Distances on and between Manifolds with Metrics| Distances on and between Manifolds with Metrics |
On a Riemannian Manifold
> s.a. statistical geometry [random points].
* From curve length: The
most common definition makes {(M, g)} a length space,
d(a, b):= inf{∫ab [gij (dxi/dt) (dxj/dt)]1/2 dt | all curves x(t)}.
* Connes distance: (applicable also to graphs)
d(x, y):= inf{ |f(x)−f(y)| | all f ∈ \(\cal F\)}, \(\cal F\):= {f : M → \(\mathbb R\) | gab (∂af) (∂b f) ≤ 1} .
@ General references: Gromov 98;
Larsen JGP(03).
@ Connes distance:
Connes JMP(95) [and non-commutative geometry];
Dimakis & Müller-Hoissen IJTP(98) [1D lattice];
Parfionov & Zapatrin JMP(00) [Lorentzian];
Martinetti a1604-proc [explicit computations];
Canarutto & Minguzzi a1902-proc.
On a Lorentzian Manifold
* Lorentzian distance:
Defined as dg(x, y)
= supremum over lengths of future-oriented causal curves from x to y,
if they exist, zero otherwhise.
@ General references:
Markowitz MPCPS(81) [conformally invariant pseudodistance];
Parfionov & Zapatrin JMP(00)gq/98 [Connes-type distance].
@ Lorentzian distance: Beem GRG(78) [homothetic maps];
Alias et al TAMS-a0802 [to a fixed point on a spacelike hypersurface];
Rennie & Whale a1412 [and generalized time functions];
Franco JPCS(18)-a1710 [in non-commutative geometry];
Minguzzi JPCS(18)-a1711 [from the family of smooth steep time functions];
> s.a. distances;
types of distances [Lorentzian length spaces].
Distances between Metric Spaces
> s.a. Gromov-Hausdorff Space.
* Hausdorff distance
between compact subsets of a metric space Z:
dHZ(A, B):= inf{ε > 0 | A ⊂ Uε(B), B ⊂ Uε(A)} .
* Hausdorff distance between compact metric spaces:
dH(X, Y):= inf{dHZ(f(X), g(Y)) | all Z, all isometric embeddings f, g} .
* Lipschitz distance between metric spaces:
dL(X, Y):= inf{|log dil f| + |log dil f −1|, all Lipschitz homeos f : X → Y} ,
or infinity if the are no such fs.
* Hausdorff-Lipschitz distance between metric spaces:
dHL(X, Y):= inf{dH(X, X1) + dL(X1, Y1) + dH(Y1, Y) | all metric spaces X1, Y1} .
@ References: Gromov 98; > s.a. discrete spacetime.
Distances between Metric Tensors on Manifolds
> s.a. riemannian geometry.
* Lorentzian metrics: One can define
separately pseudodistances for the volume elements and conformal structures,
dv(g, g'):= sup{ | ln[|g(p)|1/2 / |g'(p)|1/2] |, p ∈ M}
dcv(g, g'):= sup{ V(A Δ A') / V(A ∪ A') | p, q: V(A ∪ A') ≥ v} .
* Lorentzian geometries: There is a pseudometric for each n ∈ \(\mathbb N\),
dn(G, G') = d( {Pn(C|G)}C ∈ Cn , {Pn(C|G)}C ∈ Cn ) ,
for example,
dn(G, G') = (2/π) arccos{∑C ∈ Cn \([P_n(C|G)]^{1/2}\;[P_n(C|G')]^{1/2}\)} ;
and a distance for each l ∈ \(\mathbb R\)+,
dl(G, G'):= (2/π) arccos{∑n=0∞ ∑C ∈ Cn \([P_l(n,C|G)]^{1/2}\;[P_l(n,C|G')]^{1/2}\)},
where Pl(n,
C|G):= Pm(n)
Pn(C|G),
m = VM
/ lD, D = dim G.
@ Riemannian metrics:
Bauer et al JDG(13) [Sobolev metrics].
@ Lorentzian metrics: Eder GRG(80) [pseudodistance];
Bombelli & Sorkin pr(95);
Aguirregabiria et al GRG(01)gq;
Suvorov CQG-a2007
[between black hole spacetimes, superspace approach].
@ Lorentzian geometries: Bombelli JMP(00)gq;
Noldus CQG(04)gq/03,
PhD(04)gq.
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16 jul 2020