In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (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 p of M is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g ( v , v ) > 0 {\displaystyle g(v,v)>0} for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold M, the length of a smooth curve between two points p and q can be defined by integration, and the distance between p and q can be defined as the infimum of the lengths of all such curves; this makes M 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 x, y, and z of points on the surface depending on two auxiliary variables u and v. Thus a parametric surface is (in today's terms) a vector-valued function
r → ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) {\displaystyle {\vec {r}}(u,\,v)={\bigl (}x(u,\,v),\,y(u,\,v),\,z(u,\,v){\bigr )}}
depending on an ordered pair of real variables (u, v), and defined in an open set D in the uv-plane. One of the chief aims of Gauss'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 [ E F F G ] {\textstyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}} 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 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
s = ∫ a b ‖ d d t r → ( u ( t ) , v ( t ) ) ‖ d t = ∫ a b u ′ ( t ) 2 r → u ⋅ r → u + 2 u ′ ( t ) v ′ ( t ) r → u ⋅ r → v + v ′ ( t ) 2 r → v ⋅ r → v d t , {\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:
r → u = ∂ r → ∂ u , r → v = ∂ r → ∂ v . {\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 to the curve of the square root of the (quadratic) differential
where
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
d s 2 = [ d u d v ] [ E F F G ] [ d u d v ] {\displaystyle 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 u and v to depend on another pair of variables u′ and v′. Then the analog of (2) for the new variables is
The chain rule relates E′, F′, and G′ to E, F, and G via the matrix equation
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
J = [ ∂ u ∂ u ′ ∂ u ∂ v ′ ∂ v ∂ u ′ ∂ v ∂ v ′ ] . {\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
[ E F F G ] {\displaystyle {\begin{bmatrix}E&F\\F&G\end{bmatrix}}}
with the transformation law (3) is known as the metric tensor of the surface.
Invariance of arclength under coordinate transformations 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,
[ d u d v ] = [ ∂ u ∂ u ′ ∂ u ∂ v ′ ∂ v ∂ u ′ ∂ v ∂ v ′ ] [ d u ′ d v ′ ] {\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
d s 2 = [ d u d v ] [ E F F G ] [ d u d v ] = [ d u ′ d v ′ ] [ ∂ u ∂ u ′ ∂ u ∂ v ′ ∂ v ∂ u ′ ∂ v ∂ v ′ ] T [ E F F G ] [ ∂ u ∂ u ′ ∂ u ∂ v ′ ∂ v ∂ u ′ ∂ v ∂ v ′ ] [ d u ′ d v ′ ] = [ d u ′ d v ′ ] [ E ′ F ′ F ′ G ′ ] [ d u ′ d v ′ ] = ( d s ′ ) 2 . {\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}}}
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 M can be written in the form
p = p 1 r → u + p 2 r → v {\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:
a = a 1 r → u + a 2 r → v b = b 1 r → u + b 2 r → v {\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,
a ⋅ b = a 1 b 1 r → u ⋅ r → u + a 1 b 2 r → u ⋅ r → v + a 2 b 1 r → v ⋅ r → u + a 2 b 2 r → v ⋅ r → v = a 1 b 1 E + a 1 b 2 F + a 2 b 1 F + a 2 b