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: XX* (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(yx, yx)1/2 = || yx || .

* Generalizations: Degenerate and/or non-positive metrics.
* Remark: If the signature has k minus signs, g ∈ GL(n, \(\mathbb R\)) / SO(k, nk).
@ 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 TpM (inner product/mapping); Equivalent to a choice of orthonormal frame at each pM, 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 (M1, h1; M2, h2) \(\mapsto\) (M1 × M2, π1* h1 ⊗ π2* h2) .
* Line element: Of the form ds2 = gab(x) dxa dxb + hij(y) dyi dyj.
* Connection: Γabc = same as those of gab; Γijk = same as those of hij; All Γs with mixed indices vanish.
* Curvature: Rabcd and Rab = those of gab; R ijkl and Rij = those of hij; All mixed Rs vanish; R = gabRab + hijRij.
* Warped Product: Obtained by

(M1, h1; M2, h2) \(\mapsto\) (M1 × M2, π1* h1 ⊗ exp{2θ} π2* h2) .

@ Warped product: Choi JMP(00)mp/02.

"No metric, No nothing" – J. Stachel

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