Editing Metric tensor (section)

{{Short description|Structure defining distance on a manifold}}
{{about|metric structures on manifolds|the specific case of spacetime of relativity|Metric tensor (general relativity)}}

In the [[mathematics|mathematical]] field of [[differential geometry]], a '''metric tensor''' (or simply '''metric''') is an additional [[Mathematical structure|structure]] on a [[manifold]] {{mvar|M}} (such as a [[surface (mathematics)|surface]]) that allows defining distances and angles, just as the [[inner product]] on a [[Euclidean space]] allows defining distances and angles there.  More precisely, a metric tensor at a point {{mvar|p}} of {{mvar|M}} is a [[bilinear form]] defined on the [[tangent space]] at {{mvar|p}} (that is, a [[bilinear function]] that maps pairs of [[tangent vector]]s to [[real number]]s), and a metric field on {{mvar|M}} consists of a metric tensor at each point {{mvar|p}} of {{mvar|M}} that varies smoothly with {{mvar|p}}.

A metric tensor {{mvar|g}} is ''positive-definite'' if {{math|''g''(''v'', ''v'') > 0}} for every nonzero vector {{mvar|v}}.  A manifold equipped with a positive-definite metric tensor is known as a [[Riemannian manifold]].  Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold.  On a Riemannian manifold {{mvar|M}}, the length of a smooth curve between two points {{mvar|p}} and {{mvar|q}} can be defined by integration, and the [[metric (mathematics)|distance]] between {{mvar|p}} and {{mvar|q}} can be defined as the [[infimum]] of the lengths of all such curves; this makes {{mvar|M}} a [[metric space]].  Conversely, the metric tensor itself is the [[derivative]] of the distance function (taken in a suitable manner).{{fact|date=August 2022}}

While the notion of a metric tensor was known in some sense to mathematicians such as [[Gauss]] from the early 19th century, it was not until the early 20th century that its properties as a [[tensor]] were understood by, in particular, [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]], who first codified the notion of a tensor.  The metric tensor is an example of a [[tensor field]].

The components of a metric tensor in a [[coordinate basis]] take on the form of a [[symmetric matrix]] whose entries transform [[covariance and contravariance of vectors|covariantly]] under changes to the coordinate system.  Thus a metric tensor is a covariant [[symmetric tensor]].  From the [[Coordinate-free|coordinate-independent]] point of view, a metric tensor field is defined to be a [[nondegenerate form|nondegenerate]] [[symmetric bilinear form]] on each tangent space that varies [[smooth function|smoothly]] from point to point.