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(∂a f)(∂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]; Martinetti a1604-proc [explicit computations].

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 a1710 [in non-commutative geometry]; Minguzzi a1711 [from the family of smooth steep time functions]; > s.a. 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 | AUε(B), BUε(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 : XY} ,

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: Can define separately pseudodistances for the volume elements and conformal structures,

dv(g, g'):= sup{ | ln[|g(p)|1/2/|g'(p)|1/2] |, pM}

dcv(g, g'):= sup{ V(A Δ A') / V(AA') | p, q: V(AA') ≥ v} .

* Lorentzian geometries: There is a pseudometric for each n ∈ \(\mathbb N\),

dn(G, G') = d({Pn(C|G)}CCn , {Pn(C|G)}CCn) ,

for example,

dn(G, G') = (2/π) arccos{CCn [Pn(C|G)]1/2 [Pn(C|G')]1/2} ;

and a distance for each l ∈ \(\mathbb R\)+,

dl(G, G'):= (2/π) arccos{n=0CCn [Pl(n,C|G)]1/2 [Pl(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.
@ Lorentzian geometries: Bombelli JMP(00)gq; Noldus CQG(04)gq/03, PhD(04)gq.

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