Metric tensor

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

A metric tensor Template:Mvar is positive-definite if g(v, v) > 0 for every nonzero vector Template:Mvar. 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 Template:Mvar, the length of a smooth curve between two points Template:Mvar and Template:Mvar can be defined by integration, and the distance between Template:Mvar and Template:Mvar can be defined as the infimum of the lengths of all such curves; this makes Template:Mvar a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).

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

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

Introduction

Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates Template:Mvar, Template:Mvar, and Template:Mvar of points on the surface depending on two auxiliary variables Template:Mvar and Template:Mvar. Thus a parametric surface is (in today's terms) a vector-valued function

$\overrightarrow{r}(u,v)=(x(u,v),y(u,v),z(u,v))$

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

One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

The metric tensor is $\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]$ in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.

Arc length

If the variables Template:Mvar and Template:Mvar are taken to depend on a third variable, Template:Mvar, taking values in an interval [a, b], then Template:Vec(u(t), v(t)) will trace out a parametric curve in parametric surface Template:Mvar. The arc length of that curve is given by the integral

$\begin{array}{rl}s& =\underset{a}{\overset{b}{\int }}\Vert \frac{d}{dt}\overrightarrow{r}(u(t),v(t))\Vert dt\\ & =\underset{a}{\overset{b}{\int }}\sqrt{u^{\prime }(t{)}^2{\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_u+2u^{\prime }(t)v^{\prime }(t){\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_v+v^{\prime }(t{)}^2{\overrightarrow{r}}_v\cdot {\overrightarrow{r}}_v}dt,\end{array}$

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

${\overrightarrow{r}}_u=\frac{\partial \overrightarrow{r}}{\partial u},{\overrightarrow{r}}_v=\frac{\partial \overrightarrow{r}}{\partial v}.$

The integrand is the restriction to the curve of the square root of the (quadratic) differential

where

The quantity Template:Mvar in (1) is called the line element, while ds² is called the first fundamental form of Template:Mvar. Intuitively, it represents the principal part of the square of the displacement undergone by Template:Vec(u, v) when Template:Mvar is increased by Template:Mvar units, and Template:Mvar is increased by Template:Mvar units.

Using matrix notation, the first fundamental form becomes

$d{s}^2=\left[\begin{array}{cc}du& dv\end{array}\right]\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{c}du\\ dv\end{array}\right]$

Coordinate transformations

Suppose now that a different parameterization is selected, by allowing Template:Mvar and Template:Mvar to depend on another pair of variables u′ and v′. Then the analog of (2) for the new variables is Template:NumBlk

The chain rule relates E′, F′, and G′ to Template:Mvar, Template:Mvar, and Template:Mvar via the matrix equation

where the superscript T denotes the matrix transpose. The matrix with the coefficients Template:Mvar, Template:Mvar, and Template:Mvar arranged in this way therefore transforms by the Jacobian matrix of the coordinate change

$J=\left[\begin{array}{cc}\frac{\partial u}{\partial u^{\prime }}& \frac{\partial u}{\partial v^{\prime }}\\ \frac{\partial v}{\partial u^{\prime }}& \frac{\partial v}{\partial v^{\prime }}\end{array}\right].$

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

$\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]$

with the transformation law (3) is known as the metric tensor of the surface.

Invariance of arclength under coordinate transformations

first observed the significance of a system of coefficients Template:Mvar, Template:Mvar, and Template:Mvar, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of Template:Mvar, Template:Mvar, and Template:Mvar. Indeed, by the chain rule,

$\left[\begin{array}{c}du\\ dv\end{array}\right]=\left[\begin{array}{cc}\frac{\partial u}{\partial u^{\prime }}& \frac{\partial u}{\partial v^{\prime }}\\ \frac{\partial v}{\partial u^{\prime }}& \frac{\partial v}{\partial v^{\prime }}\end{array}\right]\left[\begin{array}{c}d{u}^{\prime }\\ d{v}^{\prime }\end{array}\right]$

so that

$\begin{array}{rl}d{s}^2& =\left[\begin{array}{cc}du& dv\end{array}\right]\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{c}du\\ dv\end{array}\right]\\ & =\left[\begin{array}{cc}d{u}^{\prime }& d{v}^{\prime }\end{array}\right]{\left[\begin{array}{cc}\frac{\partial u}{\partial u^{\prime }}& \frac{\partial u}{\partial v^{\prime }}\\ \frac{\partial v}{\partial u^{\prime }}& \frac{\partial v}{\partial v^{\prime }}\end{array}\right]}^{\mathsf{T}}\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{cc}\frac{\partial u}{\partial u^{\prime }}& \frac{\partial u}{\partial v^{\prime }}\\ \frac{\partial v}{\partial u^{\prime }}& \frac{\partial v}{\partial v^{\prime }}\end{array}\right]\left[\begin{array}{c}d{u}^{\prime }\\ d{v}^{\prime }\end{array}\right]\\ & =\left[\begin{array}{cc}d{u}^{\prime }& d{v}^{\prime }\end{array}\right]\left[\begin{array}{cc}{E}^{\prime }& F^{\prime }\\ F^{\prime }& G^{\prime }\end{array}\right]\left[\begin{array}{c}d{u}^{\prime }\\ d{v}^{\prime }\end{array}\right]\\ & =(d{s}^{\prime }{)}^2.\end{array}$

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 vectors 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 Template:Mvar can be written in the form

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

for suitable real numbers p₁ and p₂. If two tangent vectors are given:

$\begin{array}{rl}\mathbf{𝐚}& =a_1{\overrightarrow{r}}_u+a_2{\overrightarrow{r}}_v\\ \mathbf{𝐛}& =b_1{\overrightarrow{r}}_u+b_2{\overrightarrow{r}}_v\end{array}$

then using the bilinearity of the dot product,

$\begin{array}{rl}\mathbf{𝐚}\cdot \mathbf{𝐛}& =a_1{b}_1{\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_u+a_1{b}_2{\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_v+a_2{b}_1{\overrightarrow{r}}_v\cdot {\overrightarrow{r}}_u+a_2{b}_2{\overrightarrow{r}}_v\cdot {\overrightarrow{r}}_v\\ & =a_1{b}_{1}E+a_1{b}_{2}F+a_2{b}_{1}F+a_2{b}_{2}G.\\ & =\left[\begin{array}{cc}{a}_1& a_2\end{array}\right]\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right].\end{array}$

This is plainly a function of the four variables a₁, b₁, a₂, and b₂. It is more profitably viewed, however, as a function that takes a pair of arguments a = [a₁ a₂] and b = [b₁ b₂] which are vectors in the Template:Mvar-plane. That is, put

$g(\mathbf{𝐚},\mathbf{𝐛})=a_1{b}_{1}E+a_1{b}_{2}F+a_2{b}_{1}F+a_2{b}_{2}G.$

This is a symmetric function in a and b, meaning that

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

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

$\begin{array}{rl}g(\lambda \mathbf{𝐚}+\mu {\mathbf{𝐚}}^{\prime },\mathbf{𝐛})& =\lambda g(\mathbf{𝐚},\mathbf{𝐛})+\mu g({\mathbf{𝐚}}^{\prime },\mathbf{𝐛}),\text{and}\\ g(\mathbf{𝐚},\lambda \mathbf{𝐛}+\mu {\mathbf{𝐛}}^{\prime })& =\lambda g(\mathbf{𝐚},\mathbf{𝐛})+\mu g(\mathbf{𝐚},{\mathbf{𝐛}}^{\prime })\end{array}$

for any vectors a, a′, b, and b′ in the Template:Mvar plane, and any real numbers Template:Mvar and Template:Mvar.

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

$\Vert \mathbf{𝐚}\Vert =\sqrt{g(\mathbf{𝐚},\mathbf{𝐚})}$

and the angle Template:Mvar between two vectors a and b is calculated by

$\cos (\theta )=\frac{g(\mathbf{𝐚},\mathbf{𝐛})}{\Vert \mathbf{𝐚}\Vert \Vert \mathbf{𝐛}\Vert }.$

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 Template:Mvar is parameterized by the function Template:Vec(u, v) over the domain Template:Mvar in the Template:Mvar-plane, then the surface area of Template:Mvar is given by the integral

$\underset{D}{\iint }|{\overrightarrow{r}}_u\times {\overrightarrow{r}}_v|dudv$

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

$\begin{array}{rl}& \underset{D}{\iint }\sqrt{({\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_u)({\overrightarrow{r}}_v\cdot {\overrightarrow{r}}_v)-{({\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_v)}^2}dudv\\ =& \underset{D}{\iint }\sqrt{EG-F^2}dudv\\ =& \underset{D}{\iint }\sqrt{\det \left[\begin{array}{cc}E& F\\ F& G\end{array}\right]}dudv\end{array}$

where det is the determinant.

Definition

Let Template:Mvar be a smooth manifold of dimension Template:Mvar; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space ${\mathbb{ℝ}}^{n+1}$. At each point p ∈ M there is a vector space T_pM, called the tangent space, consisting of all tangent vectors to the manifold at the point Template:Mvar. A metric tensor at Template:Mvar is a function g_p(X_p, Y_p) which takes as inputs a pair of tangent vectors X_p and Y_p at Template:Mvar, and produces as an output a real number (scalar), so that the following conditions are satisfied:

• g_p is bilinear. 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 Template:Mvar and Template:Mvar and Template:Mvar are real numbers, then $$\begin{array}{rl}{g}_p(a{U}_p+b{V}_p,Y_p)& =a{g}_p(U_p,Y_p)+b{g}_p(V_p,Y_p),\text{and}\\ g_p(Y_p,a{U}_p+b{V}_p)& =a{g}_p(Y_p,U_p)+b{g}_p(Y_p,V_p).\end{array}$$
• g_p is symmetric. A function of two vector arguments is symmetric provided that for all vectors X_p and Y_p, $$g_p(X_p,Y_p)=g_p(Y_p,X_p).$$
• g_p is nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector X_p ≠ 0, the function $$Y_p\mapsto g_p(X_p,Y_p)$$ obtained by holding X_p constant and allowing Y_p to vary is not identically zero. 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 Template:Mvar on Template:Mvar assigns to each point Template:Mvar of Template:Mvar a metric tensor g_p in the tangent space at Template:Mvar in a way that varies smoothly with Template:Mvar. More precisely, given any open subset Template:Mvar of manifold Template:Mvar and any (smooth) vector fields Template:Mvar and Template:Mvar on Template:Mvar, the real function $$g(X,Y)(p)=g_p(X_p,Y_p)$$ is a smooth function of Template:Mvar.

Components of the metric

The components of the metric in any basis of vector fields, or frame, f = (X₁, ..., X_n) are given by Template:NumBlk The n² functions g_ij[f] form the entries of an n × n symmetric matrix, G[f]. If

$v=\sum _{i=1}^n{v}^i{X}_i,w=\sum _{i=1}^n{w}^i{X}_i$

are two vectors at p ∈ U, then the value of the metric applied to Template:Mvar and Template:Mvar is determined by the coefficients (4) by bilinearity:

$g(v,w)=\sum _{i,j=1}^n{v}^i{w}^{j}g(X_i,X_j)=\sum _{i,j=1}^n{v}^i{w}^j{g}_{ij}[\mathbf{𝐟}]$

Denoting the matrix (g_ij[f]) by G[f] and arranging the components of the vectors Template:Mvar and Template:Mvar into column vectors 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 transpose of the vectors v[f] and w[f], respectively. Under a change of basis of the form

$\mathbf{𝐟}\mapsto {\mathbf{𝐟}}^{\prime }=(\sum _k{X}_k{a}_{k1},\dots ,\sum _k{X}_k{a}_{kn})=\mathbf{𝐟}A$

for some invertible n × n matrix A = (a_ij), the matrix of components of the metric changes by Template:Mvar 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 Template:Mvar real-valued functions (x¹, ..., xⁿ), giving a local coordinate system on an open set Template:Mvar in Template:Mvar, determines a basis of vector fields on Template:Mvar

$\mathbf{𝐟}=(X_1=\frac{\partial }{\partial x^1},\dots ,X_n=\frac{\partial }{\partial x^n}).$

The metric Template:Mvar has components relative to this frame given by

$g_{ij}\left[\mathbf{𝐟}\right]=g(\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j}).$

Relative to a new system of local coordinates, say

$y^i=y^i(x^1,x^2,\dots ,x^n),i=1,2,\dots ,n$

the metric tensor will determine a different matrix of coefficients,

$g_{ij}\left[{\mathbf{𝐟}}^{\prime }\right]=g(\frac{\partial }{\partial y^i},\frac{\partial }{\partial y^j}).$

This new system of functions is related to the original g_ij(f) by means of the chain rule

$\frac{\partial }{\partial y^i}=\sum _{k=1}^n\frac{\partial x^k}{\partial y^i}\frac{\partial }{\partial x^k}$

so that

$g_{ij}\left[{\mathbf{𝐟}}^{\prime }\right]=\sum _{k,l=1}^n\frac{\partial x^k}{\partial y^i}{g}_{kl}\left[\mathbf{𝐟}\right]\frac{\partial x^l}{\partial y^j}.$

Or, in terms of the matrices G[f] = (g_ij[f]) and G[f′] = (g_ij[f′]),

$G\left[{\mathbf{𝐟}}^{\prime }\right]={((Dy{)}^{-1})}^{\mathsf{T}}G\left[\mathbf{𝐟}\right](Dy{)}^{-1}$

where Template:Mvar denotes the Jacobian matrix of the coordinate change.

Signature of a metric

Associated to any metric tensor is the quadratic form defined in each tangent space by

$q_m(X_m)=g_m(X_m,X_m),X_m\in T_{m}M.$

If q_m is positive for all non-zero X_m, then the metric is positive-definite at Template:Mvar. If the metric is positive-definite at every m ∈ M, then Template:Mvar is called a Riemannian metric. More generally, if the quadratic forms q_m have constant signature independent of Template:Mvar, then the signature of Template:Mvar is this signature, and Template:Mvar is called a pseudo-Riemannian metric. If Template:Mvar is connected, then the signature of Template:Mvar does not depend on Template:Mvar.

By Sylvester's law of inertia, a basis of tangent vectors X_i can be chosen locally so that the quadratic form diagonalizes in the following manner

$q_m\left(\sum _i{\xi }^i{X}_i\right)={\left({\xi }^1\right)}^2+{\left({\xi }^2\right)}^2+\cdots +{\left({\xi }^p\right)}^2-{\left({\xi }^{p+1}\right)}^2-\cdots -{\left({\xi }^n\right)}^2$

for some Template:Mvar between 1 and Template:Mvar. Any two such expressions of Template:Mvar (at the same point Template:Mvar of Template:Mvar) will have the same number Template:Mvar of positive signs. The signature of Template:Mvar is the pair of integers (p, n − p), signifying that there are Template:Mvar 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 Template:Mvar positive and n − p negative eigenvalues.

Certain metric signatures which arise frequently in applications are:

• If Template:Mvar has signature (n, 0), then Template:Mvar is a Riemannian metric, and Template:Mvar is called a Riemannian manifold. Otherwise, Template:Mvar is a pseudo-Riemannian metric, and Template:Mvar is called a pseudo-Riemannian manifold (the term semi-Riemannian is also used).
• If Template:Mvar is four-dimensional with signature (1, 3) or (3, 1), then the metric is called Lorentzian. More generally, a metric tensor in dimension Template:Mvar other than 4 of signature (1, n − 1) or (n − 1, 1) is sometimes also called Lorentzian.
• If Template:Mvar is 2n-dimensional and Template:Mvar has signature (n, n), then the metric is called ultrahyperbolic.

Inverse metric

Let f = (X₁, ..., X_n) be a basis of vector fields, and as above let G[f] be the matrix of coefficients

$g_{ij}[\mathbf{𝐟}]=g(X_i,X_j).$

One can consider the inverse matrix 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 Template:Mvar via

The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix Template:Mvar. 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 functionals.

To see this, suppose that Template:Mvar is a covector field. To wit, for each point Template:Mvar, Template:Mvar determines a function α_p defined on tangent vectors at Template:Mvar so that the following linearity condition holds for all tangent vectors X_p and Y_p, and all real numbers Template:Mvar and Template:Mvar:

${\alpha }_p(a{X}_p+b{Y}_p)=a{\alpha }_p\left(X_p\right)+b{\alpha }_p\left(Y_p\right).$

As Template:Mvar varies, Template:Mvar is assumed to be a smooth function in the sense that

$p\mapsto {\alpha }_p\left(X_p\right)$

is a smooth function of Template:Mvar for any smooth vector field Template:Mvar.

Any covector field Template:Mvar has components in the basis of vector fields f. These are determined by

${\alpha }_i=\alpha \left(X_i\right),i=1,2,\dots ,n.$

Denote the row vector of these components by

$\alpha [\mathbf{𝐟}]=\left[\begin{array}{cccc}{\alpha }_1& {\alpha }_2& \dots & {\alpha }_n\end{array}\right].$

Under a change of f by a matrix Template:Mvar, 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 Template:Mvar and Template:Mvar 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

$\begin{array}{rl}& \alpha [\mathbf{𝐟}A]G[\mathbf{𝐟}A{]}^{-1}\beta [\mathbf{𝐟}A{]}^{\mathsf{T}}\\ =& (\alpha [\mathbf{𝐟}]A)(A^{-1}G[\mathbf{𝐟}{]}^{-1}{\left(A^{-1}\right)}^{\mathsf{T}})(A^{\mathsf{T}}\beta [\mathbf{𝐟}{]}^{\mathsf{T}})\\ =& \alpha [\mathbf{𝐟}]G[\mathbf{𝐟}{]}^{-1}\beta [\mathbf{𝐟}{]}^{\mathsf{T}}.\end{array}$

So that the right-hand side of equation (6) 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 Template:Mvar and Template:Mvar have been raised to indicate the transformation law (5).

Raising and lowering indices

In a basis of vector fields f = (X₁, ..., X_n), any smooth tangent vector field Template:Mvar can be written in the form

for some uniquely determined smooth functions v¹, ..., vⁿ. Upon changing the basis f by a nonsingular matrix Template:Mvar, the coefficients vⁱ change in such a way that equation (7) remains true. That is,

$X=\mathbf{𝐟}\mathbf{𝐀}v[\mathbf{𝐟}\mathbf{𝐀}]=\mathbf{𝐟}v[\mathbf{𝐟}].$

Consequently, v[fA] = A⁻¹v[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 Template:Mvar. The contravariance of the components of v[f] is notationally designated by placing the indices of vⁱ[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 = (X₁, ..., X_n) define the dual basis to be the linear functionals (θ¹[f], ..., θⁿ[f]) such that

${\theta }^i[\mathbf{𝐟}](X_j)=\{\begin{array}{ll}1& \mathrm{i}\mathrm{f}i=j\\ 0& \mathrm{i}\mathrm{f}i=j.\end{array}$

That is, θⁱ[f](X_j) = δ_jⁱ, the Kronecker delta. Let

$\theta [\mathbf{𝐟}]=\left[\begin{array}{c}{\theta }^1[\mathbf{𝐟}]\\ {\theta }^2[\mathbf{𝐟}]\\ ⋮\\ {\theta }^n[\mathbf{𝐟}]\end{array}\right].$

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 Template:Mvar on tangent vectors can be expanded in terms of the dual basis Template:Mvar

where a[f] denotes the row vector [ a₁[f] ... a_n[f] ]. The components a_i transform when the basis f is replaced by fA in such a way that equation (8) 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⁻¹θ[f], it follows that a[fA] = a[f]A. That is, the components Template:Mvar transform covariantly (by the matrix Template:Mvar 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 linear functional on the tangent space at Template:Mvar. This operation takes a vector X_p at a point Template:Mvar and produces a covector g_p(X_p, −). In a basis of vector fields f, if a vector field Template:Mvar 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: Template:Mvar transforms covariantly. The operation of associating to the (contravariant) components of a vector field v[f] = [ v¹[f] v²[f] ... vⁿ[f] ]^T the (covariant) components of the covector field a[f] = [ a₁[f] a₂[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] = [ a₁[f] a₂[f] ... a_n[f] ] are the components of a covector in the dual basis θ[f], then the column vector Template:NumBlk 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 Template:Mvar. The operation (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, (9) is

$v^i[\mathbf{𝐟}]=\sum _{k=1}^n{g}^{ik}[\mathbf{𝐟}]a_k[\mathbf{𝐟}].$

Induced metric

Let Template:Mvar be an open set in ℝⁿ, and let Template:Mvar be a continuously differentiable function from Template:Mvar into the Euclidean space ℝ^m, where m > n. The mapping Template:Mvar is called an immersion if its differential is injective at every point of Template:Mvar. The image of Template:Mvar is called an immersed submanifold. More specifically, for m = 3, which means that the ambient Euclidean space is ℝ³, the induced metric tensor is called the first fundamental form.

Suppose that Template:Mvar is an immersion onto the submanifold M ⊂ R^m. The usual Euclidean dot product in ℝ^m is a metric which, when restricted to vectors tangent to Template:Mvar, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.

Suppose that Template:Mvar is a tangent vector at a point of Template:Mvar, say

$v=v^1{\mathbf{𝐞}}_1+\dots +v^n{\mathbf{𝐞}}_n$

where e_i are the standard coordinate vectors in ℝⁿ. When Template:Mvar is applied to Template:Mvar, the vector Template:Mvar goes over to the vector tangent to Template:Mvar given by

${\phi }_{*}(v)=\sum _{i=1}^n\sum _{a=1}^m{v}^i\frac{\partial {\phi }^a}{\partial x^i}{\mathbf{𝐞}}_a.$

(This is called the pushforward of Template:Mvar along Template:Mvar.) Given two such vectors, Template:Mvar and Template:Mvar, the induced metric is defined by

$g(v,w)={\phi }_{*}(v)\cdot {\phi }_{*}(w).$

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields e is given by

$G(\mathbf{𝐞})=(D\phi {)}^{\mathsf{T}}(D\phi )$

where Template:Mvar is the Jacobian matrix:

$D\phi =\left[\begin{array}{cccc}\frac{\partial {\phi }^1}{\partial x^1}& \frac{\partial {\phi }^1}{\partial x^2}& \dots & \frac{\partial {\phi }^1}{\partial x^n}\\ \frac{\partial {\phi }^2}{\partial x^1}& \frac{\partial {\phi }^2}{\partial x^2}& \dots & \frac{\partial {\phi }^2}{\partial x^n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial {\phi }^m}{\partial x^1}& \frac{\partial {\phi }^m}{\partial x^2}& \dots & \frac{\partial {\phi }^m}{\partial x^n}\end{array}\right].$

Intrinsic definitions of a metric

The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function

from the fiber product of the tangent bundle of Template:Mvar with itself to R such that the restriction of Template:Mvar to each fiber is a nondegenerate bilinear mapping

$g_p:{\mathrm{T}}_{p}M\times {\mathrm{T}}_{p}M\to \mathbf{𝐑}.$

The mapping (10) is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether Template:Mvar can support such a structure.

Metric as a section of a bundle

By the universal property of the tensor product, any bilinear mapping (10) gives rise naturally to a section g_⊗ of the dual of the tensor product bundle of TM with itself

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

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 Template:Mvar 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 braiding map.

Since Template:Mvar is finite-dimensional, there is a natural isomorphism

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

so that g_⊗ is regarded also as a section of the bundle T*M ⊗ T*M of the cotangent bundle T*M with itself. Since Template:Mvar is symmetric as a bilinear mapping, it follows that g_⊗ is a symmetric tensor.

Metric in a vector bundle

More generally, one may speak of a metric in a vector bundle. If Template:Mvar is a vector bundle over a manifold Template:Mvar, then a metric is a mapping

$g:E{\times }_{M}E\to \mathbf{𝐑}$

from the fiber product of Template:Mvar 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 section of the tensor product bundle E* ⊗ E*.

Tangent–cotangent isomorphism

The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. This isomorphism is obtained by setting, for each tangent vector X_p ∈ T_pM,

$S_g{X}_p\stackrel{\text{def}}{=}g(X_p,-),$

the linear functional on T_pM which sends a tangent vector Y_p at Template:Mvar to g_p(X_p,Y_p). That is, in terms of the pairing [−, −] between T_pM and its dual space TM,

$[S_g{X}_p,Y_p]=g_p(X_p,Y_p)$

for all tangent vectors X_p and Y_p. The mapping S_g is a linear transformation from T_pM to TM. It follows from the definition of non-degeneracy that the kernel of S_g is reduced to zero, and so by the rank–nullity theorem, S_g is a linear isomorphism. Furthermore, S_g is a symmetric linear transformation in the sense that

$[S_g{X}_p,Y_p]=[S_g{Y}_p,X_p]$

for all tangent vectors X_p and Y_p.

Conversely, any linear isomorphism S : T_pM → TM defines a non-degenerate bilinear form on T_pM by means of

$g_S(X_p,Y_p)=[S{X}_p,Y_p].$

This bilinear form is symmetric if and only if Template:Mvar is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on T_pM and symmetric linear isomorphisms of T_pM to the dual TM.

As Template:Mvar varies over Template:Mvar, S_g defines a section of the bundle Hom(TM, T*M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as Template:Mvar: it is continuous, differentiable, smooth, or real-analytic according as Template:Mvar. The mapping S_g, which associates to every vector field on Template:Mvar a covector field on Template:Mvar 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 defines a linear mapping

$S_g^{-1}:{\mathrm{T}}^{*}M\to \mathrm{T}M$

which is nonsingular and symmetric in the sense that

$[S_g^{-1}\alpha ,\beta ]=[S_g^{-1}\beta ,\alpha ]$

for all covectors Template:Mvar, Template:Mvar. Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map

${\mathrm{T}}^{*}M\otimes {\mathrm{T}}^{*}M\to \mathbf{𝐑}$

or by the double dual isomorphism to a section of the tensor product

$\mathrm{T}M\otimes \mathrm{T}M.$

Arclength and the line element

Suppose that Template:Mvar is a Riemannian metric on Template:Mvar. In a local coordinate system xⁱ, i = 1, 2, …, n, the metric tensor appears as a matrix, 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 parametric curve in Template:Mvar, for a ≤ t ≤ b. The arclength of the curve is defined by

$L=\underset{a}{\overset{b}{\int }}\sqrt{\sum _{i,j=1}^n{g}_{ij}(\gamma (t))(\frac{d}{dt}{x}^i\circ \gamma (t))(\frac{d}{dt}{x}^j\circ \gamma (t))}dt.$

In connection with this geometrical application, the quadratic differential form

$d{s}^2=\sum _{i,j=1}^n{g}_{ij}(p)d{x}^{i}d{x}^j$

is called the first fundamental form associated to the metric, while Template:Mvar is the line element. When ds² is pulled back to the image of a curve in Template:Mvar, 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

$L=\underset{a}{\overset{b}{\int }}\sqrt{|\sum _{i,j=1}^n{g}_{ij}(\gamma (t))(\frac{d}{dt}{x}^i\circ \gamma (t))(\frac{d}{dt}{x}^j\circ \gamma (t))|}dt.$

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:

$E=\frac{1}{2}\underset{a}{\overset{b}{\int }}\sum _{i,j=1}^n{g}_{ij}(\gamma (t))(\frac{d}{dt}{x}^i\circ \gamma (t))(\frac{d}{dt}{x}^j\circ \gamma (t))dt.$

This usage comes from physics, specifically, classical mechanics, where the integral Template:Mvar 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 equations may be obtained by applying variational principles 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.

Canonical measure and volume form

In analogy with the case of surfaces, a metric tensor on an Template:Mvar-dimensional paracompact manifold Template:Mvar gives rise to a natural way to measure the Template:Mvar-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 Template:Mvar on the space C₀(M) of compactly supported continuous functions on Template:Mvar. More precisely, if Template:Mvar is a manifold with a (pseudo-)Riemannian metric tensor Template:Mvar, then there is a unique positive Borel measure μ_g such that for any coordinate chart (U, φ), $$\mathrm{\Lambda }f=\underset{U}{\int }fd{\mu }_g=\underset{\phi (U)}{\int }f\circ {\phi }^{-1}(x)\sqrt{|\det g|}dx$$ for all Template:Mvar supported in Template:Mvar. 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 Jacobian change of variables. It extends to a unique positive linear functional on C₀(M) by means of a partition of unity.

If Template:Mvar is also oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system (x¹, ..., xⁿ) the volume form is represented as $$\omega =\sqrt{|\det g|}d{x}^1\wedge \cdots \wedge d{x}^n$$ where the dxⁱ are the coordinate differentials and ∧ denotes the exterior product in the algebra of differential forms. 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 metric tensor. In the usual Cartesian (x, y) coordinates, we can write

$g=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right].$

The length of a curve reduces to the formula:

$L=\underset{a}{\overset{b}{\int }}\sqrt{(dx{)}^2+(dy{)}^2}.$

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

Polar coordinates (r, θ):

$\begin{array}{rl}x& =r\cos \theta \\ y& =r\sin \theta \\ J& =\left[\begin{array}{cc}\cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \end{array}\right].\end{array}$

So

$g=J^{\mathsf{T}}J=\left[\begin{array}{cc}{\cos }^2\theta +{\sin }^2\theta & -r\sin \theta \cos \theta +r\sin \theta \cos \theta \\ -r\cos \theta \sin \theta +r\cos \theta \sin \theta & r^2{\sin }^2\theta +r^2{\cos }^2\theta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& r^2\end{array}\right]$

by trigonometric identities.

In general, in a Cartesian coordinate system xⁱ on a Euclidean space, the partial derivatives ∂ / ∂xⁱ are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δ_ij in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates qⁱ 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 ℝ³ comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. In standard spherical coordinates (θ, φ), with θ the colatitude, the angle measured from the Template:Mvar-axis, and Template:Mvar the angle from the Template:Mvar-axis in the Template:Mvar-plane, the metric takes the form

$g=\left[\begin{array}{cc}1& 0\\ 0& {\sin }^2\theta \end{array}\right].$

This is usually written in the form

$d{s}^2=d{\theta }^2+{\sin }^2\theta d{\phi }^2.$

Lorentzian metrics from relativity

In flat Minkowski space (special relativity), with coordinates

$r^{\mu }\to (x^0,x^1,x^2,x^3)=(ct,x,y,z),$

the metric is, depending on choice of metric signature,

$g=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right]\text{or}g=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$

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

In this case, the spacetime interval is written as

$d{s}^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2=d{r}^{\mu }d{r}_{\mu }=g_{\mu \nu }d{r}^{\mu }d{r}^{\nu }.$

The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates

$(x^0,x^1,x^2,x^3)=(ct,r,\theta ,\phi ),$

we can write the metric as

$g_{\mu \nu }=\left[\begin{array}{cccc}(1-\frac{2GM}{r{c}^2})& 0& 0& 0\\ 0& -{(1-\frac{2GM}{r{c}^2})}^{-1}& 0& 0\\ 0& 0& -r^2& 0\\ 0& 0& 0& -r^2{\sin }^2\theta \end{array}\right],$

where Template:Mvar (inside the matrix) is the gravitational constant and Template:Mvar represents the total 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

References

• .
• translated by A. M. Hiltebeitel and J. C. Morehead; "Disquisitiones generales circa superficies curvas", Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores Vol. VI (1827), pp. 99–146.
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• (to appear).