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

where denotes the Jacobian of the the embedding and its transpose.

1 Examples

1.1 The Euclidean metric

Given a two-dimensional Euclidean metric tensor:

The length of a curve reduces to the familiar calculusFor other uses of the term calculus see calculus (disambiguation Calculus is a branch of mathematics, developed from algebra and geometry, involving two major complementary ideas: The first, called differential calculus is a theory about rates of change, formula:

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

Polar coordinates:

Cylindrical coordinates:

Spherical coordinates:

2 See also

pseudo-Riemannian metric

Riemannian geometry

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Topics: Metric Tensor, Examples, The Euclidean Metric, See Also...

Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows one to define of various notions as the length

Riemannian geometryIn mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.

Riemannian submanifoldA Riemannian submanifold ''N of a Riemannian manifold M is a smooth manifold equipped with the induced Riemannian metric from M''. The image of an isometric immersion is a Riemannian submanifold. Riemannian geometry.

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