Metric
Tensors| Metric
Tensors |
On a Vector Space
$ As inner product: A
symmetric bilinear map g: X × X → 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 
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, R)
/ SO(k, n–k).
@ Generalizations: Fernández AACA(01)mp/02,
AACA(01)mp/02 [metric "extensor"].
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?
> Types and examples:
see 2D, 3D, 4D
manifolds; lorentzian and riemannian
geometry; types
of metric.
> Related topics:
see metric decomposition, matching and perturbations.
Tensor Products of Metrics
$ Def: Obtained by (M1, h1; M2, h2) (M1 × M2, 
1* h1 
2* h2)
.
* Line element: Of the
form ds2 = gab(x)
dxa dxb + hij(y)
dyi dyj.
* Connection: 
abc =
same as those of gab; ijk =
same as those of hij; All 's
with mixed indices vanish.
* Curvature: Rabcd and Rab =
those of gab; R ijkl and Rij =
those of hij; All mixed R's
vanish; R = gabRab + hijRij.
* Warped Product: Obtained by
(M1, h1; M2, h2) (M1 × M2, 1* h1 exp{2
}2* 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
5 jul 2008