M'' gives an abstract formulation of "lowering the index" on a vector field. The inverse of ''S''<sub>''g''</sub> is a mapping T*''M'' → T''M'' which, analogously, gives an abstract formulation of "raising the index" on a covector field.
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| The inverse ''S''<sub>''g''</sub><sup>−1</sup> defines a linear mapping
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| :<math>S_g^{-1} : T^*M \to TM</math>
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| which is nonsingular and symmetric in the sense that
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| :<math>[S_g^{-1}\alpha,\beta] = [S_g^{-1}\beta,\alpha]</math>
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| for all covectors α, β. Such a nonsingular symmetric mapping gives rise (by the [[tensor-hom adjunction]]) to a map
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| :<math>T^*M\otimes T^*M \to \mathbf{R}</math>
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| or by the [[Double dual|double dual isomorphism]] to a section of the tensor product
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| :<math>TM\otimes TM.</math>
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| ==Arclength and the line element==
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| Suppose that ''g'' is a Riemannian metric on ''M''. In a local coordinate system ''x''<sup>''i''</sup>, ''i'' = 1,2,…,''n'', the metric tensor appears as a [[matrix (math)|matrix]], denoted here by '''G''', whose entries are the components ''g''<sub>''ij''</sub> of the metric tensor relative to the coordinate vector fields.
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| Let γ(''t'') be a piecewise differentiable [[parametric curve]] in ''M'', for ''a'' ≤''t'' ≤ ''b''. The '''[[arclength]]''' of the curve is defined by
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| :<math>L = \int_a^b \sqrt{ \sum_{i,j=1}^n g_{ij}(\gamma(t))\left({d\over dt}x^i\circ\gamma(t)\right)\left({d\over dt}x^j\circ\gamma(t)\right)}\,dt.</math>
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| In connection with this geometrical application, the [[quadratic form|quadratic]] [[differential form]]
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| :<math>ds^2 = \sum_{i,j=1}^n g_{ij}(p)dx^i dx^j</math>
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| is called the [[first fundamental form]] associated to the metric, while ''ds'' is the [[line element]]. When ''ds''<sup>2</sup> is [[pullback (differential geometry)|pulled back]] to the image of a curve in ''M'', it represents the square of the differential with respect to arclength.
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| For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define
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| :<math>L = \int_a^b \sqrt{ \left|\sum_{i,j=1}^ng_{ij}(\gamma(t))\left({d\over dt}x^i\circ\gamma(t)\right)\left({d\over dt}x^j\circ\gamma(t)\right)\right|}\,dt \ .</math>
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| Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.
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| ===The energy, variational principles and geodesics===
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| Given a segment of a curve, another frequently defined quantity is the (kinetic) '''energy''' of the curve:
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| :<math>E = \frac{1}{2} \int_a^b \sum_{i,j=1}^ng_{ij}(\gamma(t))\left({d\over dt}x^i\circ\gamma(t)\right)\left({d\over dt}x^j\circ\gamma(t)\right)\,dt. \ </math>
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| This usage comes from [[physics]], specifically, [[classical mechanics]], where the integral ''E'' can be seen to directly correspond to the [[kinetic energy]] of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of [[Maupertuis principle]], the metric tensor can be seen to correspond to the mass tensor of a moving particle.
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| In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the [[geodesic equation]]s may be obtained by applying [[variational principle]]s to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the [[principle of least action]]: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.<ref>{{harvnb|Sternberg|1983}}</ref>
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| ==Canonical measure and volume form==
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| In analogy with the case of surfaces, a metric tensor on an ''n''-dimensional paracompact manifold ''M'' gives rise to a natural way to measure the ''n''-dimensional [[volume]] of subsets of the manifold. The resulting natural positive [[Borel measure]] allows one to develop a theory of integrating functions on the manifold by means of the associated [[Lebesgue integral]].
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| A measure can be defined, by the [[Riesz representation theorem]], by giving a positive [[linear functional]] Λ on the space ''C''<sub>0</sup>(''M'') of [[compact support|compactly supported]] [[continuous function]]s on ''M''. More precisely, if ''M'' is a manifold with a (pseudo-)Riemannian metric tensor ''g'', then there is a unique positive [[Borel measure]] μ<sub>''g''</sub> such that for any [[coordinate chart]] (''U'',φ),
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| :<math>\Lambda f = \int_U f\,d\mu_g = \int_{\varphi(U)} f\circ\varphi^{-1}(x) \sqrt{|\det g|}\,dx</math>
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| for all ''ƒ'' supported in ''U''. Here det ''g'' is the [[determinant]] of the matrix formed by the components of the metric tensor in the coordinate chart. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by [[integration by substitution|Jacobian change of variables]]. It extends to a unique positive linear functional on ''C''<sub>0</sub>(''M'') by means of a [[partition of unity]].
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| If ''M'' is in addition [[orientation (mathematics)|oriented]], then it is possible to define a natural [[volume form]] from the metric tensor. In a [[right-handed coordinate system|positively oriented coordinate system]] (''x''<sup>''1''</sup>,...,''x''<sup>''n''</sup>) the volume form is represented as
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| :<math>\omega = \sqrt{|\det g|}\, dx^1\wedge\cdots\wedge dx^n</math>
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| where the ''dx''<sup>''i''</sup> are the [[coordinate differential]]s and the wedge ∧ denotes the [[exterior product]] in the algebra of [[differential form]]s. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.
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| ==Examples==
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| ===The ''Euclidean metric''===
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| The most familiar example is that of elementary [[Euclidean geometry]]: the two-dimensional [[Euclidean distance|Euclidean]] metric tensor. In the usual <math>x</math>-<math>y</math> coordinates, we can write
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| :<math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}. \ </math>
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| The length of a curve reduces to the formula:
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| :<math>L = \int_a^b \sqrt{ (dx)^2 + (dy)^2}. \ </math>
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| The Euclidean metric in some other common coordinate systems can be written as follows.
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| [[Polar coordinates]]: <math>(r, \theta) \ </math>
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| :<math>x = r \cos\theta</math>
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| :<math>y = r \sin\theta</math>
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| :<math>J = \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix}.</math>
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| So
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| :<math>g = J^\mathrm{T}J = \begin{bmatrix}\cos^2\theta+\sin^2\theta & -r\sin\theta \cos\theta + r\sin\theta\cos\theta \\ -r\cos\theta\sin\theta + r\cos\theta\sin\theta & r^2 \sin^2\theta + r^2\cos^2\theta\end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & r^2\end{bmatrix} \ </math>
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| by [[trigonometric identity|trigonometric identities]].
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| In general, in a [[Cartesian coordinate system]] ''x''<sup>''i''</sup> on a [[Euclidean space]], the partial derivatives <math>\partial/\partial x^i</math> are [[orthonormal]] with respect to the Euclidean metric. Thus the metric tensor is the [[Kronecker delta]] δ<sub>''ij''</sub> in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates <math>q^i</math> is given by:
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| ::<math>g_{ij} = \sum_{kl}\delta_{kl}{\partial x^k \over \partial q^i} {\partial x^l \over \partial q^j} = \sum_k\frac{\partial x^k}{\partial q^i}\frac{\partial x^k}{\partial q^j}.</math>
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| ====The round metric on a sphere====
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| The unit sphere in '''R'''<sup>3</sup> comes equipped with a natural metric induced from the ambient Euclidean metric. In standard spherical coordinates <math>(\theta,\phi)</math>, with <math>\theta</math> the co-latitude, the angle measured from the z axis, and <math>\phi</math> the angle from the x axis in the xy plane, the metric takes the form
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| :<math>g = \left[\begin{array}{cc} 1 & 0 \\ 0 & \sin^2 \theta\end{array}\right].</math>
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| This is usually written in the form
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| :<math>ds^2 = d\theta^2 + \sin^2\theta\,d\phi^2.</math>
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| ===''Lorentzian metrics from relativity''===
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| {{main|Metric tensor (general relativity)}}
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| In flat [[Minkowski space]] ([[special relativity]]), with coordinates <math>r^\mu \rightarrow (x^0, x^1, x^2, x^3)=(ct, x, y, z) \ ,</math> the metric is
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| :<math>g = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}. \ </math>
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| For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a [[Spacetime interval|timelike]] curve, the length formula gives the [[proper time]] along the curve.
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| In this case, the [[spacetime interval]] is written as
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| :<math>ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = dr^\mu dr_\mu = g_{\mu \nu} dr^\mu dr^\nu\ </math>.
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| The [[Schwarzschild metric]] describes the spacetime around a spherically symmetric body, such as a planet, or a [[black hole]]. With coordinates <math>(x^0, x^1, x^2, x^3)=(ct, r, \theta, \phi) </math>, we can write the metric as
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| :<math>G = (g_{\mu\nu}) = \begin{bmatrix} (1-\frac{2GM}{rc^2}) & 0 & 0 & 0\\ 0 & -(1-\frac{2GM}{r c^2})^{-1} & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2 \sin^2 \theta \end{bmatrix}\,</math>
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| where ''G'' (inside the matrix) is the [[gravitational constant]] and ''M'' represents the total mass-energy content of the central object.
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| ==See also==
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| * [[Basic introduction to the mathematics of curved spacetime]]
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| * [[Clifford algebra]]
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| * [[Finsler manifold]]
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| * [[List of coordinate charts]]
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| * [[Ricci calculus]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
| |
| * {{Citation | last1=Dodson | first1=C. T. J. | last2=Poston | first2=T. | title=Tensor geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-3-540-52018-4 | id={{MathSciNet | id = 1223091}} | year=1991 | volume=130}}
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| * {{Citation | last1=Gallot | first1=Sylvestre | last2=Hullin | first2=Dominique | last3=Lafontaine | first3=Jacques | title=Riemannian Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-3-540-20493-0 | year=2004}}.
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| * {{citation|last=Gauss|first=Carl Friedrich|authorlink=Carl Friedrich Gauss|title=General Investigations of Curved Surfaces|url= http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABR1255|publication-date=1965|year=1827|publisher = Raven Press|publication-place=New York}} translated by A.M.Hiltebeitel and J.C.Morehead; [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN35283028X_0006_2NS "Disquisitiones generales circa superficies curvas"], ''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. '''VI''' (1827), pp. 99–146.
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| * {{citation|first1=S.W.|last1=Hawking|authorlink1=Stephen Hawking|first2=G.F.R.|last2=Ellis|title=The large scale structure of space-time|publisher=Cambridge University Press|year=1973}}.
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| * {{citation|title=Schaum's Outline of Theory and Problems of Tensor Calculus|first=David|last=Kay|publisher=McGraw-Hill|year=1988|isbn=978-0-07-033484-7}}.
| |
| * {{citation|first=Morris|last=Kline|authorlink=Morris Kline|title= Mathematical thought from ancient to modern times, Volume 3|year=1990|publisher=Oxford University Press}}.
| |
| * {{citation|first=John|last=Lee|title=Riemannian manifolds|publisher=Springer Verlag|year=1997|isbn=978-0-387-98322-6}}.
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| * {{citation|first=Peter W.|last=Michor|title=Topics in Differential Geometry|series=Graduate Studies in Mathematics|volume=Vol. 93|publisher=American Mathematical Society|publication-place=Providence|year=2008}} (''to appear'').
| |
| * {{Citation
| |
| | first1=Charles W.
| |
| | last1=Misner
| |
| | authorlink1=Charles W. Misner
| |
| | first2=Kip S.
| |
| | last2=Thorne
| |
| | authorlink2=Kip S. Thorne
| |
| | first3=John A.
| |
| | last3=Wheeler
| |
| | authorlink3=John Archibald Wheeler
| |
| | title=Gravitation
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| | publisher= W. H. Freeman
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| | year=1973
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| | isbn=0-7167-0344-0
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| }}
| |
| * {{citation | first1=Gregorio|last1=Ricci-Curbastro|authorlink1=Gregorio Ricci-Curbastro|first2=Tullio|last2=Levi-Civita|authorlink2=Tullio Levi-Civita|title=Méthodes de calcul différentiel absolu et leurs applications|journal=Mathematische Annalen|year=1900|volume=54|pages=125–201|issn=1432-1807 | doi=10.1007/BF01454201|issue=1|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0054&DMDID=DMDLOG_0011&L=1}}
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| * {{citation | last = Sternberg | first = S. |authorlink=Shlomo Sternberg| year = 1983 | title = Lectures on Differential Geometry | edition = 2nd | publisher = Chelsea Publishing Co. | location = New York | isbn = 0-8218-1385-4}}
| |
| * {{Citation
| |
| | last=Vaughn
| |
| | first=Michael T.
| |
| | title=Introduction to mathematical physics
| |
| | publisher=Wiley-VCH Verlag GmbH & Co.
| |
| | location=Weinheim
| |
| | isbn=978-3-527-40627-2
| |
| | id={{MathSciNet | id = 2324500}}
| |
| | year=2007}}
| |
| * {{Citation | last1=Wells | first1=Raymond | author1-link=Raymond O'Neil Wells, Jr. | title=Differential Analysis on Complex Manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1980}}
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| ==External links==
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| * [http://www.black-holes.org/numrel1.html Caltech Tutorial on Relativity] — A simple introduction to the basics of metrics in the context of relativity.
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| {{tensors}}
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| [[Category:Riemannian geometry]]
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| [[Category:Tensors]]
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| [[Category:Concepts in physics]]
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| [[Category:Differential geometry]]
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| [[Category:Metric tensors|*1]]
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