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 xⁱ is chosen, the metric tensor appears as a matrix, conventionally notated as 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:

The angle θ between two tangent vectors, and , is defined as:

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

G = J^TJ

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

Examples

The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

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

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

Polar coordinates: (x¹,x²) = (r,θ)

Cylindrical coordinates: (x¹,x²,x³) = (r,θ,z)

Spherical coordinates: (x¹,x²,x³) = (r,φ,θ)

Flat Minkowski space: (x¹,x²,x³,x⁴) = (t,x,y,z)

See also

• pseudo-Riemannian metric