Metric
Tensors |

**On a Vector Space**

$ __As inner product__: A
metric tensor on a vector space *X* is a symmetric bilinear map *g*:
*X* × *X* → \(\mathbb R\) (rank-2 covariant tensor); Usually required
to be non-degenerate, and in fact (except for spacetime metrics) positive-definite.

$ __As mapping__: An isomorphism
*g*: *X* → *X** (index raising/lowering), by *v* \(\mapsto\)*g*(*v*, ).

$ __As orthonormal frame__: A choice of basis for *X*, up to gauge.

* __Relationships__: Any
such metric induces a norm by ||*x*||:= *g*(*x*, *x*)^{1/2}, and a distance by

*d*(*x*, *y*):= *g*(*y*–*x*,
*y*–*x*)^{1/2} = || *y*–*x* ||
.

* __Generalizations__: Degenerate and/or non-positive metrics.

* __Remark__: If the signature
has *k* minus
signs, *g* ∈ GL(*n*, \(\mathbb R\)) / SO(*k*, *n*–*k*).

@ __Generalizations__: Fernández AACA(01)mp/02,
AACA(01)mp/02 [metric "extensor"]; Hammond IJMPD(13) [non-symmetric, and spin].

> __Online resources__: see MathWorld page; Wikipedia page.

**Metric Tensor Field on a Manifold** > s.a. connection;
curvature.

* __Idea__: Globally, one
of the ways in which one specifies the geometry of a differentiable manifold;
Locally, a structure which gives all line elements at *p* congruent to
any given line element at *q*; It implies conformal,
projective and affine structures.

$ __Defs__: A smooth assignment of
a metric tensor on each T_{p}*M* (inner
product/mapping); Equivalent to a choice of orthonormal frame at each *p* ∈ *M*, up to gauge.

* __Question__: If (*M*,* d*)
is such that *d* is a smooth function of 2 variables,
can we define a metric *g* on *M*?

@ __References__: Mendez a1507
[analysis based on Takagi's factorization of the metric tensor].

> __Types and examples__:
see 2D, 3D, 4D
manifolds; lorentzian and riemannian
geometry; types of metrics.

> __Related topics__:
see metric decomposition, matching and perturbations.

**Space of Metrics on a Manifold** > see lorentzian geometries;
riemannian geometries.

@ __General references__: Demmel & Nink PRD(15)-a1506 [connections and geodesics].

@ __Characterization of metrics__: Hervik CQG(11)-a1107 [*ε*-property];
> s.a. distance between metrics.

**Tensor Products of Metrics**

$ __Def__: Obtained by (*M*_{1},* h*_{1};* M*_{2}, *h*_{2}) \(\mapsto\) (*M*_{1 }× *M*_{2}, *π*_{1}* *h*_{1 }⊗ *π*_{2}* *h*_{2})
.

* __Line element__: Of the
form d*s*^{2} = *g*_{ab}(*x*)
d*x*^{a} d*x*^{b} + *h*_{ij}(*y*)
d*y*^{i} d*y*^{j}.

* __Connection__: Γ^{a}_{bc} =
same as those of *g*_{ab}; Γ^{i}_{jk} =
same as those of *h*_{ij}; All Γs
with mixed indices vanish.

* __Curvature__: *R*^{a}_{bcd} and *R*_{ab} =
those of *g*_{ab}; *R*^{ i}_{jkl} and *R*_{ij} =
those of *h*_{ij}; All mixed *R*s
vanish; *R* = *g*^{ab}*R*_{ab} + *h*^{ij}*R*_{ij}.

* __Warped Product__: Obtained by

(*M*_{1}, *h*_{1}; *M*_{2}, *h*_{2}) \(\mapsto\) (*M*_{1 }× *M*_{2}, *π*_{1}* *h*_{1 }⊗ exp{2*θ*} *π*_{2}* *h*_{2})
.

@ __Warped product__: Choi JMP(00)mp/02.

"No metric, No nothing" – J. Stachel

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2016