Editing Metric tensor (section)

==Intrinsic definitions of a metric==
The notion of a metric can be defined [[Differential geometry#Intrinsic versus extrinsic|intrinsically]] using the language of [[fiber bundle]]s and [[vector bundle]]s. In these terms, a '''metric tensor''' is a function

{{NumBlk|:|<math>g : \mathrm{T}M\times_M \mathrm{T}M\to \mathbf{R}</math>|{{EquationRef|10}}}}

from the [[fiber product]] of the [[tangent bundle]] of {{mvar|M}} with itself to {{math|'''R'''}} such that the restriction of {{mvar|g}} to each fiber is a nondegenerate bilinear mapping

:<math>g_p : \mathrm{T}_pM\times \mathrm{T}_pM \to \mathbf{R}.</math>

The mapping ({{EquationNote|10}}) is required to be [[continuous function|continuous]], and often [[continuously differentiable]], [[smooth function|smooth]], or [[real analytic]], depending on the case of interest, and whether {{mvar|M}} can support such a structure.

===Metric as a section of a bundle===
By the [[Tensor product#Universal property|universal property of the tensor product]], any bilinear mapping ({{EquationNote|10}}) gives rise [[natural transformation|naturally]] to a [[section (fiber bundle)|section]] {{math|''g''<sub>⊗</sub>}} of the [[dual space|dual]] of the [[tensor product bundle]] of {{math|T''M''}} with itself

:<math>g_\otimes \in \Gamma\left((\mathrm{T}M \otimes \mathrm{T}M)^*\right).</math>

The section {{math|''g''<sub>⊗</sub>}} is defined on simple elements of {{math|T''M'' ⊗ T''M''}} by

:<math>g_\otimes(v \otimes w) = g(v, w)</math>

and is defined on arbitrary elements of {{math|T''M'' ⊗ T''M''}} by extending linearly to linear combinations of simple elements. The original bilinear form {{mvar|g}} is symmetric if and only if
:<math>g_\otimes \circ \tau = g_\otimes</math>
where
:<math>\tau : \mathrm{T}M \otimes \mathrm{T}M \stackrel{\cong}{\to} TM \otimes TM</math>
is the [[tensor product#Tensor powers and braiding|braiding map]].

Since {{mvar|M}} is finite-dimensional, there is a [[natural isomorphism]]

:<math>(\mathrm{T}M \otimes \mathrm{T}M)^* \cong \mathrm{T}^*M \otimes \mathrm{T}^*M,</math>

so that {{math|''g''<sub>⊗</sub>}} is regarded also as a section of the bundle {{math|T*''M'' ⊗ T*''M''}} of the [[cotangent bundle]] {{math|T*''M''}} with itself. Since {{mvar|g}} is symmetric as a bilinear mapping, it follows that {{math|''g''<sub>⊗</sub>}} is a [[symmetric tensor]].

===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''*}}.

===Tangent–cotangent isomorphism===
{{see also|Musical isomorphism}}
The metric tensor gives a [[natural isomorphism]] from the [[tangent bundle]] to the [[cotangent bundle]], sometimes called the [[musical isomorphism]].<ref>For the terminology "musical isomorphism", see {{harvtxt|Gallot|Hulin|Lafontaine|2004|p=75}}. See also {{harvtxt|Lee|1997|pp=27–29}}</ref> This isomorphism is obtained by setting, for each tangent vector {{math|''X''<sub>''p''</sub> ∈ T<sub>''p''</sub>''M''}},

:<math>S_gX_p\, \stackrel\text{def}{=}\, g(X_p, -),</math>

the [[linear functional]] on {{math|T<sub>''p''</sub>''M''}} which sends a tangent vector {{math|''Y''<sub>''p''</sub>}} at {{mvar|p}} to {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>,''Y''<sub>''p''</sub>)}}. That is, in terms of the pairing {{math|[−, −]}} between {{math|T<sub>''p''</sub>''M''}} and its [[dual space]] {{math|T{{su|b=''p''|p=∗}}''M''}},

:<math>[S_gX_p, Y_p] = g_p(X_p, Y_p)</math>

for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}. The mapping {{math|''S''<sub>''g''</sub>}} is a [[linear transformation]] from {{math|T<sub>''p''</sub>''M''}} to {{math|T{{su|b=''p''|p=∗}}''M''}}. It follows from the definition of non-degeneracy that the [[kernel (set theory)|kernel]] of {{math|''S''<sub>''g''</sub>}} is reduced to zero, and so by the [[rank–nullity theorem]], {{math|''S''<sub>''g''</sub>}} is a [[linear isomorphism]]. Furthermore, {{math|''S''<sub>''g''</sub>}} is a [[symmetric linear transformation]] in the sense that

:<math>[S_gX_p, Y_p] = [S_gY_p, X_p] </math>

for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}.

Conversely, any linear isomorphism {{math|''S'' : T<sub>''p''</sub>''M'' → T{{su|b=''p''|p=∗}}''M''}} defines a non-degenerate bilinear form on {{math|T<sub>''p''</sub>''M''}} by means of

:<math>g_S(X_p, Y_p) = [SX_p, Y_p]\,.</math>

This bilinear form is symmetric if and only if {{mvar|S}} is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on {{math|T<sub>''p''</sub>''M''}} and symmetric linear isomorphisms of {{math|T<sub>''p''</sub>''M''}} to the dual {{math|T{{su|b=''p''|p=∗}}''M''}}.

As {{mvar|p}} varies over {{mvar|M}}, {{math|''S''<sub>''g''</sub>}} defines a section of the bundle {{math|Hom(T''M'', T*''M'')}} of [[vector bundle morphism|vector bundle isomorphisms]] of the tangent bundle to the cotangent bundle. This section has the same smoothness as {{mvar|g}}: it is continuous, differentiable, smooth, or real-analytic according as {{mvar|g}}. The mapping {{math|''S''<sub>''g''</sub>}}, which associates to every vector field on {{mvar|M}} a covector field on {{mvar|M}} gives an abstract formulation of "lowering the index" on a vector field. The inverse of {{math|''S''<sub>''g''</sub>}} is a mapping {{math|T*''M'' → T''M''}} which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverse {{math|''S''{{su|b=''g''|p=−1}}}} defines a linear mapping
:<math>S_g^{-1} : \mathrm{T}^*M \to \mathrm{T}M</math>

which is nonsingular and symmetric in the sense that
:<math>\left[S_g^{-1}\alpha, \beta\right] = \left[S_g^{-1}\beta, \alpha\right]</math>

for all covectors {{mvar|α}}, {{mvar|β}}. Such a nonsingular symmetric mapping gives rise (by the [[tensor-hom adjunction]]) to a map
:<math>\mathrm{T}^*M \otimes \mathrm{T}^*M \to \mathbf{R}</math>

or by the [[Double dual|double dual isomorphism]] to a section of the tensor product
:<math>\mathrm{T}M \otimes \mathrm{T}M.</math>