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{{Short description|Structure defining distance on a manifold}}
{{about|metric structures on manifolds|the specific case of spacetime of relativity|Metric tensor (general relativity)}}

In the [[mathematics|mathematical]] field of [[differential geometry]], a '''metric tensor''' (or simply '''metric''') is an additional [[Mathematical structure|structure]] on a [[manifold]] {{mvar|M}} (such as a [[surface (mathematics)|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 {{mvar|p}} of {{mvar|M}} is a [[bilinear form]] defined on the [[tangent space]] at {{mvar|p}} (that is, a [[bilinear function]] that maps pairs of [[tangent vector]]s to [[real number]]s), and a metric field on {{mvar|M}} consists of a metric tensor at each point {{mvar|p}} of {{mvar|M}} that varies smoothly with {{mvar|p}}.

A metric tensor {{mvar|g}} is ''positive-definite'' if {{math|''g''(''v'', ''v'') > 0}} for every nonzero vector {{mvar|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 {{mvar|M}}, the length of a smooth curve between two points {{mvar|p}} and {{mvar|q}} can be defined by integration, and the [[metric (mathematics)|distance]] between {{mvar|p}} and {{mvar|q}} can be defined as the [[infimum]] of the lengths of all such curves; this makes {{mvar|M}} a [[metric space]].  Conversely, the metric tensor itself is the [[derivative]] of the distance function (taken in a suitable manner).{{fact|date=August 2022}}

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 [[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 field is defined to be a [[nondegenerate form|nondegenerate]] [[symmetric bilinear form]] on each tangent space that varies [[smooth function|smoothly]] from point to point.

==Introduction==
[[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]] {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} of points on the surface depending on two auxiliary variables {{mvar|u}} and {{mvar|v}}. Thus a parametric surface is (in today's terms) a [[vector-valued function]]

:<math>\vec{r}(u,\,v) = \bigl( x(u,\,v),\, y(u,\,v),\, z(u,\,v) \bigr)</math>

depending on an [[ordered pair]] of real variables {{math|(''u'', ''v'')}}, and defined in an [[open set]] {{mvar|D}} in the {{mvar|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 [[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. 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 <math display=inline> \begin{bmatrix} E & F \\ F & G \end{bmatrix} </math> in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.
<!-- Since what? -->

===Arc length===
If the variables {{mvar|u}} and {{mvar|v}} are taken to depend on a third variable, {{mvar|t}}, taking values in an [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}}, then {{math|{{vec|''r''}}(''u''(''t''), ''v''(''t''))}} will trace out a [[parametric curve]] in parametric surface {{mvar|M}}. The [[arc length]] of that curve is given by the [[integral]]

: <math> \begin{align}
  s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\[5pt]
    &= \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{align}</math>

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>\vec{r}_u = \frac{\partial \vec{r}}{\partial u}\,, \quad \vec{r}_v = \frac{\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]]

{{NumBlk|:|<math>(ds)^2 = E \,(du)^2 + 2F \,du\, dv + G\, (dv)^2 ,</math>|{{EquationRef|1}}}}

where

{{NumBlk|:|<math>
  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 .
</math>|{{EquationRef|2}}}}

The quantity {{mvar|ds}} in ({{EquationNote|1}}) is called the [[line element]], while {{math|''ds''<sup>2</sup>}} is called the [[first fundamental form]] of {{mvar|M}}. Intuitively, it represents the [[principal part]] of the square of the displacement undergone by {{math|{{vec|''r''}}(''u'', ''v'')}} when {{mvar|u}} is increased by {{mvar|du}} units, and {{mvar|v}} is increased by {{mvar|dv}} units.

Using matrix notation, the first fundamental form becomes
:<math>ds^2 =
  \begin{bmatrix} du & dv \end{bmatrix}
  \begin{bmatrix} E & F \\ F & G \end{bmatrix}
  \begin{bmatrix} du \\ dv \end{bmatrix}
</math>

===Coordinate transformations===
Suppose now that a different parameterization is selected, by allowing {{mvar|u}} and {{mvar|v}} to depend on another pair of variables {{math|''u''′}} and {{math|''v''′}}. Then the analog of ({{EquationNote|2}}) for the new variables is
{{NumBlk|:|<math>
  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'}.
</math>|{{EquationRef|2'}}}}

The [[chain rule]] relates {{math|''E''′}}, {{math|''F''′}}, and {{math|''G''′}} to {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} via the [[matrix (mathematics)|matrix]] equation

{{NumBlk|:|<math>\begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix} =
  \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
  \end{bmatrix}^\mathsf{T}
  \begin{bmatrix} E & F \\ F & G \end{bmatrix}
  \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
  \end{bmatrix}
</math>|{{EquationRef|3}}}}

where the superscript T denotes the [[matrix transpose]]. The matrix with the coefficients {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} arranged in this way therefore transforms by the [[Jacobian matrix]] of the coordinate change

:<math>
  J = \begin{bmatrix}
    \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\
    \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
\end{bmatrix}\,.</math>

A matrix which transforms in this way is one kind of what is called a [[tensor]]. The matrix

:<math>\begin{bmatrix} E & F \\ F & G \end{bmatrix}</math>

with the transformation law ({{EquationNote|3}}) is known as the metric tensor of the surface.

===Invariance of arclength under coordinate transformations===
{{harvtxt|Ricci-Curbastro|Levi-Civita|1900}} first observed the significance of a system of coefficients {{mvar|E}}, {{mvar|F}}, and {{mvar|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 {{mvar|E}}, {{mvar|F}}, and {{mvar|G}}. Indeed, by the chain rule,

:<math>\begin{bmatrix} du \\ dv \end{bmatrix} =
  \begin{bmatrix}
    \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\
    \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
  \end{bmatrix}
  \begin{bmatrix} du' \\ dv' \end{bmatrix}
</math>

so that

:<math>\begin{align}
  ds^2
  &=
    \begin{bmatrix} du & dv \end{bmatrix}
    \begin{bmatrix} E & F \\ F & G \end{bmatrix}
    \begin{bmatrix} du \\ dv \end{bmatrix} \\[6pt]
  &=
    \begin{bmatrix} du' & dv' \end{bmatrix}
    \begin{bmatrix}
      \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
      \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
    \end{bmatrix}^\mathsf{T}
    \begin{bmatrix} E & F \\ F & G \end{bmatrix}
    \begin{bmatrix}
      \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt]
      \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'}
    \end{bmatrix}
    \begin{bmatrix} du' \\ dv' \end{bmatrix} \\[6pt]
  &=
    \begin{bmatrix} du' & dv' \end{bmatrix}
    \begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix}
    \begin{bmatrix} du' \\ dv' \end{bmatrix}\\[6pt]
  &= (ds')^2 \,.
\end{align}</math>

===Length and angle===
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]](non-euclidean geometry) of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface {{mvar|M}} can be written in the form

:<math>\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v</math>

for suitable real numbers {{math|''p''<sub>1</sub>}} and {{math|''p''<sub>2</sub>}}. If two tangent vectors are given:

:<math>\begin{align}
  \mathbf{a} &= a_1\vec{r}_u + a_2\vec{r}_v \\
  \mathbf{b} &= b_1\vec{r}_u + b_2\vec{r}_v
\end{align}</math>

then using the [[bilinear form|bilinearity]] of the dot product,

:<math>\begin{align}
  \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 + a_2b_1 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v \\[8pt]
    &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\[8pt]
    &= \begin{bmatrix} a_1 & a_2 \end{bmatrix}
         \begin{bmatrix} E & F \\ F & G \end{bmatrix}
         \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \,.
\end{align}</math>

This is plainly a function of the four variables {{math|''a''<sub>1</sub>}}, {{math|''b''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}}, and {{math|''b''<sub>2</sub>}}. It is more profitably viewed, however, as a function that takes a pair of arguments {{math|'''a''' {{=}} [''a''<sub>1</sub> ''a''<sub>2</sub>]}} and {{math|'''b''' {{=}} [''b''<sub>1</sub> ''b''<sub>2</sub>]}} which are vectors in the {{mvar|uv}}-plane. That is, put

:<math>g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,.</math>

This is a [[symmetric function]] in {{math|'''a'''}} and {{math|'''b'''}}, meaning that

:<math>g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.</math>

It is also [[bilinear form|bilinear]], meaning that it is [[linear functional|linear]] in each variable {{math|'''a'''}} and {{math|'''b'''}} separately. That is,

:<math>\begin{align}
  g\left(\lambda\mathbf{a} + \mu\mathbf{a}', \mathbf{b}\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}', \mathbf{b}\right),\quad\text{and} \\
  g\left(\mathbf{a}, \lambda\mathbf{b} + \mu\mathbf{b}'\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}, \mathbf{b}'\right)
\end{align}</math>

for any vectors {{math|'''a'''}}, {{math|'''a'''′}}, {{math|'''b'''}}, and {{math|'''b'''′}} in the {{mvar|uv}} plane, and any real numbers {{mvar|μ}} and {{mvar|λ}}.

In particular, the length of a tangent vector {{math|'''a'''}} is given by

:<math> \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}</math>

and the angle {{mvar|θ}} between two vectors {{math|'''a'''}} and {{math|'''b'''}} is calculated by

:<math>\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.</math>

===Area===
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 {{mvar|M}} is parameterized by the function {{math|{{vec|''r''}}(''u'', ''v'')}} over the domain {{mvar|D}} in the {{mvar|uv}}-plane, then the surface area of {{mvar|M}} is given by the integral

:<math>\iint_D \left|\vec{r}_u \times \vec{r}_v\right|\,du\,dv</math>

where {{math|×}} 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

:<math>\begin{align}
       &\iint_D \sqrt{\left(\vec{r}_u\cdot\vec{r}_u\right) \left(\vec{r}_v\cdot\vec{r}_v\right) - \left(\vec{r}_u\cdot\vec{r}_v\right)^2}\,du\,dv \\[5pt]
   ={} &\iint_D \sqrt{EG - F^2}\,du\,dv\\[5pt]
   ={} &\iint_D \sqrt{\det \begin{bmatrix} E & F \\ F & G \end{bmatrix}}\, du\, dv
\end{align}</math>

where {{math|det}} is the [[determinant]].

==Definition==
Let {{mvar|M}} be a [[smooth manifold]] of dimension {{mvar|n}}; for instance a [[surface (differential geometry)|surface]] (in the case {{math|1=''n'' = 2}}) or [[hypersurface]] in the [[Cartesian space]] <math>\R^{n+1}</math>. At each point {{math|''p'' ∈ ''M''}} there is a [[vector space]] {{math|T<sub>''p''</sub>''M''}}, called the [[tangent space]], consisting of all tangent vectors to the manifold at the point {{mvar|p}}. A metric tensor at {{mvar|p}} is a function {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, ''Y''<sub>''p''</sub>)}} which takes as inputs a pair of tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}} at {{mvar|p}}, and produces as an output a [[real number]] ([[scalar (mathematics)|scalar]]), so that the following conditions are satisfied:
* {{math|''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 {{math|''U''<sub>''p''</sub>}}, {{math|''V''<sub>''p''</sub>}}, {{math|''Y''<sub>''p''</sub>}} are three tangent vectors at {{mvar|p}} and {{mvar|a}} and {{mvar|b}} are real numbers, then <math display="block">\begin{align}
  g_p(aU_p + bV_p, Y_p) &= ag_p(U_p, Y_p) + bg_p(V_p, Y_p) \,, \quad \text{and} \\
  g_p(Y_p, aU_p + bV_p) &= ag_p(Y_p, U_p) + bg_p(Y_p, V_p) \,.
\end{align}</math>
* {{math|''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 {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}, <math display="block">g_p(X_p, Y_p) = g_p(Y_p, X_p)\,.</math>
* {{math|''g''<sub>''p''</sub>}} is [[nondegenerate]]. A bilinear function is nondegenerate provided that, for every tangent vector {{math|''X''<sub>''p''</sub> ≠ 0}}, the function <math display="block">Y_p \mapsto g_p(X_p,Y_p)</math> obtained by holding {{math|''X''<sub>''p''</sub>}} constant and allowing {{math|''Y''<sub>''p''</sub>}} to vary is not [[identically zero]]. That is, for every {{math|''X''<sub>''p''</sub> ≠ 0}} there exists a {{math|''Y''<sub>''p''</sub>}} such that {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, ''Y''<sub>''p''</sub>) ≠ 0}}.

A metric tensor field {{mvar|g}} on {{mvar|M}} assigns to each point {{mvar|p}} of {{mvar|M}} a metric tensor {{math|''g''<sub>''p''</sub>}} in the tangent space at {{mvar|p}} in a way that varies [[smooth function|smoothly]] with {{mvar|p}}. More precisely, given any [[open set|open subset]] {{mvar|U}} of manifold {{mvar|M}} and any (smooth) [[vector field]]s {{mvar|X}} and {{mvar|Y}} on {{mvar|U}}, the real function
<math display="block">g(X, Y)(p) = g_p(X_p, Y_p)</math>
is a smooth function of {{mvar|p}}.

==Components of the metric==
{{Hatnote|This section assumes some familiarity with [[coordinate vector]]s.}}
The components of the metric in any [[basis of a vector space|basis]] of [[vector field]]s, or [[frame bundle|frame]], {{math|'''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>
{{NumBlk|:|<math>g_{ij}[\mathbf{f}] = g\left(X_i, X_j\right).</math>|{{EquationRef|4}}}}
The {{math|''n''<sup>2</sup>}} functions {{math|''g''<sub>''ij''</sub>['''f''']}} form the entries of an {{math|''n'' × ''n''}} [[symmetric matrix]], {{math|''G''['''f''']}}. If
:<math>v = \sum_{i=1}^n v^iX_i \,, \quad w = \sum_{i=1}^n w^iX_i</math>
are two vectors at {{math|''p'' ∈ ''U''}}, then the value of the metric applied to {{mvar|v}} and {{mvar|w}} is determined by the coefficients ({{EquationNote|4}}) by bilinearity:

:<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>

Denoting the [[matrix (mathematics)|matrix]] {{math|(''g''<sub>''ij''</sub>['''f'''])}} by {{math|''G''['''f''']}} and arranging the components of the vectors {{mvar|v}} and {{mvar|w}} into [[column vector]]s {{math|'''v'''['''f''']}} and {{math|'''w'''['''f''']}},

:<math>g(v,w) = \mathbf{v}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]</math>

where {{math|'''v'''['''f''']}}<sup>T</sup> and {{math|'''w'''['''f''']}}<sup>T</sup> denote the [[matrix transpose|transpose]] of the vectors {{math|'''v'''['''f''']}} and {{math|'''w'''['''f''']}}, respectively. Under a [[change of basis]] of the form

:<math>\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A</math>

for some [[invertible matrix|invertible]] {{math|''n'' × ''n''}} matrix {{math|''A'' {{=}} (''a''<sub>''ij''</sub>)}}, the matrix of components of the metric changes by {{mvar|A}} as well. That is,

:<math>G[\mathbf{f}A] = A^\mathsf{T} G[\mathbf{f}]A</math>

or, in terms of the entries of this matrix,

:<math>g_{ij}[\mathbf{f}A] = \sum_{k,l=1}^n a_{ki}g_{kl}[\mathbf{f}]a_{lj} \, .</math>

For this reason, the system of quantities {{math|''g''<sub>''ij''</sub>['''f''']}} is said to transform covariantly with respect to changes in the frame {{math|'''f'''}}.

===Metric in coordinates===
A system of {{mvar|n}} real-valued functions {{math|(''x''<sup>1</sup>, ..., ''x''<sup>''n''</sup>)}}, giving a local [[coordinates|coordinate system]] on an [[open set]] {{mvar|U}} in {{mvar|M}}, determines a basis of vector fields on {{mvar|U}}
:<math>\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.</math>

The metric {{mvar|g}} has components relative to this frame given by
:<math>g_{ij}\left[\mathbf{f}\right] = 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}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).</math>

This new system of functions is related to the original {{math|''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}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.</math>

Or, in terms of the matrices {{math|''G''['''f'''] {{=}} (''g''<sub>''ij''</sub>['''f'''])}} and {{math|''G''['''f'''′] {{=}} (''g''<sub>''ij''</sub>['''f'''′])}},
:<math>G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}</math>

where {{mvar|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 {{math|''q''<sub>''m''</sub>}} is positive for all non-zero {{math|''X''<sub>''m''</sub>}}, then the metric is [[definite bilinear form|positive-definite]] at {{mvar|m}}. If the metric is positive-definite at every {{math|''m'' ∈ ''M''}}, then {{mvar|g}} is called a [[Riemannian metric]]. More generally, if the quadratic forms {{math|''q''<sub>''m''</sub>}} have constant [[signature of a quadratic form|signature]] independent of {{mvar|m}}, then the signature of {{mvar|g}} is this signature, and {{mvar|g}} is called a [[pseudo-Riemannian metric]].<ref>{{harvnb|Dodson|Poston|1991|loc=Chapter VII §3.04}}</ref> If {{mvar|M}} is [[connected space|connected]], then the signature of {{mvar|q<sub>m</sub>}} does not depend on {{mvar|m}}.<ref>{{harvnb|Vaughn|2007|loc=§3.4.3}}</ref>

By [[Sylvester's law of inertia]], a basis of tangent vectors {{math|''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) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^{p+1}\right)^2-\cdots-\left(\xi^n\right)^2</math>

for some {{mvar|p}} between 1 and {{mvar|n}}. Any two such expressions of {{mvar|q}} (at the same point {{mvar|m}} of {{mvar|M}}) will have the same number {{mvar|p}} of positive signs. The signature of {{mvar|g}} is the pair of integers {{math|(''p'', ''n'' − ''p'')}}, signifying that there are {{mvar|p}} positive signs and {{math|''n'' − ''p''}} negative signs in any such expression. Equivalently, the metric has signature {{math|(''p'', ''n'' − ''p'')}} if the matrix {{math|''g''<sub>''ij''</sub>}} of the metric has {{mvar|p}} positive and {{math|''n'' − ''p''}} negative [[eigenvalue]]s.

Certain metric signatures which arise frequently in applications are:
* If {{mvar|g}} has signature {{math|(''n'', 0)}}, then {{mvar|g}} is a Riemannian metric, and {{mvar|M}} is called a [[Riemannian manifold]]. Otherwise, {{mvar|g}} is a pseudo-Riemannian metric, and {{mvar|M}} is called a [[pseudo-Riemannian manifold]] (the term semi-Riemannian is also used).
* If {{mvar|M}} is four-dimensional with signature {{math|(1, 3)}} or {{math|(3, 1)}}, then the metric is called [[Lorentzian metric|Lorentzian]]. More generally, a metric tensor in dimension {{mvar|n}} other than 4 of signature {{math|(1, ''n'' − 1)}} or {{math|(''n'' − 1, 1)}} is sometimes also called Lorentzian.
* If {{mvar|M}} is {{math|2''n''}}-dimensional and {{mvar|g}} has signature {{math|(''n'', ''n'')}}, then the metric is called [[ultrahyperbolic metric|ultrahyperbolic]].

===Inverse metric===
Let {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} be a basis of vector fields, and as above let {{math|''G''['''f''']}} be the matrix of coefficients
:<math>g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.</math>
One can consider the [[inverse matrix]] {{math|''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 {{math|'''f'''}} is changed by a matrix {{mvar|A}} via

{{NumBlk|:|<math>G[\mathbf{f}A]^{-1} = A^{-1}G[\mathbf{f}]^{-1}\left(A^{-1}\right)^\mathsf{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 {{mvar|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 {{mvar|α}} is a covector field. To wit, for each point {{mvar|p}}, {{mvar|α}} determines a function {{math|''α''<sub>''p''</sub>}} defined on tangent vectors at {{mvar|p}} so that the following [[linear transformation|linearity]] condition holds for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}, and all real numbers {{mvar|a}} and {{mvar|b}}:

:<math>\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.</math>

As {{mvar|p}} varies, {{mvar|α}} is assumed to be a [[smooth function]] in the sense that

:<math>p \mapsto \alpha_p \left(X_p\right)</math>

is a smooth function of {{mvar|p}} for any smooth vector field {{mvar|X}}.

Any covector field {{mvar|α}} has components in the basis of vector fields {{math|'''f'''}}. These are determined by

:<math>\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.</math>

Denote the [[row vector]] of these components by

:<math>\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.</math>

Under a change of {{math|'''f'''}} by a matrix {{mvar|A}}, {{math|''α''['''f''']}} changes by the rule

:<math>\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.</math>

That is, the row vector of components {{math|''α''['''f''']}} transforms as a ''covariant'' vector.

For a pair {{mvar|α}} and {{mvar|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}]^\mathsf{T}.</math>|{{EquationRef|6}}}}

The resulting definition, although it involves the choice of basis {{math|'''f'''}}, does not actually depend on {{math|'''f'''}} in an essential way. Indeed, changing basis to {{math|'''f'''''A''}} gives

:<math>\begin{align}
      &\alpha[\mathbf{f}A] G[\mathbf{f}A]^{-1} \beta[\mathbf{f}A]^\mathsf{T} \\
  ={} &\left(\alpha[\mathbf{f}]A\right) \left(A^{-1}G[\mathbf{f}]^{-1} \left(A^{-1}\right)^\mathsf{T}\right) \left(A^\mathsf{T}\beta[\mathbf{f}]^\mathsf{T}\right) \\
  ={} &\alpha[\mathbf{f}] G[\mathbf{f}]^{-1} \beta[\mathbf{f}]^\mathsf{T}.
\end{align}
</math>

So that the right-hand side of equation ({{EquationNote|6}}) is unaffected by changing the basis {{math|'''f'''}} to any other basis {{math|'''f'''''A''}} whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix {{math|''G''['''f''']}} are denoted by {{math|''g''<sup>''ij''</sup>}}, where the indices {{mvar|i}} and {{mvar|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 {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}, any smooth tangent vector field {{mvar|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 {{math|''v''<sup>1</sup>, ..., ''v''<sup>''n''</sup>}}. Upon changing the basis {{math|'''f'''}} by a nonsingular matrix {{mvar|A}}, the coefficients {{math|''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, {{math|''v''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''v''['''f''']}}. In other words, the components of a vector transform ''contravariantly'' (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix {{mvar|A}}. The contravariance of the components of {{math|''v''['''f''']}} is notationally designated by placing the indices of {{math|''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 {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} define the [[dual basis]] to be the [[linear functional]]s {{math|(''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, {{math|''θ''<sup>''i''</sup>['''f'''](''X''<sub>''j''</sub>) {{=}} ''δ''<sub>''j''</sub><sup>''i''</sup>}}, 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 {{math|'''f''' ↦ '''f'''''A''}} for a nonsingular matrix {{math|''A''}}, {{math|''θ''['''f''']}} transforms via

:<math>\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].</math>

Any linear functional {{mvar|α}} on tangent vectors can be expanded in terms of the dual basis {{mvar|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}] \\[8pt]
         &= \big\lbrack\begin{array}{cccc}a_1[\mathbf{f}] & a_2[\mathbf{f}] & \dots & a_n[\mathbf{f}]\end{array}\big\rbrack \theta[\mathbf{f}] \\[8pt]
         &= a[\mathbf{f}] \theta[\mathbf{f}]
\end{align}</math>|{{EquationRef|8}}}}

where {{math|''a''['''f''']}} denotes the [[row vector]] {{math|[ ''a''<sub>1</sub>['''f'''] ... ''a''<sub>''n''</sub>['''f'''] ]}}. The components {{math|''a''<sub>''i''</sub>}} transform when the basis {{math|'''f'''}} is replaced by {{math|'''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 {{math|''θ''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''θ''['''f''']}}, it follows that {{math|1=''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}. That is, the components {{mvar|a}} transform ''covariantly'' (by the matrix {{mvar|A}} rather than its inverse). The covariance of the components of {{math|''a''['''f''']}} is notationally designated by placing the indices of {{math|''a''<sub>''i''</sub>['''f''']}} in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding {{math|''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 {{math|''Y''<sub>''p''</sub>}} defines a [[linear functional]] on the tangent space at {{mvar|p}}. This operation takes a vector {{math|''X''<sub>''p''</sub>}} at a point {{mvar|p}} and produces a covector {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, −)}}. In a basis of vector fields {{math|'''f'''}}, if a vector field {{mvar|X}} has components {{math|''v''['''f''']}}, then the components of the covector field {{math|''g''(''X'', −)}} in the dual basis are given by the entries of the row vector
:<math>a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].</math>

Under a change of basis {{math|'''f''' ↦ '''f'''''A''}}, the right-hand side of this equation transforms via
:<math>
  v[\mathbf{f}A]^\mathsf{T} G[\mathbf{f}A] =
    v[\mathbf{f}]^\mathsf{T} \left(A^{-1}\right)^\mathsf{T} A^\mathsf{T} G[\mathbf{f}]A =
    v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]A
</math>

so that {{math|''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}: {{mvar|a}} transforms covariantly. The operation of associating to the (contravariant) components of a vector field {{math|''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 {{math|''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 {{math|''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 {{math|''θ''['''f''']}}, then the column vector
{{NumBlk|:|<math>v[\mathbf{f}] = G^{-1}[\mathbf{f}]a[\mathbf{f}]^\mathsf{T}</math>|{{EquationRef|9}}}}
has components which transform contravariantly:
:<math>v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].</math>

Consequently, the quantity {{math|''X'' {{=}} '''f'''''v''['''f''']}} does not depend on the choice of basis {{math|'''f'''}} in an essential way, and thus defines a vector field on {{mvar|M}}. The operation ({{EquationNote|9}}) associating to the (covariant) components of a covector {{math|''a''['''f''']}} the (contravariant) components of a vector {{math|''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 {{mvar|U}} be an [[open set]] in {{math|'''ℝ'''<sup>''n''</sup>}}, and let {{mvar|φ}} be a [[continuously differentiable]] function from {{mvar|U}} into the [[Euclidean space]] {{math|'''ℝ'''<sup>''m''</sup>}}, where {{math|''m'' > ''n''}}. The mapping {{mvar|φ}} is called an [[immersion (mathematics)|immersion]] if its differential is [[injective]] at every point of {{mvar|U}}. The image of {{mvar|φ}} is called an [[immersed submanifold]]. More specifically, for {{math|1=''m'' = 3}}, which means that the ambient Euclidean space is {{math|'''ℝ'''<sup>''3''</sup>}}, the induced metric tensor is called the [[first fundamental form]].

Suppose that {{mvar|φ}} is an immersion onto the submanifold {{math|''M'' ⊂ '''R'''<sup>''m''</sup>}}. The usual Euclidean [[dot product]] in {{math|'''ℝ'''<sup>''m''</sup>}} is a metric which, when restricted to vectors tangent to {{mvar|M}}, gives a means for taking the dot product of these tangent vectors. This is called the '''induced metric'''.

Suppose that {{mvar|v}} is a tangent vector at a point of {{mvar|U}}, say
:<math>v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n</math>

where {{math|'''e'''<sub>''i''</sub>}} are the standard coordinate vectors in {{math|'''ℝ'''<sup>''n''</sup>}}. When {{mvar|φ}} is applied to {{mvar|U}}, the vector {{mvar|v}} goes over to the vector tangent to {{mvar|M}} given by
:<math>\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.</math>

(This is called the [[pushforward (differential)|pushforward]] of {{mvar|v}} along {{mvar|φ}}.) Given two such vectors, {{mvar|v}} and {{mvar|w}}, the induced metric is defined by
:<math>g(v,w) = \varphi_*(v)\cdot \varphi_*(w).</math>

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields {{math|'''e'''}} is given by
:<math>G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)</math>

where {{mvar|Dφ}} is the Jacobian matrix:
:<math>D\varphi = \begin{bmatrix}
  \frac{\partial\varphi^1}{\partial x^1} & \frac{\partial\varphi^1}{\partial x^2} &
    \dots  & \frac{\partial\varphi^1}{\partial x^n} \\[1ex]
  \frac{\partial\varphi^2}{\partial x^1} & \frac{\partial\varphi^2}{\partial x^2} &
    \dots  & \frac{\partial\varphi^2}{\partial x^n} \\
  \vdots                                 & \vdots                                 &
    \ddots & \vdots \\
  \frac{\partial\varphi^m}{\partial x^1} & \frac{\partial\varphi^m}{\partial x^2} &
    \dots  & \frac{\partial\varphi^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 : \mathrm{T}M\times_M \mathrm{T}M\to \mathbf{R}</math>|{{EquationRef|10}}}}

from the [[fiber product]] of the [[tangent bundle]] of {{mvar|M}} with itself to {{math|'''R'''}} such that the restriction of {{mvar|g}} to each fiber is a nondegenerate bilinear mapping

:<math>g_p : \mathrm{T}_pM\times \mathrm{T}_pM \to \mathbf{R}.</math>

The mapping ({{EquationNote|10}}) 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 {{mvar|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|10}}) gives rise [[natural transformation|naturally]] to a [[section (fiber bundle)|section]] {{math|''g''<sub>⊗</sub>}} of the [[dual space|dual]] of the [[tensor product bundle]] of {{math|T''M''}} with itself

:<math>g_\otimes \in \Gamma\left((\mathrm{T}M \otimes \mathrm{T}M)^*\right).</math>

The section {{math|''g''<sub>⊗</sub>}} is defined on simple elements of {{math|T''M'' ⊗ T''M''}} by

:<math>g_\otimes(v \otimes w) = g(v, w)</math>

and is defined on arbitrary elements of {{math|T''M'' ⊗ T''M''}} by extending linearly to linear combinations of simple elements. The original bilinear form {{mvar|g}} is symmetric if and only if
:<math>g_\otimes \circ \tau = g_\otimes</math>
where
:<math>\tau : \mathrm{T}M \otimes \mathrm{T}M \stackrel{\cong}{\to} TM \otimes TM</math>
is the [[tensor product#Tensor powers and braiding|braiding map]].

Since {{mvar|M}} is finite-dimensional, there is a [[natural isomorphism]]

:<math>(\mathrm{T}M \otimes \mathrm{T}M)^* \cong \mathrm{T}^*M \otimes \mathrm{T}^*M,</math>

so that {{math|''g''<sub>⊗</sub>}} is regarded also as a section of the bundle {{math|T*''M'' ⊗ T*''M''}} of the [[cotangent bundle]] {{math|T*''M''}} with itself. Since {{mvar|g}} is symmetric as a bilinear mapping, it follows that {{math|''g''<sub>⊗</sub>}} is a [[symmetric tensor]].

===Metric in a vector bundle===
{{see also|metric (vector bundle)}}
More generally, one may speak of a metric in a [[vector bundle]]. If {{mvar|E}} is a vector bundle over a manifold {{mvar|M}}, then a metric is a mapping

:<math>g : E\times_M E\to \mathbf{R}</math>

from the [[fiber product]] of {{mvar|E}} to {{math|'''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|''E''* ⊗ ''E''*}}.

===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 {{math|''X''<sub>''p''</sub> ∈ T<sub>''p''</sub>''M''}},

:<math>S_gX_p\, \stackrel\text{def}{=}\, g(X_p, -),</math>

the [[linear functional]] on {{math|T<sub>''p''</sub>''M''}} which sends a tangent vector {{math|''Y''<sub>''p''</sub>}} at {{mvar|p}} to {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>,''Y''<sub>''p''</sub>)}}. That is, in terms of the pairing {{math|[−, −]}} between {{math|T<sub>''p''</sub>''M''}} and its [[dual space]] {{math|T{{su|b=''p''|p=∗}}''M''}},

:<math>[S_gX_p, Y_p] = g_p(X_p, Y_p)</math>

for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}. The mapping {{math|''S''<sub>''g''</sub>}} is a [[linear transformation]] from {{math|T<sub>''p''</sub>''M''}} to {{math|T{{su|b=''p''|p=∗}}''M''}}. It follows from the definition of non-degeneracy that the [[kernel (set theory)|kernel]] of {{math|''S''<sub>''g''</sub>}} is reduced to zero, and so by the [[rank–nullity theorem]], {{math|''S''<sub>''g''</sub>}} is a [[linear isomorphism]]. Furthermore, {{math|''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 {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}.

Conversely, any linear isomorphism {{math|''S'' : T<sub>''p''</sub>''M'' → T{{su|b=''p''|p=∗}}''M''}} defines a non-degenerate bilinear form on {{math|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 {{mvar|S}} is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on {{math|T<sub>''p''</sub>''M''}} and symmetric linear isomorphisms of {{math|T<sub>''p''</sub>''M''}} to the dual {{math|T{{su|b=''p''|p=∗}}''M''}}.

As {{mvar|p}} varies over {{mvar|M}}, {{math|''S''<sub>''g''</sub>}} defines a section of the bundle {{math|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 {{mvar|g}}: it is continuous, differentiable, smooth, or real-analytic according as {{mvar|g}}. The mapping {{math|''S''<sub>''g''</sub>}}, which associates to every vector field on {{mvar|M}} a covector field on {{mvar|M}} gives an abstract formulation of "lowering the index" on a vector field. The inverse of {{math|''S''<sub>''g''</sub>}} is a mapping {{math|T*''M'' → T''M''}} which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverse {{math|''S''{{su|b=''g''|p=−1}}}} defines a linear mapping
:<math>S_g^{-1} : \mathrm{T}^*M \to \mathrm{T}M</math>

which is nonsingular and symmetric in the sense that
:<math>\left[S_g^{-1}\alpha, \beta\right] = \left[S_g^{-1}\beta, \alpha\right]</math>

for all covectors {{mvar|α}}, {{mvar|β}}. Such a nonsingular symmetric mapping gives rise (by the [[tensor-hom adjunction]]) to a map
:<math>\mathrm{T}^*M \otimes \mathrm{T}^*M \to \mathbf{R}</math>

or by the [[Double dual|double dual isomorphism]] to a section of the tensor product
:<math>\mathrm{T}M \otimes \mathrm{T}M.</math>

==Arclength and the line element==
Suppose that {{mvar|g}} is a Riemannian metric on {{mvar|M}}. In a local coordinate system {{math|''x''<sup>''i''</sup>}}, {{math|''i'' {{=}} 1, 2, …, ''n''}}, the metric tensor appears as a [[matrix (math)|matrix]], denoted here by {{math|'''G'''}}, whose entries are the components {{math|''g''<sub>''ij''</sub>}} of the metric tensor relative to the coordinate vector fields.

Let {{math|''γ''(''t'')}} be a piecewise-differentiable [[parametric curve]] in {{mvar|M}}, for {{math|''a'' ≤ ''t'' ≤ ''b''}}. The [[arclength]] of the curve is defined by

:<math>L = \int_a^b \sqrt{ \sum_{i,j=1}^n g_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right) \left(\frac{d}{dt} x^j \circ \gamma(t)\right)}\,dt \,.</math>

In connection with this geometrical application, the [[quadratic form|quadratic]] [[differential form]]

:<math>ds^2 = \sum_{i,j=1}^n g_{ij}(p) dx^i dx^j</math>

is called the [[first fundamental form]] associated to the metric, while {{mvar|ds}} is the [[line element]]. When {{math|''ds''<sup>2</sup>}} is [[pullback (differential geometry)|pulled back]] to the image of a curve in {{mvar|M}}, it represents the square of the differential with respect to arclength.

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

:<math>L = \int_a^b \sqrt{ \left|\sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\right|}\,dt \, .</math>

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.

===The energy, variational principles and geodesics===
Given a segment of a curve, another frequently defined quantity is the (kinetic) '''energy''' of the curve:

:<math>E = \frac{1}{2} \int_a^b \sum_{i,j=1}^ng_{ij}(\gamma(t)) \left(\frac{d}{dt}x^i \circ \gamma(t)\right)\left(\frac{d}{dt}x^j \circ \gamma(t)\right)\,dt \,. </math>

This usage comes from [[physics]], specifically, [[classical mechanics]], where the integral {{mvar|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.

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>

==Canonical measure and volume form==
In analogy with the case of surfaces, a metric tensor on an {{mvar|n}}-dimensional paracompact manifold {{mvar|M}} gives rise to a natural way to measure the {{mvar|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]].

A measure can be defined, by the [[Riesz representation theorem]], by giving a [[positive linear functional]] {{mvar|Λ}} on the space {{math|''C''<sub>0</sub>(''M'')}} of [[compact support|compactly supported]] [[continuous function]]s on {{mvar|M}}. More precisely, if {{mvar|M}} is a manifold with a (pseudo-)Riemannian metric tensor {{mvar|g}}, then there is a unique positive [[Borel measure]] {{math|''μ''<sub>''g''</sub>}} such that for any [[coordinate chart]] {{math|(''U'', ''φ'')}},
<math display="block">\Lambda f = \int_U f \, d\mu_g = \int_{\varphi(U)} f \circ \varphi^{-1}(x) \sqrt{\left|\det g\right|}\,dx</math>
for all {{mvar|f}} supported in {{mvar|U}}. Here {{math|det ''g''}} is the [[determinant]] of the matrix formed by the components of the metric tensor in the coordinate chart. That {{math|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 {{math|''C''<sub>0</sub>(''M'')}} by means of a [[partition of unity]].

If {{mvar|M}} is also [[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]] {{math|(''x''<sup>''1''</sup>, ..., ''x''<sup>''n''</sup>)}} the volume form is represented as
<math display="block">\omega = \sqrt{\left|\det g\right|} \, dx^1 \wedge \cdots \wedge dx^n</math>
where the {{math|''dx''<sup>''i''</sup>}} are the [[coordinate differential]]s and {{math|∧}} 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.

==Examples==

===Euclidean metric===
The most familiar example is that of elementary [[Euclidean geometry]]: the two-dimensional [[Euclidean distance|Euclidean]] metric tensor. In the usual [[Cartesian coordinate system|Cartesian]] {{math|(''x'', ''y'')}} coordinates, we can write

:<math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \,. </math>

The length of a curve reduces to the formula:

:<math>L = \int_a^b \sqrt{ (dx)^2 + (dy)^2} \,. </math>

The Euclidean metric in some other common coordinate systems can be written as follows.

[[Polar coordinates]] {{math|(''r'', ''θ'')}}:
:<math>\begin{align}
  x &= r \cos\theta \\
  y &= r \sin\theta \\
  J &= \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix} \,.
\end{align}</math>

So
:<math>g = J^\mathsf{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>
by [[trigonometric identity|trigonometric identities]].

In general, in a [[Cartesian coordinate system]] {{math|''x''<sup>''i''</sup>}} on a [[Euclidean space]], the partial derivatives {{math|∂ / ∂''x<sup>i</sup>''}} 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<sup>i</sup>''}} is given by
:<math>g_{ij} =
  \sum_{kl}\delta_{kl}\frac{\partial x^k}{\partial q^i} \frac{\partial x^l}{\partial q^j} =
  \sum_k\frac{\partial x^k}{\partial q^i}\frac{\partial x^k}{\partial q^j}.
</math>

====The round metric on a sphere====
The unit sphere in {{math|'''ℝ'''<sup>3</sup>}} comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the [[#Induced_metric|induced metric section]]. In standard spherical coordinates {{math|(''θ'', ''φ'')}}, with {{math|''θ''}} the [[colatitude]], the angle measured from the {{mvar|z}}-axis, and {{mvar|φ}} the angle from the {{mvar|x}}-axis in the {{mvar|xy}}-plane, the metric takes the form

:<math>g = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2 \theta\end{bmatrix} \,.</math>

This is usually written in the form

:<math>ds^2 = d\theta^2 + \sin^2\theta\,d\varphi^2\,.</math>

===Lorentzian metrics from relativity===
{{main|Metric tensor (general relativity)}}
In flat [[Minkowski space]] ([[special relativity]]), with coordinates
:<math>r^\mu \rightarrow \left(x^0, x^1, x^2, x^3\right) = (ct, x, y, z) \, ,</math>

the metric is, depending on choice of [[metric signature]],
:<math>g = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \quad \text{or} \quad g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \,. </math>

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.

In this case, the [[spacetime interval]] is written as
:<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>

The [[Schwarzschild metric]] describes the spacetime around a spherically symmetric body, such as a planet, or a [[black hole]]. With coordinates
:<math>\left(x^0, x^1, x^2, x^3\right) = (ct, r, \theta, \varphi) \,,</math>

we can write the metric as
:<math>g_{\mu\nu} =
  \begin{bmatrix}
    \left(1 - \frac{2GM}{rc^2}\right) & 0 & 0 & 0 \\
    0 & -\left(1 - \frac{2GM}{r c^2}\right)^{-1} & 0 & 0 \\
    0 & 0 & -r^2 & 0 \\
    0 & 0 & 0 & -r^2 \sin^2 \theta
  \end{bmatrix}\,,
</math>

where {{mvar|G}} (inside the matrix) is the [[gravitational constant]] and {{mvar|M}} represents the total [[mass–energy equivalence|mass–energy]] content of the central object.

==See also==
* [[Riemannian manifold]]
* [[Pseudo-Riemannian manifold]]
* [[Basic introduction to the mathematics of curved spacetime]]
* [[Clifford algebra]]
* [[Finsler manifold]]
* [[List of coordinate charts]]
* [[Ricci calculus]]
* [[Tissot's indicatrix]], a technique to visualize the metric tensor

==Notes==
{{Reflist}}

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{{Riemannian geometry}}
{{tensors}}
{{Manifolds}}

[[Category:Riemannian geometry]]
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[[Category:Concepts in physics]]
[[Category:Differential geometry]]
[[Category:Metric tensors|*1]]