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. Once a local coordinate system

x^i

is chosen, the metric tensor appears as a matrix, conventionally denoted G. The notation

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 notation for implicit sums. The length of a segment of a curve parameterized by t, from a to b, is defined as:

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

The angle

\theta

between two tangent vectors,

U=u^i{\partial\over \partial x_i}

and

V=v^i{\partial\over \partial x_i}

, is defined as:

\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

G = J^T J

where

J

denotes the Jacobian of the embedding and

J^T

its transpose.

Examples

The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

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

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

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:

(x^1, x^2)=(r, \theta)

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

Cylindrical coordinates:

(x^1, x^2, x^3)=(r, \theta, z)

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

Spherical coordinates:

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

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

Flat Minkowski space:

(x^1, x^2, x^3, x^4)=(t, x, y, z)

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

Metric Tensor

Examples

The Euclidean metric

See also