# Metric tensor

(Redirected from Riemannian metric)

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.

Once a local coordinate system [itex] x^i [itex] is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation [itex]g_{ij}[itex] is conventionally used for the components of the metric tensor (i.e. the elements of the matrix). In the following, we use the Einstein notation for implicit sums.

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

[itex]L = \int_a^b \sqrt{ g_{ij}{dx^i\over dt}{dx^j\over dt}}dt[itex]

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

[itex]

\cos \theta = \frac{g_{ij}u^iv^j} {\sqrt{ \left| g_{ij}u^iu^j \right| \left| g_{ij}v^iv^j \right|}} [itex]

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

[itex]G = J^T J[itex]

where [itex]J [itex] denotes the Jacobian of the embedding and [itex]J^T [itex] its transpose.

## Examples

### The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

[itex]g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}[itex]

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

[itex]L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2} [itex]

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

Polar coordinates: [itex](x^1, x^2)=(r, \theta)[itex]

[itex]g = \begin{bmatrix} 1 & 0 \\ 0 & (x^1)^2\end{bmatrix}[itex]

Cylindrical coordinates: [itex](x^1, x^2, x^3)=(r, \theta, z)[itex]

[itex]g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & 1\end{bmatrix}[itex]

Spherical coordinates: [itex](x^1, x^2, x^3)=(r, \phi, \theta)[itex]

[itex]g = \begin{bmatrix} 1 & 0 & 0\\ 0 & (x^1)^2 & 0 \\ 0 & 0 & (x^1\sin x^2)^2\end{bmatrix}[itex]

Flat Minkowski space: [itex](x^0, x^1, x^2, x^3)=(t, x, y, z)[itex]

[itex]g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}[itex]

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