Editing Metric tensor (section)

==Definition==
Let {{mvar|M}} be a [[smooth manifold]] of dimension {{mvar|n}}; for instance a [[surface (differential geometry)|surface]] (in the case {{math|1=''n'' = 2}}) or [[hypersurface]] in the [[Cartesian space]] <math>\R^{n+1}</math>. At each point {{math|''p'' ∈ ''M''}} there is a [[vector space]] {{math|T<sub>''p''</sub>''M''}}, called the [[tangent space]], consisting of all tangent vectors to the manifold at the point {{mvar|p}}. A metric tensor at {{mvar|p}} is a function {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, ''Y''<sub>''p''</sub>)}} which takes as inputs a pair of tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}} at {{mvar|p}}, and produces as an output a [[real number]] ([[scalar (mathematics)|scalar]]), so that the following conditions are satisfied:
* {{math|''g''<sub>''p''</sub>}} is [[bilinear form|bilinear]]. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if {{math|''U''<sub>''p''</sub>}}, {{math|''V''<sub>''p''</sub>}}, {{math|''Y''<sub>''p''</sub>}} are three tangent vectors at {{mvar|p}} and {{mvar|a}} and {{mvar|b}} are real numbers, then <math display="block">\begin{align}
  g_p(aU_p + bV_p, Y_p) &= ag_p(U_p, Y_p) + bg_p(V_p, Y_p) \,, \quad \text{and} \\
  g_p(Y_p, aU_p + bV_p) &= ag_p(Y_p, U_p) + bg_p(Y_p, V_p) \,.
\end{align}</math>
* {{math|''g''<sub>''p''</sub>}} is [[symmetric function|symmetric]].<ref>In several formulations of [[classical unified field theories]], the metric tensor was allowed to be non-symmetric; however, the antisymmetric part of such a tensor plays no role in the contexts described here, so it will not be further considered.</ref> A function of two vector arguments is symmetric provided that for all vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}, <math display="block">g_p(X_p, Y_p) = g_p(Y_p, X_p)\,.</math>
* {{math|''g''<sub>''p''</sub>}} is [[nondegenerate]]. A bilinear function is nondegenerate provided that, for every tangent vector {{math|''X''<sub>''p''</sub> ≠ 0}}, the function <math display="block">Y_p \mapsto g_p(X_p,Y_p)</math> obtained by holding {{math|''X''<sub>''p''</sub>}} constant and allowing {{math|''Y''<sub>''p''</sub>}} to vary is not [[identically zero]]. That is, for every {{math|''X''<sub>''p''</sub> ≠ 0}} there exists a {{math|''Y''<sub>''p''</sub>}} such that {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, ''Y''<sub>''p''</sub>) ≠ 0}}.

A metric tensor field {{mvar|g}} on {{mvar|M}} assigns to each point {{mvar|p}} of {{mvar|M}} a metric tensor {{math|''g''<sub>''p''</sub>}} in the tangent space at {{mvar|p}} in a way that varies [[smooth function|smoothly]] with {{mvar|p}}. More precisely, given any [[open set|open subset]] {{mvar|U}} of manifold {{mvar|M}} and any (smooth) [[vector field]]s {{mvar|X}} and {{mvar|Y}} on {{mvar|U}}, the real function
<math display="block">g(X, Y)(p) = g_p(X_p, Y_p)</math>
is a smooth function of {{mvar|p}}.