# Metric tensor

In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. In other terms, given a smooth manifold, we make a choice of positive-definite quadratic form on the manifold's tangent spaces which varies smoothly from point to point. The manifold, equipped with the metric tensor (the varying choice of quadratic form), is called a Riemannian manifold and in this context the metric tensor is often called a Riemannian metric.

Once a local coordinate system $\displaystyle x^i \$ is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation $\displaystyle g_{ij} \$ is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein summation notation for implicit sums: please read that article if the equations in this article seem incorrect at first sight.

The length of a segment of a curve parameterized by t, from a to b, is defined as:

$\displaystyle L = \int_a^b \sqrt{ g_{ij}{dx^i\over dt}{dx^j\over dt}}dt \$

The angle $\displaystyle \theta \$ between two tangent vectors, $\displaystyle U=u^i{\partial\over \partial x_i} \$ and $\displaystyle V=v^i{\partial\over \partial x_i} \$ , is defined as:

$\displaystyle \cos \theta = \frac{g_{ij}u^iv^j} {\sqrt{ \left| g_{ij}u^iu^j \right| \left| g_{ij}v^iv^j \right|}} \$

The induced metric tensor for a smooth embedding of a manifold into Euclidean space can be computed by the formula

$\displaystyle G = J^T J \$

where $\displaystyle J \$ denotes the Jacobian of the embedding and $\displaystyle J^T \$ its transpose.

## Examples

### The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

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

The length of a curve reduces to the familiar calculus formula:

$\displaystyle L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2} \$

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

Polar coordinates: $\displaystyle (x^1, x^2)=(r, \theta) \$

$\displaystyle g = \begin{bmatrix} 1 & 0 \\ 0 & (x^1)^2\end{bmatrix} \$

Cylindrical coordinates: $\displaystyle (x^1, x^2, x^3)=(r, \theta, z) \$

$\displaystyle g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & 1\end{bmatrix} \$

Spherical coordinates: $\displaystyle (x^1, x^2, x^3)=(r, \phi, \theta) \$

$\displaystyle g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & (x^1\sin x^2)^2\end{bmatrix} \$

Flat Minkowski space: $\displaystyle (x^0, x^1, x^2, x^3)=(t, x, y, z) \$

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

## The tangent-cotangent isomorphism

In tensor analysis, the metric tensor is often used to provide a canonical isomorphism from the tangent space to the cotangent space: given a manifold M, v ∈ TpM and a metric tensor g on M, we have that g(v, . ), the mapping that sends another given vector w ∈ TpM to g(v,w), is an element of the dual space Tp*M. The nondegeneracy of the metric tensor makes it a one-to-one correspondence, and the fact that g itself is a tensor means that this identification is independent of coordinates. In component terminology, it means that one can identify covariant and contravariant objects i.e. "raise and lower indices."

This has a nice physical interpretation which is often glossed over. The metric tensor obviously has to do with measurement. We may ask, what is the scale for these measurements? A choice of basis defines the system of units on our manifold. The notions of contravariance and covariance correspond to quantities whose components transform "inversely" or "with" the coordinate system, hence the names. For example, consider R3 with the standard coordinate chart. If we transform the coordinate system by scaling the unit distance (say meters) down by a factor of 1000, the displacement vector (1,2,3) becomes (1000,2000,3000). On the other hand, if (1,2,3) represents a dual vector (for example, electric field strength), an object which takes a displacement vector and yields a scalar (in the example: potential difference in, say, volts), then the transformed coordinates become (0.001,0.002,0.003). What does the Euclidean metric on R3 do? (1,2,3) becoming (1000,2000,3000) makes sense because scaling down by 1000 takes meters to millimeters. For the field strength vector, (1,2,3) becoming (0.001, 0.002, 0.003) is a reflection of field strength going from volts per meter to volts per millimeter.

But what is the contravariant version of the field strength? How can we make a field strength vector's coordinates go from (1,2,3) to (1000,2000,3000)? The solution is to view the scale-down-by-1000 transformation as affecting the volts on the units V/m instead of the meters so that our new strength is measured in millivolts per meter. The metric tensor tells us precisely that we are still dealing with the same object, that is, it identifies the scaling of the basis vectors for the units "in the denominator" with a corresponding inverse change "in the numerator." Although somewhat trivial for R3, for general manifolds M it is very important since one can only define things locally. One can also imagine, for example, defining "funny units" on R3 which vary from point to point.