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 hilbert space;
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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send feedback and suggestions to bombelli at olemiss.edu – modified 13 feb 2021