Editing Metric tensor (section)

===Metric in a vector bundle===
{{see also|metric (vector bundle)}}
More generally, one may speak of a metric in a [[vector bundle]]. If {{mvar|E}} is a vector bundle over a manifold {{mvar|M}}, then a metric is a mapping

:<math>g : E\times_M E\to \mathbf{R}</math>

from the [[fiber product]] of {{mvar|E}} to {{math|'''R'''}} which is bilinear in each fiber:

:<math>g_p : E_p \times E_p\to \mathbf{R}.</math>

Using duality as above, a metric is often identified with a [[section (fiber bundle)|section]] of the [[tensor product]] bundle {{math|''E''* ⊗ ''E''*}}.