Metric Tensors| 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 TpM (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
(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 = gab
Rab
+ hij
Rij.
* 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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send feedback and suggestions to bombelli at olemiss.edu – modified 13 feb 2021