Metric tensor

In the field of, a metric tensor is a type of function which takes as input a pair of s $v$ and $w$ at a point of a surface (or higher dimensional ) and produces a   $g(v, w)$ in a way that generalizes many of the familiar properties of the  of  in. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through, the metric tensor allows one to define and compute the length of curves on the manifold.

A metric tensor is called positive-definite if it assigns a positive value $g(v, v) > 0$ to every nonzero vector $v$. A manifold equipped with a positive-definite metric tensor is known as a. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a, 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, meaning that it has a $d(p, q)$ whose value at a pair of points $p$ and $q$ is the distance from $p$ to $q$. Conversely, the metric tensor itself is the of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.

While the notion of a metric tensor was known in some sense to mathematicians such as from the early 19th century, it was not until the early 20th century that its properties as a  were understood by, in particular,  and, who first codified the notion of a tensor. The metric tensor is an example of a.

The components of a metric tensor in a take on the form of a  whose entries transform  under changes to the coordinate system. Thus a metric tensor is a covariant. From the point of view, a metric tensor field is defined to be a   on each tangent space that varies  from point to point.

Introduction
in his 1827  (General investigations of curved surfaces) considered a surface, with the $x$, $y$, 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


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

depending on an of real variables $(u, v)$, and defined in an  $D$ in the $uv$-plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.

One natural such invariant quantity is the drawn along the surface. Another is the between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the 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.

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


 * $$ \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}$$

where $$ \left\| \cdot \right\| $$ represents the. Here the has been applied, and the subscripts denote s:


 * $$\vec{r}_u = \frac{\partial \vec{r}}{\partial u}\,, \quad \vec{r}_v = \frac{\partial \vec{r}}{\partial v}\,.$$

The integrand is the restriction to the curve of the square root of the

where

The quantity $$ in ($$) is called the, while $ds^{2}$ is called the of $ds$. Intuitively, it represents the of the square of the displacement undergone by $r(u, v)$ when $$ is increased by $M$ units, and $u$ is increased by $du$ units.

Using matrix notation, the first fundamental form becomes
 * $$ds^2 =

\begin{bmatrix} du & dv \end{bmatrix} \begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} du \\ dv \end{bmatrix} $$

Coordinate transformations
Suppose now that a different parameterization is selected, by allowing $v$ and $dv$ to depend on another pair of variables $u′$ and $v′$. Then the analog of ($u$) for the new variables is

The relates $E′$, $F′$, and $G′$ to $v$, $$, and $$ via the  equation

where the superscript T denotes the. The matrix with the coefficients $E$, $F$, and $G$ arranged in this way therefore transforms by the of the coordinate change



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}\,.$$

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


 * $$\begin{bmatrix} E & F \\ F & G \end{bmatrix}$$

with the transformation law ($$) is known as the metric tensor of the surface.

Invariance of arclength under coordinate transformations
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 ($$) 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,


 * $$\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} $$

so that


 * $$\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}$$

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 s to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface $$ can be written in the form


 * $$\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v$$

for suitable real numbers $p_{1}$ and $p_{2}$. If two tangent vectors are given:


 * $$\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}$$

then using the of the dot product,


 * $$\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 + b_1a_2 \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 + b_1a_2 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}$$

This is plainly a function of the four variables $a_{1}$, $b_{1}$, $a_{2}$, and $b_{2}$. It is more profitably viewed, however, as a function that takes a pair of arguments $a = [a_{1} a_{2}]$ and $b = [b_{1} b_{2}]$ which are vectors in the $E$-plane. That is, put


 * $$g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G \,.$$

This is a in $a$ and $b$, meaning that


 * $$g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.$$

It is also, meaning that it is in each variable $a$ and $b$ separately. That is,


 * $$\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}$$

for any vectors $a$, $a′$, $b$, and $b′$ in the $F$ plane, and any real numbers $G$ and $M$.

In particular, the length of a tangent vector $a$ is given by


 * $$ \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}$$

and the angle $uv$ between two vectors $a$ and $b$ is calculated by


 * $$\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.$$

Area
The is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface $uv$ is parameterized by the function $r(u, v)$ over the domain $μ$ in the $λ$-plane, then the surface area of $θ$ is given by the integral


 * $$\iint_D \left|\vec{r}_u \times \vec{r}_v\right|\,du\,dv$$

where $×$ denotes the, and the absolute value denotes the length of a vector in Euclidean space. By for the cross product, the integral can be written


 * $$\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}$$

where $det$ is the.

Definition
Let $M$ be a of dimension $D$; for instance a  (in the case $n = 2$) or  in the  $ℝ^{n + 1}$. At each point $p ∈ M$ there is a $T_{p}M$, called the, consisting of all tangent vectors to the manifold at the point $uv$. A metric tensor at $M$ is a function $g_{p}(X_{p}, Y_{p})$ which takes as inputs a pair of tangent vectors $X_{p}$ and $Y_{p}$ at $M$, and produces as an output a, so that the following conditions are satisfied: g_p\left(aU_p + bV_p, Y_p\right) &= ag_p\left(U_p, Y_p\right) + bg_p\left(V_p, Y_p\right) \,, \quad \text{and} \\ g_p\left(Y_p, aU_p + bV_p\right) &= ag_p\left(Y_p, U_p\right) + bg_p\left(Y_p, V_p\right) \,. \end{align}$$
 * $g_{p}$ is . A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if $U_{p}$, $V_{p}$, $Y_{p}$ are three tangent vectors at $n$ and $p$ and $p$ are real numbers, then
 * $$\begin{align}
 * $g_{p}$ is . A function of two vector arguments is symmetric provided that for all vectors $X_{p}$ and $Y_{p}$,
 * $$g_p\left(X_p, Y_p\right) = g_p\left(Y_p, X_p\right)\,.$$
 * $g_{p}$ is . A bilinear function is nondegenerate provided that, for every tangent vector $X_{p} ≠ 0$, the function
 * $$Y_p\mapsto g_p\left(X_p,Y_p\right)$$
 * obtained by holding $X_{p}$ constant and allowing $Y_{p}$ to vary is not . That is, for every $X_{p} ≠ 0$ there exists a $Y_{p}$ such that $g_{p}(X_{p}, Y_{p}) ≠ 0$.

A metric tensor field $p$ on $p$ assigns to each point $a$ of $b$ a metric tensor $g_{p}$ in the tangent space at $g$ in a way that varies with $M$. More precisely, given any $p$ of manifold $M$ and any (smooth) s $p$ and $p$ on $U$, the real function


 * $$g(X, Y)(p) = g_p\left(X_p, Y_p\right)$$

is a smooth function of $M$.

Components of the metric
The components of the metric in any of s, or, $f = (X_{1}, ..., X_{n})$ are given by

The $n^{2}$ functions $g_{ij}[f]$ form the entries of an $n × n$, $G[f]$. If
 * $$v = \sum_{i=1}^n v^iX_i \,, \quad w = \sum_{i=1}^n w^iX_i$$

are two vectors at $p ∈ U$, then the value of the metric applied to $X$ and $Y$ is determined by the coefficients ($U$) by bilinearity:


 * $$g(v, w) = \sum_{i,j=1}^n v^iw^jg\left(X_i,X_j\right) = \sum_{i,j=1}^n v^iw^jg_{ij}[\mathbf{f}]$$

Denoting the $(g_{ij}[f])$ by $G[f]$ and arranging the components of the vectors $p$ and $$ into s $v[f]$ and $w[f]$,


 * $$g(v,w) = \mathbf{v}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]$$

where $v[f]$T and $w[f]$T denote the of the vectors $v[f]$ and $w[f]$, respectively. Under a of the form


 * $$\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A$$

for some $n × n$ matrix $A = (a_{ij})$, the matrix of components of the metric changes by $v$ as well. That is,


 * $$G[\mathbf{f}A] = A^\mathsf{T} G[\mathbf{f}]A$$

or, in terms of the entries of this matrix,


 * $$g_{ij}[\mathbf{f}A] = \sum_{k,l=1}^n a_{ki}g_{kl}[\mathbf{f}]a_{lj} \, .$$

For this reason, the system of quantities $g_{ij}[f]$ is said to transform covariantly with respect to changes in the frame $f$.

Metric in coordinates
A system of $w$ real-valued functions $(x^{1}, ..., x^{n})$, giving a on an  $$ in $v$, determines a basis of vector fields on $w$
 * $$\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.$$

The metric $A$ has components relative to this frame given by
 * $$g_{ij}\left[\mathbf{f}\right] = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \,.$$

Relative to a new system of local coordinates, say
 * $$y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n$$

the metric tensor will determine a different matrix of coefficients,
 * $$g_{ij}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).$$

This new system of functions is related to the original $g_{ij}(f)$ by means of the
 * $$\frac{\partial}{\partial y^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}$$

so that
 * $$g_{ij}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.$$

Or, in terms of the matrices $G[f] = (g_{ij}[f])$ and $G[f′] = (g_{ij}[f′])$,
 * $$G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}$$

where $n$ denotes the of the coordinate change.

Signature of a metric
Associated to any metric tensor is the defined in each tangent space by


 * $$q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM.$$

If $q_{m}$ is positive for all non-zero $X_{m}$, then the metric is at $U$. If the metric is positive-definite at every $m ∈ M$, then $M$ is called a. More generally, if the quadratic forms $q_{m}$ have constant independent of $U$, then the signature of $g$ is this signature, and $Dy$ is called a. If $m$ is, then the signature of $g$ does not depend on $m$.

By, a basis of tangent vectors $X_{i}$ can be chosen locally so that the quadratic form diagonalizes in the following manner


 * $$q_m\left(\sum_i\xi^iX_i\right) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^{p+1}\right)^2-\cdots-\left(\xi^n\right)^2$$

for some $g$ between 1 and $g$. Any two such expressions of $M$ (at the same point $q_{m}$ of $m$) will have the same number $p$ of positive signs. The signature of $n$ is the pair of integers $(p, n − p)$, signifying that there are $q$ positive signs and $n − p$ negative signs in any such expression. Equivalently, the metric has signature $(p, n − p)$ if the matrix $g_{ij}$ of the metric has $m$ positive and $n − p$ negative s.

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

Inverse metric
Let $f = (X_{1}, ..., X_{n})$ be a basis of vector fields, and as above let $G[f]$ be the matrix of coefficients
 * $$g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.$$

One can consider the $G[f]^{−1}$, 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 $M$ via

The inverse metric transforms , or with respect to the inverse of the change of basis matrix $M$. 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) fields; that is, fields of s.

To see this, suppose that $n$ is a covector field. To wit, for each point $M$, $g$ determines a function $α_{p}$ defined on tangent vectors at $A$ so that the following condition holds for all tangent vectors $X_{p}$ and $Y_{p}$, and all real numbers $$ and $A$:


 * $$\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.$$

As $α$ varies, $p$ is assumed to be a in the sense that


 * $$p \mapsto \alpha_p \left(X_p\right)$$

is a smooth function of $α$ for any smooth vector field $p$.

Any covector field $a$ has components in the basis of vector fields $f$. These are determined by


 * $$\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.$$

Denote the of these components by


 * $$\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.$$

Under a change of $f$ by a matrix $b$, $α[f]$ changes by the rule


 * $$\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.$$

That is, the row vector of components $α[f]$ transforms as a covariant vector.

For a pair $p$ and $α$ of covector fields, define the inverse metric applied to these two covectors by

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 $fA$ gives


 * $$\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} $$

So that the right-hand side of equation ($p$) is unaffected by changing the basis $f$ to any other basis $fA$ 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 $g^{ij}$, where the indices $X$ and $α$ have been raised to indicate the transformation law ($A$).

Raising and lowering indices
In a basis of vector fields $f = (X_{1}, ..., X_{n})$, any smooth tangent vector field $α$ can be written in the form

for some uniquely determined smooth functions $v^{1}, ..., v^{n}$. Upon changing the basis $f$ by a nonsingular matrix $β$, the coefficients $v^{i}$ change in such a way that equation ($$) remains true. That is,


 * $$X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}]\,.$$

Consequently, $v[fA] = A^{−1}v[f]$. 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 $$. The contravariance of the components of $v[f]$ is notationally designated by placing the indices of $v^{i}[f]$ in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields $f = (X_{1}, ..., X_{n})$ define the to be the s $(θ^{1}[f], ..., θ^{n}[f])$ such that


 * $$\theta^i[\mathbf{f}](X_j) = \begin{cases} 1 & \mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}$$

That is, $θ^{i}[f](X_{j}) = δ_{j}^{i}$, the. Let


 * $$\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}] \\ \theta^2[\mathbf{f}] \\ \vdots \\ \theta^n[\mathbf{f}]\end{bmatrix}.$$

Under a change of basis $f ↦ fA$ for a nonsingular matrix $A$, $θ[f]$ transforms via


 * $$\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].$$

Any linear functional $i$ on tangent vectors can be expanded in terms of the dual basis $j$

where $a[f]$ denotes the $[ a_{1}[f] ... a_{n}[f] ]$. The components $a_{i}$ transform when the basis $f$ is replaced by $fA$ in such a way that equation ($$) continues to hold. That is,


 * $$\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]$$

whence, because $θ[fA] = A^{−1}θ[f]$, it follows that $1=a[fA] = a[f]A$. That is, the components $X$ transform covariantly (by the matrix $$ rather than its inverse). The covariance of the components of $a[f]$ is notationally designated by placing the indices of $a_{i}[f]$ in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding $X_{p}$ fixed, the function


 * $$g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p)$$

of tangent vector $Y_{p}$ defines a on the tangent space at $A$. This operation takes a vector $X_{p}$ at a point $$ and produces a covector $g_{p}(X_{p}, −)$. In a basis of vector fields $f$, if a vector field $A$ has components $v[f]$, then the components of the covector field $g(X, −)$ in the dual basis are given by the entries of the row vector
 * $$a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].$$

Under a change of basis $f ↦ fA$, the right-hand side of this equation transforms via

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 $$

so that $a[fA] = a[f]A$: $α$ transforms covariantly. The operation of associating to the (contravariant) components of a vector field $v[f] = [ v^{1}[f] v^{2}[f] ... v^{n}[f] ]$T the (covariant) components of the covector field $a[f] = [ a_{1}[f] a_{2}[f] … a_{n}[f] ]$, where
 * $$a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]$$

is called lowering the index.

To raise the index, one applies the same construction but with the inverse metric instead of the metric. If $a[f] = [ a_{1}[f] a_{2}[f] ... a_{n}[f] ]$ are the components of a covector in the dual basis $θ[f]$, then the column vector

has components which transform contravariantly:
 * $$v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].$$

Consequently, the quantity $X = fv[f]$ does not depend on the choice of basis $f$ in an essential way, and thus defines a vector field on $θ$. The operation ($$) associating to the (covariant) components of a covector $a[f]$ the (contravariant) components of a vector $v[f]$ given is called raising the index. In components, ($$) is
 * $$v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}] a_k[\mathbf{f}].$$

Induced metric
Let $a$ be an in $ℝ^{n}$, and let $A$ be a  function from $p$ into the  $ℝ^{m}$, where $m > n$. The mapping $p$ is called an if its differential is  at every point of $X$. The image of $a$ is called an. More specifically, for $m$=3, which means that the ambient is $ℝ^{3}$, the induced metric tensor is called the.

Suppose that $$ is an immersion onto the submanifold $M ⊂ R^{m}$. The usual Euclidean in $ℝ^{m}$ 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 $$ is a tangent vector at a point of $$, say
 * $$v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n$$

where $e_{i}$ are the standard coordinate vectors in $ℝ^{n}$. When $U$ is applied to $φ$, the vector $U$ goes over to the vector tangent to $φ$ given by
 * $$\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.$$

(This is called the of $U$ along $φ$.) Given two such vectors, $φ$ and $M$, the induced metric is defined by
 * $$g(v,w) = \varphi_*(v)\cdot \varphi_*(w).$$

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields $e$ is given by
 * $$G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)$$

where $v$ is the Jacobian matrix:
 * $$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}.$$

Intrinsic definitions of a metric
The notion of a metric can be defined intrinsically using the language of s and s. In these terms, a metric tensor is a function

from the of the  of $U$ with itself to $R$ such that the restriction of $φ$ to each fiber is a nondegenerate bilinear mapping


 * $$g_p : \mathrm{T}_pM\times \mathrm{T}_pM \to \mathbf{R}.$$

The mapping ($U$) is required to be, and often , , or , depending on the case of interest, and whether $v$ can support such a structure.

Metric as a section of a bundle
By the, any bilinear mapping ($M$) gives rise to a  $g_{⊗}$ of the  of the  of $TM$ with itself


 * $$g_\otimes \in \Gamma\left((\mathrm{T}M \otimes \mathrm{T}M)^*\right).$$

The section $g_{⊗}$ is defined on simple elements of $TM ⊗ TM$ by


 * $$g_\otimes(v \otimes w) = g(v, w)$$

and is defined on arbitrary elements of $TM ⊗ TM$ by extending linearly to linear combinations of simple elements. The original bilinear form $v$ is symmetric if and only if
 * $$g_\otimes \circ \tau = g_\otimes$$

where
 * $$\tau : \mathrm{T}M \otimes \mathrm{T}M \stackrel{\cong}{\to} TM \otimes TM$$

is the.

Since $φ$ is finite-dimensional, there is a


 * $$(\mathrm{T}M \otimes \mathrm{T}M)^* \cong \mathrm{T}^*M \otimes \mathrm{T}^*M,$$

so that $g_{⊗}$ is regarded also as a section of the bundle $T*M ⊗ T*M$ of the $T*M$ with itself. Since $v$ is symmetric as a bilinear mapping, it follows that $g_{⊗}$ is a.

Metric in a vector bundle
More generally, one may speak of a metric in a. If $w$ is a vector bundle over a manifold $Dφ$, then a metric is a mapping


 * $$g : E\times_M E\to \mathbf{R}$$

from the of $$ to $R$ which is bilinear in each fiber:


 * $$g_p : E_p \times E_p\to \mathbf{R}.$$

Using duality as above, a metric is often identified with a of the  bundle $E* ⊗ E*$. (See .)

Tangent–cotangent isomorphism
The metric tensor gives a from the  to the, sometimes called the. This isomorphism is obtained by setting, for each tangent vector $X_{p} ∈ T_{p}M$,


 * $$S_gX_p\, \stackrel\text{def}{=}\, g(X_p, -),$$

the on $T_{p}M$ which sends a tangent vector $Y_{p}$ at $M$ to $g_{p}(X_{p},Y_{p})$. That is, in terms of the pairing $[−, −]$ between $T_{p}M$ and its $T∗ pM$,


 * $$[S_gX_p, Y_p] = g_p(X_p, Y_p)$$

for all tangent vectors $X_{p}$ and $Y_{p}$. The mapping $S_{g}$ is a from $T_{p}M$ to $T∗ pM$. It follows from the definition of non-degeneracy that the of $S_{g}$ is reduced to zero, and so by the, $S_{g}$ is a. Furthermore, $S_{g}$ is a in the sense that


 * $$[S_gX_p, Y_p] = [S_gY_p, X_p] $$

for all tangent vectors $X_{p}$ and $Y_{p}$.

Conversely, any linear isomorphism $S : T_{p}M → T∗ pM$ defines a non-degenerate bilinear form on $T_{p}M$ by means of


 * $$g_S(X_p, Y_p) = [SX_p, Y_p]\,.$$

This bilinear form is symmetric if and only if $g$ is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on $T_{p}M$ and symmetric linear isomorphisms of $T_{p}M$ to the dual $T∗ pM$.

As $$ varies over $M$, $S_{g}$ defines a section of the bundle $Hom(TM, T*M)$ of of the tangent bundle to the cotangent bundle. This section has the same smoothness as $$: it is continuous, differentiable, smooth, or real-analytic according as $g$. The mapping $S_{g}$, which associates to every vector field on $M$ a covector field on $g$ gives an abstract formulation of "lowering the index" on a vector field. The inverse of $S_{g}$ is a mapping $T*M → TM$ which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverse $S−1 g$ defines a linear mapping
 * $$S_g^{-1} : \mathrm{T}^*M \to \mathrm{T}M$$

which is nonsingular and symmetric in the sense that
 * $$\left[S_g^{-1}\alpha, \beta\right] = \left[S_g^{-1}\beta, \alpha\right]$$

for all covectors $E$, $M$. Such a nonsingular symmetric mapping gives rise (by the ) to a map
 * $$\mathrm{T}^*M \otimes \mathrm{T}^*M \to \mathbf{R}$$

or by the to a section of the tensor product
 * $$\mathrm{T}M \otimes \mathrm{T}M.$$

Arclength and the line element
Suppose that $E$ is a Riemannian metric on $p$. In a local coordinate system $x^{i}$, $i = 1, 2, …, n$, the metric tensor appears as a, denoted here by $G$, whose entries are the components $g_{ij}$ of the metric tensor relative to the coordinate vector fields.

Let $γ(t)$ be a piecewise-differentiable in $S$, for $a ≤ t ≤ b$. The of the curve is defined by


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

In connection with this geometrical application, the


 * $$ds^2 = \sum_{i,j=1}^n g_{ij}(p) dx^i dx^j$$

is called the associated to the metric, while $p$ is the. When $ds^{2}$ is to the image of a curve in $M$, 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


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

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.

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


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

This usage comes from, specifically, , where the integral $g$ can be seen to directly correspond to the of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of, 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 s may be obtained by applying s to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the : 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.

Canonical measure and volume form
In analogy with the case of surfaces, a metric tensor on an $g$-dimensional paracompact manifold $M$ gives rise to a natural way to measure the $M$-dimensional of subsets of the manifold. The resulting natural positive allows one to develop a theory of integrating functions on the manifold by means of the associated.

A measure can be defined, by the, by giving a positive $α$ on the space $C_{0}(M)$ of  s on $β$. More precisely, if $g$ is a manifold with a (pseudo-)Riemannian metric tensor $M$, then there is a unique positive $μ_{g}$ such that for any  $(U, φ)$,
 * $$\Lambda f = \int_U f\,d\mu_g = \int_{\varphi(U)} f\circ\varphi^{-1}(x) \sqrt{|\det g|}\,dx$$

for all $M$ supported in $ds$. Here $det g$ is the of the matrix formed by the components of the metric tensor in the coordinate chart. That $M$ is well-defined on functions supported in coordinate neighborhoods is justified by. It extends to a unique positive linear functional on $C_{0}(M)$ by means of a.

If $E$ is in addition, then it is possible to define a natural from the metric tensor. In a $(x^{1}, ..., x^{n})$ the volume form is represented as
 * $$\omega = \sqrt{|\det g|}\, dx^1\wedge\cdots\wedge dx^n$$

where the $dx^{i}$ are the s and $∧$ denotes the in the algebra of 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.

Euclidean metric
The most familiar example is that of elementary : the two-dimensional metric tensor. In the usual $(x, y)$ coordinates, we can write


 * $$g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \,. $$

The length of a curve reduces to the formula:


 * $$L = \int_a^b \sqrt{ (dx)^2 + (dy)^2} \,. $$

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

$(r, θ)$:
 * $$\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}$$

So
 * $$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} $$ by.

In general, in a $x^{i}$ on a, the partial derivatives $∂ / ∂x^{i}$ are  with respect to the Euclidean metric. Thus the metric tensor is the δij in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates $q^{i}$ is given by
 * $$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}. $$

The round metric on a sphere
The unit sphere in $ℝ^{3}$ comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the. In standard spherical coordinates $(θ, φ)$, with $θ$ the, the angle measured from the $n$-axis, and $M$ the angle from the $n$-axis in the $Λ$-plane, the metric takes the form


 * $$g = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2 \theta\end{bmatrix} \,.$$

This is usually written in the form


 * $$ds^2 = d\theta^2 + \sin^2\theta\,d\varphi^2\,.$$

Lorentzian metrics from relativity
In flat, with coordinates
 * $$r^\mu \rightarrow \left(x^0, x^1, x^2, x^3\right) = (ct, x, y, z) \, ,$$

the metric is, depending on choice of ,
 * $$g = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \quad \text{or} \quad g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \,. $$

For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a curve, the length formula gives the  along the curve.

In this case, the is written as
 * $$ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = dr^\mu dr_\mu = g_{\mu \nu} dr^\mu dr^\nu\,. $$

The describes the spacetime around a spherically symmetric body, such as a planet, or a. With coordinates
 * $$\left(x^0, x^1, x^2, x^3\right) = (ct, r, \theta, \varphi) \,,$$

we can write the metric as
 * $$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}\,, $$

where $M$ (inside the matrix) is the and $M$ represents the total  content of the central object.