Metric tensor

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

${\displaystyle {\begin{aligned}s&=\int _{a}^{b}\left\|{\frac {d}{dt}}{\vec {r}}(u(t),v(t))\right\|\,dt\\[5pt]&=\int _{a}^{b}{\sqrt {u'(t)^{2}\,{\vec {r}}_{u}\cdot {\vec {r}}_{u}+2u'(t)v'(t)\,{\vec {r}}_{u}\cdot {\vec {r}}_{v}+v'(t)^{2}\,{\vec {r}}_{v}\cdot {\vec {r}}_{v}}}\,dt\,,\end{aligned}}}$

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

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

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

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

(1)

where

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

(2)

The quantity ds in (1) is called the line element, while ds² is called the first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by r→(u, v) when u is increased by du units, and v is increased by dv units.

Using matrix notation, the first fundamental form becomes

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

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

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

(2')

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

${\displaystyle {\begin{bmatrix}E'&F'\\F'&G'\end{bmatrix}}={\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}}$

(3)

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

${\displaystyle J={\begin{bmatrix}{\frac {\partial u}{\partial u'}}&{\frac {\partial u}{\partial v'}}\\{\frac {\partial v}{\partial u'}}&{\frac {\partial v}{\partial v'}}\end{bmatrix}}\,.}$

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

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

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

Ricci-Curbastro & Levi-Civita (1900) first observed the significance of a system of coefficients E, F, and G, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule,

${\displaystyle {\begin{bmatrix}du\\dv\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}}$

so that

${\displaystyle {\begin{aligned}ds^{2}&={\begin{bmatrix}du&dv\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}du\\dv\end{bmatrix}}\\[6pt]&={\begin{bmatrix}du'&dv'\end{bmatrix}}{\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\[6pt]{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}{\dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\[6pt]{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial v'}}\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}\\[6pt]&={\begin{bmatrix}du'&dv'\end{bmatrix}}{\begin{bmatrix}E'&F'\\F'&G'\end{bmatrix}}{\begin{bmatrix}du'\\dv'\end{bmatrix}}\\[6pt]&=(ds')^{2}\,.\end{aligned}}}$

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 M can be written in the form

${\displaystyle \mathbf {p} =p_{1}{\vec {r}}_{u}+p_{2}{\vec {r}}_{v}}$

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

${\displaystyle {\begin{aligned}\mathbf {a} &=a_{1}{\vec {r}}_{u}+a_{2}{\vec {r}}_{v}\\\mathbf {b} &=b_{1}{\vec {r}}_{u}+b_{2}{\vec {r}}_{v}\end{aligned}}}$

then using the bilinearity of the dot product,

${\displaystyle {\begin{aligned}\mathbf {a} \cdot \mathbf {b} &=a_{1}b_{1}{\vec {r}}_{u}\cdot {\vec {r}}_{u}+a_{1}b_{2}{\vec {r}}_{u}\cdot {\vec {r}}_{v}+b_{1}a_{2}{\vec {r}}_{v}\cdot {\vec {r}}_{u}+a_{2}b_{2}{\vec {r}}_{v}\cdot {\vec {r}}_{v}\\[8pt]&=a_{1}b_{1}E+a_{1}b_{2}F+b_{1}a_{2}F+a_{2}b_{2}G\\[8pt]&={\begin{bmatrix}a_{1}&a_{2}\end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\end{bmatrix}}\,.\end{aligned}}}$

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 uv-plane. That is, put

${\displaystyle g(\mathbf {a} ,\mathbf {b} )=a_{1}b_{1}E+a_{1}b_{2}F+b_{1}a_{2}F+a_{2}b_{2}G\,.}$

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

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

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

${\displaystyle {\begin{aligned}g\left(\lambda \mathbf {a} +\mu \mathbf {a} ',\mathbf {b} \right)&=\lambda g(\mathbf {a} ,\mathbf {b} )+\mu g\left(\mathbf {a} ',\mathbf {b} \right),\quad {\text{and}}\\g\left(\mathbf {a} ,\lambda \mathbf {b} +\mu \mathbf {b} '\right)&=\lambda g(\mathbf {a} ,\mathbf {b} )+\mu g\left(\mathbf {a} ,\mathbf {b} '\right)\end{aligned}}}$

for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ.

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

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

and the angle θ between two vectors a and b is calculated by

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

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

${\displaystyle \iint _{D}\left|{\vec {r}}_{u}\times {\vec {r}}_{v}\right|\,du\,dv}$

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

${\displaystyle {\begin{aligned}&\iint _{D}{\sqrt {\left({\vec {r}}_{u}\cdot {\vec {r}}_{u}\right)\left({\vec {r}}_{v}\cdot {\vec {r}}_{v}\right)-\left({\vec {r}}_{u}\cdot {\vec {r}}_{v}\right)^{2}}}\,du\,dv\\[5pt]={}&\iint _{D}{\sqrt {EG-F^{2}}}\,du\,dv\\[5pt]={}&\iint _{D}{\sqrt {\det {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}}\,du\,dv\end{aligned}}}$

where det is the determinant.

A system of n real-valued functions (x¹, ..., xⁿ), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U

${\displaystyle \mathbf {f} =\left(X_{1}={\frac {\partial }{\partial x^{1}}},\dots ,X_{n}={\frac {\partial }{\partial x^{n}}}\right)\,.}$

The metric g has components relative to this frame given by

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

Relative to a new system of local coordinates, say

${\displaystyle y^{i}=y^{i}(x^{1},x^{2},\dots ,x^{n}),\quad i=1,2,\dots ,n}$

the metric tensor will determine a different matrix of coefficients,

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

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

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

so that

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

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

${\displaystyle G\left[\mathbf {f} '\right]=\left((Dy)^{-1}\right)^{\mathsf {T}}G\left[\mathbf {f} \right](Dy)^{-1}}$

where Dy denotes the Jacobian matrix of the coordinate change.

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

${\displaystyle q_{m}(X_{m})=g_{m}(X_{m},X_{m})\,,\quad X_{m}\in T_{m}M.}$

If q_m is positive for all non-zero X_m, then the metric is positive-definite at m. If the metric is positive-definite at every m ∈ M, then g is called a Riemannian metric. More generally, if the quadratic forms q_m have constant signature independent of m, then the signature of g is this signature, and g is called a pseudo-Riemannian metric.[4] If M is connected, then the signature of q_m does not depend on m.[5]

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

${\displaystyle q_{m}\left(\sum _{i}\xi ^{i}X_{i}\right)=\left(\xi ^{1}\right)^{2}+\left(\xi ^{2}\right)^{2}+\cdots +\left(\xi ^{p}\right)^{2}-\left(\xi ^{p+1}\right)^{2}-\cdots -\left(\xi ^{n}\right)^{2}}$

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

Certain metric signatures which arise frequently in applications are:

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

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

${\displaystyle g_{ij}[\mathbf {f} ]=g\left(X_{i},X_{j}\right)\,.}$

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 A via

${\displaystyle G[\mathbf {f} A]^{-1}=A^{-1}G[\mathbf {f} ]^{-1}\left(A^{-1}\right)^{\mathsf {T}}.}$

(5)

The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix A. Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals.

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

${\displaystyle \alpha _{p}\left(aX_{p}+bY_{p}\right)=a\alpha _{p}\left(X_{p}\right)+b\alpha _{p}\left(Y_{p}\right)\,.}$

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

${\displaystyle p\mapsto \alpha _{p}\left(X_{p}\right)}$

is a smooth function of p for any smooth vector field X.

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

${\displaystyle \alpha _{i}=\alpha \left(X_{i}\right)\,,\quad i=1,2,\dots ,n\,.}$

Denote the row vector of these components by

${\displaystyle \alpha [\mathbf {f} ]={\big \lbrack }{\begin{array}{cccc}\alpha _{1}&\alpha _{2}&\dots &\alpha _{n}\end{array}}{\big \rbrack }\,.}$

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

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

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

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

${\displaystyle {\tilde {g}}(\alpha ,\beta )=\alpha [\mathbf {f} ]G[\mathbf {f} ]^{-1}\beta [\mathbf {f} ]^{\mathsf {T}}.}$

(6)

The resulting definition, although it involves the choice of basis f, does not actually depend on f in an essential way. Indeed, changing basis to fA gives

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

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 i and j have been raised to indicate the transformation law (5).

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

${\displaystyle X=v^{1}[\mathbf {f} ]X_{1}+v^{2}[\mathbf {f} ]X_{2}+\dots +v^{n}[\mathbf {f} ]X_{n}=\mathbf {f} {\begin{bmatrix}v^{1}[\mathbf {f} ]\\v^{2}[\mathbf {f} ]\\\vdots \\v^{n}[\mathbf {f} ]\end{bmatrix}}=\mathbf {f} v[\mathbf {f} ]}$

(7)

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

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

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 A. 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

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

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

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

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

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

Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ

${\displaystyle {\begin{aligned}\alpha &=a_{1}[\mathbf {f} ]\theta ^{1}[\mathbf {f} ]+a_{2}[\mathbf {f} ]\theta ^{2}[\mathbf {f} ]+\cdots +a_{n}[\mathbf {f} ]\theta ^{n}[\mathbf {f} ]\\[8pt]&={\big \lbrack }{\begin{array}{cccc}a_{1}[\mathbf {f} ]&a_{2}[\mathbf {f} ]&\dots &a_{n}[\mathbf {f} ]\end{array}}{\big \rbrack }\theta [\mathbf {f} ]\\[8pt]&=a[\mathbf {f} ]\theta [\mathbf {f} ]\end{aligned}}}$

(8)

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,

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

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

${\displaystyle g_{p}(X_{p},-):Y_{p}\mapsto g_{p}(X_{p},Y_{p})}$

of tangent vector Y_p defines a linear functional on the tangent space at p. This operation takes a vector X_p at a point p and produces a covector g_p(X_p, −). In a basis of vector fields f, if a vector field X has components v[f], then the components of the covector field g(X, −) in the dual basis are given by the entries of the row vector

${\displaystyle a[\mathbf {f} ]=v[\mathbf {f} ]^{\mathsf {T}}G[\mathbf {f} ].}$

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

${\displaystyle v[\mathbf {f} A]^{\mathsf {T}}G[\mathbf {f} A]=v[\mathbf {f} ]^{\mathsf {T}}\left(A^{-1}\right)^{\mathsf {T}}A^{\mathsf {T}}G[\mathbf {f} ]A=v[\mathbf {f} ]^{\mathsf {T}}G[\mathbf {f} ]A}$

so that a[fA] = a[f]A: a 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

${\displaystyle a_{i}[\mathbf {f} ]=\sum _{k=1}^{n}v^{k}[\mathbf {f} ]g_{ki}[\mathbf {f} ]}$

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

${\displaystyle v[\mathbf {f} ]=G^{-1}[\mathbf {f} ]a[\mathbf {f} ]^{\mathsf {T}}}$

(9)

has components which transform contravariantly:

${\displaystyle v[\mathbf {f} A]=A^{-1}v[\mathbf {f} ].}$

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 M. 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

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

Let U be an open set in ℝⁿ, and let φ be a continuously differentiable function from U into the Euclidean space ℝ^m, where m > n. The mapping φ is called an immersion if its differential is injective at every point of U. The image of φ 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 φ 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 M, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.

Suppose that v is a tangent vector at a point of U, say

${\displaystyle v=v^{1}\mathbf {e} _{1}+\dots +v^{n}\mathbf {e} _{n}}$

where e_i are the standard coordinate vectors in ℝⁿ. When φ is applied to U, the vector v goes over to the vector tangent to M given by

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

(This is called the pushforward of v along φ.) Given two such vectors, v and w, the induced metric is defined by

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

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

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

where Dφ is the Jacobian matrix:

${\displaystyle D\varphi ={\begin{bmatrix}{\frac {\partial \varphi ^{1}}{\partial x^{1}}}&{\frac {\partial \varphi ^{1}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{1}}{\partial x^{n}}}\\[1ex]{\frac {\partial \varphi ^{2}}{\partial x^{1}}}&{\frac {\partial \varphi ^{2}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{2}}{\partial x^{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial \varphi ^{m}}{\partial x^{1}}}&{\frac {\partial \varphi ^{m}}{\partial x^{2}}}&\dots &{\frac {\partial \varphi ^{m}}{\partial x^{n}}}\end{bmatrix}}.}$

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

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

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

${\displaystyle g_{\otimes }(v\otimes w)=g(v,w)}$

and is defined on arbitrary elements of TM ⊗ TM by extending linearly to linear combinations of simple elements. The original bilinear form g is symmetric if and only if

${\displaystyle g_{\otimes }\circ \tau =g_{\otimes }}$

where

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

is the braiding map.

Since M is finite-dimensional, there is a natural isomorphism

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

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 g is symmetric as a bilinear mapping, it follows that g_⊗ is a symmetric tensor.

More generally, one may speak of a metric in a vector bundle. If E is a vector bundle over a manifold M, then a metric is a mapping

${\displaystyle g:E\times _{M}E\to \mathbf {R} }$

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

${\displaystyle g_{p}:E_{p}\times E_{p}\to \mathbf {R} .}$

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

The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism.[6] This isomorphism is obtained by setting, for each tangent vector X_p ∈ T_pM,

${\displaystyle S_{g}X_{p}\,{\stackrel {\text{def}}{=}}\,g(X_{p},-),}$

the linear functional on T_pM which sends a tangent vector Y_p at p to g_p(X_p,Y_p). That is, in terms of the pairing [−, −] between T_pM and its dual space T^∗
_pM,

${\displaystyle [S_{g}X_{p},Y_{p}]=g_{p}(X_{p},Y_{p})}$

for all tangent vectors X_p and Y_p. The mapping S_g is a linear transformation from T_pM to T^∗
_pM. 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

${\displaystyle [S_{g}X_{p},Y_{p}]=[S_{g}Y_{p},X_{p}]}$

for all tangent vectors X_p and Y_p.

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

${\displaystyle g_{S}(X_{p},Y_{p})=[SX_{p},Y_{p}]\,.}$

This bilinear form is symmetric if and only if S is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on T_pM and symmetric linear isomorphisms of T_pM to the dual T^∗
_pM.

As p varies over M, 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 g: 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 M 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

${\displaystyle S_{g}^{-1}:\mathrm {T} ^{*}M\to \mathrm {T} M}$

which is nonsingular and symmetric in the sense that

${\displaystyle \left[S_{g}^{-1}\alpha ,\beta \right]=\left[S_{g}^{-1}\beta ,\alpha \right]}$

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

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

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

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

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve:

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

This usage comes from physics, specifically, classical mechanics, where the integral E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of Maupertuis' principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle.

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.[7]

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

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

The length of a curve reduces to the formula:

${\displaystyle L=\int _{a}^{b}{\sqrt {(dx)^{2}+(dy)^{2}}}\,.}$

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

Polar coordinates (r, θ):

${\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\J&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix}}\,.\end{aligned}}}$

So

${\displaystyle g=J^{\mathsf {T}}J={\begin{bmatrix}\cos ^{2}\theta +\sin ^{2}\theta &-r\sin \theta \cos \theta +r\sin \theta \cos \theta \\-r\cos \theta \sin \theta +r\cos \theta \sin \theta &r^{2}\sin ^{2}\theta +r^{2}\cos ^{2}\theta \end{bmatrix}}={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}}$

by 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

${\displaystyle g_{ij}=\sum _{kl}\delta _{kl}{\frac {\partial x^{k}}{\partial q^{i}}}{\frac {\partial x^{l}}{\partial q^{j}}}=\sum _{k}{\frac {\partial x^{k}}{\partial q^{i}}}{\frac {\partial x^{k}}{\partial q^{j}}}.}$

The 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 z-axis, and φ the angle from the x-axis in the xy-plane, the metric takes the form

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

This is usually written in the form

${\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\,.}$

In flat Minkowski space (special relativity), with coordinates

${\displaystyle r^{\mu }\rightarrow \left(x^{0},x^{1},x^{2},x^{3}\right)=(ct,x,y,z)\,,}$

the metric is, depending on choice of metric signature,

${\displaystyle g={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\quad {\text{or}}\quad g={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,.}$

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