Editing Metric tensor (section)

===Signature of a metric===
{{main|Metric signature}}

Associated to any metric tensor is the [[quadratic form]] defined in each tangent space by

:<math>q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM.</math>

If {{math|''q''<sub>''m''</sub>}} is positive for all non-zero {{math|''X''<sub>''m''</sub>}}, then the metric is [[definite bilinear form|positive-definite]] at {{mvar|m}}. If the metric is positive-definite at every {{math|''m'' ∈ ''M''}}, then {{mvar|g}} is called a [[Riemannian metric]]. More generally, if the quadratic forms {{math|''q''<sub>''m''</sub>}} have constant [[signature of a quadratic form|signature]] independent of {{mvar|m}}, then the signature of {{mvar|g}} is this signature, and {{mvar|g}} is called a [[pseudo-Riemannian metric]].<ref>{{harvnb|Dodson|Poston|1991|loc=Chapter VII §3.04}}</ref> If {{mvar|M}} is [[connected space|connected]], then the signature of {{mvar|q<sub>m</sub>}} does not depend on {{mvar|m}}.<ref>{{harvnb|Vaughn|2007|loc=§3.4.3}}</ref>

By [[Sylvester's law of inertia]], a basis of tangent vectors {{math|''X''<sub>''i''</sub>}} can be chosen locally so that the quadratic form diagonalizes in the following manner

:<math>q_m\left(\sum_i\xi^iX_i\right) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^{p+1}\right)^2-\cdots-\left(\xi^n\right)^2</math>

for some {{mvar|p}} between 1 and {{mvar|n}}. Any two such expressions of {{mvar|q}} (at the same point {{mvar|m}} of {{mvar|M}}) will have the same number {{mvar|p}} of positive signs. The signature of {{mvar|g}} is the pair of integers {{math|(''p'', ''n'' − ''p'')}}, signifying that there are {{mvar|p}} positive signs and {{math|''n'' − ''p''}} negative signs in any such expression. Equivalently, the metric has signature {{math|(''p'', ''n'' − ''p'')}} if the matrix {{math|''g''<sub>''ij''</sub>}} of the metric has {{mvar|p}} positive and {{math|''n'' − ''p''}} negative [[eigenvalue]]s.

Certain metric signatures which arise frequently in applications are:
* If {{mvar|g}} has signature {{math|(''n'', 0)}}, then {{mvar|g}} is a Riemannian metric, and {{mvar|M}} is called a [[Riemannian manifold]]. Otherwise, {{mvar|g}} is a pseudo-Riemannian metric, and {{mvar|M}} is called a [[pseudo-Riemannian manifold]] (the term semi-Riemannian is also used).
* If {{mvar|M}} is four-dimensional with signature {{math|(1, 3)}} or {{math|(3, 1)}}, then the metric is called [[Lorentzian metric|Lorentzian]]. More generally, a metric tensor in dimension {{mvar|n}} other than 4 of signature {{math|(1, ''n'' − 1)}} or {{math|(''n'' − 1, 1)}} is sometimes also called Lorentzian.
* If {{mvar|M}} is {{math|2''n''}}-dimensional and {{mvar|g}} has signature {{math|(''n'', ''n'')}}, then the metric is called [[ultrahyperbolic metric|ultrahyperbolic]].