SaoPaulo,July22nd-26th,2013 7thInt.MeetingonLorentzianGeometry MiguelS´anchez RecentprogressontheLorentz-Finslercorrespondence

(1)

1 Causal structure 2 Visibility and lensing 3 Causal boundary

Recent progress on the Lorentz-Finsler correspondence

Miguel S´anchez

Universidad de Granada

7th Int. Meeting on Lorentzian Geometry Sao Paulo,

July 22nd-26th, 2013

(2)

Introduction

Stationary to Randers correspondence. Equivalence:

Conformal structure of stationary spacetimes

←→ Geometry of Randers spaces

Applicability:

← Precise description of spacetime elements in terms of Finsler counterparts

→ New geometric elements and results in Randers spaces

—some of them extensible to general Finsler manifolds Broad relation

Lorentzian Geometry ←→Finsler Geometry (including the Riemannian one!)

(3)

Introduction

Applicability:

—some of them extensible to general Finsler manifolds

Broad relation

(4)

Introduction

Applicability:

—some of them extensible to general Finsler manifolds Broad relation

(5)

Starting point

Normalized standard stationary spacetime:

V = (R×M,g_L =−1dt²+π^∗ω⊗dt+dt⊗π^∗ω+π^∗g)

ω 1-form, g Riemannian metric on M,π :R×M →M projection

∂_t timelike (future-directed) Killing vector field g_L ≡ −dt²+ 2ωdt+g

Normalized: −1dt². Useful for conformal elements such as lightlike vectors/geodesics (otherwise: −Λdt²)

Global “standard” (but not unique) splitting not too restrictive: it always hold locally and [Javaloyes & — ’08]:

A spacetime is (globally) conformal to a standard stationary one iff it admits a complete timelike conformal vector field and it isdistinguishing (and, so, strongly causal and causally continuous).

(6)

Starting point

Normalized standard stationary spacetime:

V = (R×M,g_L =−1dt²+π^∗ω⊗dt+dt⊗π^∗ω+π^∗g)

ω 1-form, g Riemannian metric on M,π :R×M →M projection

∂_t timelike (future-directed) Killing vector field g_L ≡ −dt²+ 2ωdt+g

Normalized: −1dt². Useful for conformal elements such as lightlike vectors/geodesics (otherwise: −Λdt²)

Global “standard” (but not unique) splitting not too restrictive: it always hold locally and [Javaloyes & — ’08]:

A spacetime is (globally) conformal to a standard stationary one iff it admits a complete timelike conformal vector field and it isdistinguishing (and, so, strongly causal and causally continuous).

(7)

Starting point

Appearance of Finsler Geometry

With these elementsω,g construct the functionsF^± :TM →R

F^±(v) = q

g(v,v) +ω(v)²±ω(v) Finsler metrics of Randers type onM,“Fermat metrics”

Connection with the spacetime geometry(Cap, Jav. Mas. ’11): A curve γ(t) = (±t,c(t)),t ∈[a,b] is lightlike and future/past directediff F^±( ˙c) = 1. In this case:

thearrival time b−ais equal to theF^±-length ofc,Rb a F^±( ˙c) (Fermat principle) γ is a pregeodesic iffc is a geodesic for F^± (i.e., a critical point of the arrival time/length functional c 7→R

F^±( ˙c) parametrized withF^±-length).

(8)

Starting point

Appearance of Finsler Geometry

With these elementsω,g construct the functionsF^± :TM →R

F^±(v) = q

g(v,v) +ω(v)²±ω(v) Finsler metrics of Randers type onM,“Fermat metrics”

Connection with the spacetime geometry(Cap, Jav. Mas. ’11):

A curveγ(t) = (±t,c(t)),t ∈[a,b] is lightlike and future/past directediff F^±( ˙c) = 1. In this case:

thearrival time b−ais equal to theF^±-length ofc,Rb a F^±( ˙c) (Fermat principle) γ is a pregeodesic iffc is a geodesic for F^± (i.e., a critical point of the arrival time/length functional c 7→R

F^±( ˙c) parametrized withF^±-length).

(9)

Starting point

These elementary considerations suggest the possibility to relate theconformalgeometry of standard stationary spacetimes and the geometry of the corresponding class of Finsler manifolds, i.e.

Randersspaces.

Aims:

1 Causal structure ←→Finslerian distances

(Caponio, Javaloyes, — Rev. Mat. Iberoam, ’11)

2 Visibility and gravitational lensing ←→ convexity of Finsler hypersurfaces

(Caponio, Germinario, —, arxiv:1112.3892)

3 Causal boundaries ←→Cauchy, Gromov and Busemann boundaries in Finslerian (and Riemannian) settings (Flores, Herrera, — ATMP’11, Memoirs AMS’13).

(10)

Starting point

Randersspaces. Aims:

2 Visibility and gravitational lensing ←→ convexity of Finsler hypersurfaces

(11)

Starting point

2 Visibility and gravitational lensing ←→

convexity of Finsler hypersurfaces

(12)

Starting point

2 Visibility and gravitational lensing ←→

convexity of Finsler hypersurfaces

(13)

Preliminaries on Finsler distances Causal Hierarchy

Further results

FIRST PART

1. CAUSAL STRUCTURE

(14)

Further results

Notion of Finsler and Randers metric

F :TM →RFinsler metric: continuous, smooth away 0 + positively homog. strongly convex norm at each p ∈M Positively homogeneous: F(λv) =λF(v) forλ >0 Strongly convex: the second fundamental form of the unit sphere is positive definite

Randers metric: R=p

g +ω²+ω for some Riemannian g (and h :=g +ω²) and 1-form ω.

In particular, Fermat metricsF^± are Randers (and viceversa) Reversed Finsler metric: F rev(v) :=F(−v)

In particular,

for Fermat metrics (F⁺)rev(v) =F⁻(v)

if ω6= 0, Randers metrics are non-reversible(R 6=Rrev)

(15)

Further results

Notion of Finsler and Randers metric

Randers metric: R=p

In particular, Fermat metricsF^± are Randers (and viceversa)

Reversed Finsler metric: F rev(v) :=F(−v) In particular,

(16)

Further results

Notion of Finsler and Randers metric

Randers metric: R=p

In particular, Fermat metricsF^± are Randers (and viceversa) Reversed Finsler metric: F rev(v) :=F(−v)

In particular,

(17)

Further results

Notion of generalized distance

Taking infimum of lengths of curves connecting two points, each Finsler metric induces ageneralized distance d. This means:

1 all the axioms of a distance hold but symmetry

2 for sequences {x_n}: d(x,xn)→0 ⇐⇒ d(xn,x)→0 Centered at any pointx₀, there areforward balls (d(x₀,x)<r) and backward balls (d(x,x₀)<r) that may differ but generate the same topology(in the Finslerian case, the manifold topology)

Symmetrized distance: d_s(x,y) = (d(x,y) +d(y,x))/2 RemarkEven in the Finslerian case,d_s does not come from a length space.

(18)

Further results

Notion of generalized distance

2 for sequences{x_n}: d(x,xn)→0 ⇐⇒ d(xn,x)→0

Centered at any pointx₀, there areforward balls (d(x₀,x)<r) and backward balls (d(x,x₀)<r) that may differ but generate the same topology(in the Finslerian case, the manifold topology)

(19)

Further results

Notion of generalized distance

2 for sequences{x_n}: d(x,xn)→0 ⇐⇒ d(xn,x)→0 Centered at any pointx₀, there areforward balls (d(x₀,x)<r) and backward balls (d(x,x0)<r) that may differ but generate the same topology(in the Finslerian case, the manifold topology)

(20)

Further results

Notion of generalized distance

2 for sequences{x_n}: d(x,xn)→0 ⇐⇒ d(xn,x)→0 Centered at any pointx₀, there areforward balls (d(x₀,x)<r) and backward balls (d(x,x0)<r) that may differ but generate the same topology(in the Finslerian case, the manifold topology)

(21)

Further results

Finslerian Hopf-Rinow

Theorem

Let(M,F) be a (connected) Finsler manifold with generalized distance d . They are equivalent:

(a) d isforward(resp. backward) complete (Cauchy sequences).

(b) The Finsler manifold (M,F) isforward(resp. backward) geodesically complete.

(c) At some (and then all) point p∈M,exp_p (resp. exp˜ _p) is defined on all of T_pM.

(d) Heine-Borel property: every closed and forward (resp.

backward) bounded subset of(M,d) is compact.

Moreover,in this case (M,F) isconvex, i.e., every pair of points p,q ∈M can be joined by a minimizing geodesic from p to q.

(22)

Further results

Finslerian Hopf-Rinow

Remark. Relation with ds:

1 d is either forward or backward complete =⇒

2 ds satisfies Heine-Borel ⇐⇒all ¯Bs(x,r) compact=⇒

3 ds complete

(23)

Further results

The correspondence for chronological futures

Basic idea:

Forp∈M,d⁺≡d distance ofF⁺≡F:

I⁺(0,p)determined by the graph of d⁺(p,·):

{t₀} ×B⁺(p,t0) =I⁺(0,p)∩({t₀} ×M)

(24)

Further results

Ladder of causality for stationary s-p

Theorem

The slices {t₀} ×M are Cauchy hypersurfaces of(R×M,g_L) iff d⁺ is forward and backward complete.

(R×M,g) is globally hyperbolic (causal + J⁺(z)∩J⁻(z⁰) compact) iff B¯_s⁺(p,r) are compact∀p ∈M,r >0

(Heine-Borel property)

(R×M,gL) is causally simple (causal + J^±(z) closed) iff (M,F) is convex

(Full characterization of Causality, as standard stationary s-t are always causally continuous.)

(25)

Further results

Ladder of causality for stationary s-p

Theorem

(26)

Further results

Ladder of causality for stationary s-p

Theorem

(27)

Further results

Consequences for Finsler manifolds

Remark. Compactness of ¯B_s⁺(p,r) for a Randers metric

=⇒(R×M,g_L) is glob. hyp

=⇒(R×M,gL) admits a spacelike Cauchy hypersurface M_f ={(f(x),x) :x ∈M}

=⇒The spacetime admits a splitting R×M_f with Cauchy slices and Fermat metricFf ≡F −df

=⇒for somef, the new metric F −df is a forward and backward complete Randers metric (with the same pregeodesics asF) ! this property is extensible to any Finsler metric(Matveev’12).

In general, the compactness of B¯_s⁺(p,r)(Heine-Borel) is the optimal assumption for classical Finsler theorems (Myers, sphere...)

(28)

Further results

Consequences for Finsler manifolds

(29)

Further results

Consequences for Finsler manifolds

(30)

Further results

Consequences for Finsler manifolds

(31)

Further results

Consequences for Finsler manifolds

(32)

Further results

Cauchy developments

A⊂V achronalset

D⁺(A)={z ∈V :γ∩A6=∅for all γ past-inextensible causal curve starting atp}

H⁺(A)={z ∈D⁺(A) :I⁺(z)∩D⁺(A) =∅}

Proposition

For A⊂M ≡ {0} ×M (slice ofR×S ): D⁺(A) ={(t,y) : 0≤t <d⁺(x,y)∀x 6∈A}

H⁺(A) ={(t,y) :t=infx6∈Ad⁺(x,y)(=d⁺(M\A,y))} H⁺(A) is constructed from the level sets of d⁺(M\A, ·)

(33)

Further results

Cauchy developments

A⊂V achronalset

D⁺(A)={z ∈V :γ∩A6=∅for all γ past-inextensible causal curve starting atp}

H⁺(A)={z ∈D⁺(A) :I⁺(z)∩D⁺(A) =∅}

Proposition

For A⊂M ≡ {0} ×M (slice ofR×S ):

D⁺(A) ={(t,y) : 0≤t <d⁺(x,y)∀x 6∈A}

H⁺(A) ={(t,y) :t=inf_x6∈Ad⁺(x,y)(=d⁺(M\A,y))}

H⁺(A) is constructed from the level sets of d⁺(M\A, ·)

(34)

Further results

Applications to Finsler

Remark. The results on horizons also yield results on the distance function to a set in a Randers manifold. [This extends the

viewpoint for the static case (and symmetric distances) by Chrusciel, Fu, Galloway and Howard ’02.]

An example is the following translation of a result by Beem and Krolak ’98:

Theorem

Let C ⊂M a closed subset, p ∈M\C is a differentiable point of the distance from C iff p is crossed by exactly one minimizing segment generalizable to any Finsler manifold

(Sabau-Tanaka’12)

(35)

Further results

Applications to Finsler

Theorem

Let C ⊂M a closed subset, p ∈M\C is a differentiable point of the distance from C iff p is crossed by exactly one minimizing segment

generalizable to any Finsler manifold (Sabau-Tanaka’12)

(36)

Further results

Applications to Finsler

Theorem

Let C ⊂M a closed subset, p ∈M\C is a differentiable point of the distance from C iff p is crossed by exactly one minimizing segment generalizable to any Finsler manifold

(Sabau-Tanaka’12)

(37)

The physical problem

Preliminaries on convex hypersurfaces Main ideas and results

SECOND PART

2. VISIBILITY AND LENSING

(38)

Visibility of particles

Problem studied by many authors: Giannoni, Fortunato, Masiello, Perlick, Piccione...

Fermat principle(Kovner’90, Perlick’90): if a first arriving (or critical for the arrival time) causal curve connecting a point and a line (stationary trajectory) exists then it is a lightlike geodesic. BUT:

Does such a geodesic exist?

Will it remain in a (reasonably big)realistic region R×D? (where the conformally stationary model remains valid) Will massive particles (timelike geodesics)arrive in a finite proper time (length) prior to their desintegration?

Lensing: can we receive more than once particles from the same source emitted at the same or different times? Problems related toconvexity.

(39)

Visibility of particles

Fermat principle(Kovner’90, Perlick’90): if a first arriving (or critical for the arrival time) causal curve connecting a point and a line (stationary trajectory) exists then it is a lightlike geodesic.

BUT:

(40)

Visibility of particles

BUT:

(41)

Visibility of particles

BUT:

Will it remain in a (reasonably big)realistic region R×D?

(where the conformally stationary model remains valid)

Will massive particles (timelike geodesics)arrive in a finite proper time (length) prior to their desintegration?

(42)

Visibility of particles

BUT:

(where the conformally stationary model remains valid) Will massive particles (timelike geodesics)arrive in a finite proper time (length) prior to their desintegration?

(43)

Visibility of particles

BUT:

Lensing: can we receive more than once particles from the same source emitted at the same or different times?

Problems related toconvexity.

(44)

Visibility of particles

BUT:

Lensing: can we receive more than once particles from the same source emitted at the same or different times?

(45)

Previous: Riemannian convexity

(M,g_R) Riemannian,D ⊂M open domain with smooth ∂D Notions of convexity for∂D:

1 Infinitesimal convexity: second fundamental form positive semi-definite with respect to the inner normal

⇐⇒ Hessφnegative semidefinite on T(∂D) for any smooth φ:D→[0,∞) with∂D =φ⁻¹(0) andφregular on ∂D.

2 Local convexity: locally each exp(T_p(∂D)) does not touchD.

Local =⇒ infinitesimal trivially (and pointwise) Non-trivial converse (Do Carmo, Warner ’70) Bishop ’74 proved the converse but

1 his proof requiredsmoothnessC⁴(also forg)

2 Itcannot be extended to the Finslerian setting(Borisenko, Olin ’10)

(46)

Previous: Riemannian convexity

Local =⇒ infinitesimal trivially (and pointwise) Non-trivial converse (Do Carmo, Warner ’70)

Bishop ’74 proved the converse but

2 Itcannot be extended to the Finslerian setting(Borisenko, Olin ’10)

(47)

Previous: Riemannian convexity

Local =⇒ infinitesimal trivially (and pointwise) Non-trivial converse (Do Carmo, Warner ’70) Bishop ’74 proved the converse but

2 Itcannot be extended to the Finslerian setting (Borisenko,

(48)

General Finslerian results

Approach by Bartolo, Caponio, Germinario, — ’10:

previous notions extensible to Finslerian manifolds

intermediate notion: ∂D isgeometrically convexwhen: no geodesic in D connecting somep,q∈D touches ∂D.

Theorem

For any Finsler manifold, and domain D withC^1,1

loc boundary∂D:

1 The infinitesimal, geometric and local notions of convexity are equivalent.

(49)

General Finslerian results

Approach by Bartolo, Caponio, Germinario, — ’10:

previous notions extensible to Finslerian manifolds

intermediate notion: ∂D isgeometrically convexwhen: no geodesic in D connecting somep,q∈D touches ∂D.

Theorem

For any Finsler manifold, and domain D withC^1,1

loc boundary∂D:

(50)

General Finslerian results

Theorem (Bartolo, Caponio, Germinario, — ’10)

For any Finsler manifold, and domain D withC^1,1loc boundary∂D:

2 If all B¯_s^D^¯(p,r)(in particular, if Bs(p,r)∩D) are compact: ∂D is convex iff D is convex.

In the last case,if D is not contractiblethen any p,q ∈D can be connected byinfinitely many geodesicscontained in D with diverging lengths.

(51)

General Finslerian results

Theorem (Bartolo, Caponio, Germinario, — ’10)

For any Finsler manifold, and domain D withC^1,1loc boundary∂D:

2 If all B¯_s^D^¯(p,r)(in particular, if Bs(p,r)∩D) are compact: ∂D is convex iff D is convex.

In the last case,if D is not contractiblethen any p,q ∈D can be connected byinfinitely many geodesicscontained in D with diverging lengths.

(52)

Basic ideas: existence

Existenceof connecting causal geodesics in a prescribed R×D Lightlike geodesics. Optimal conditions:

1 (geometric) light-convexity ofR×∂D inR×M

⇐⇒Convexity of∂D in (M,F).

2 + completeness of the space of connecting curves:

⇐⇒Compactness of ¯B_s^D^¯(p,r) (⇐B¯_s(p,r)∩D)¯ Timelike geodesics. Reduction to the lightlike case: γ timelike geodesic in R×M with gL(γ⁰, γ⁰) =−c²

⇐⇒˜γ(s) = (cs, γ(s)) is a lightlike geodesic inR×(R×M) with the product metric (≡du²+g_L)

(53)

Basic ideas: existence

⇐⇒Compactness of ¯B_s^D^¯(p,r) (⇐B¯_s(p,r)∩D)¯

Timelike geodesics. Reduction to the lightlike case: γ timelike geodesic in R×M with gL(γ⁰, γ⁰) =−c²

(54)

Basic ideas: existence

⇐⇒Compactness of ¯B_s^D^¯(p,r) (⇐B¯_s(p,r)∩D)¯ Timelike geodesics. Reduction to the lightlike case:

γ timelike geodesic in R×M with gL(γ⁰, γ⁰) =−c²

(55)

Basic ideas: existence

⇐⇒Compactness of ¯B_s^D^¯(p,r) (⇐B¯_s(p,r)∩D)¯ Timelike geodesics. Reduction to the lightlike case:

γ timelike geodesic in R×M with gL(γ⁰, γ⁰) =−c²

(56)

Basic ideas: multiplicity

Multiplicity (lensing)

Local, around a given lightlike pregeodesic γ(t) = (t,c(t)):

existence of conjugate points for γ

c admits conjugate point as a Finsler geodesic (tidal lensing)

Global: non trivial topology of R×D D non-contractible

(topological lensing)

(57)

Basic ideas: multiplicity

Multiplicity (lensing)

Local, around a given lightlike pregeodesic γ(t) = (t,c(t)):

existence of conjugate points for γ

c admits conjugate point as a Finsler geodesic (tidal lensing)

Global: non trivial topology of R×D D non-contractible

(topological lensing)

(58)

Results: lightlike geodesics

R×M standard stationary,D ⊂M aC² domain Theorem

Assume that allB¯_s^D^¯(p,r) are compact(which happens, in

particular, whenB¯_s(p,r)∩D are compact¯ ). They are equivalent:

1 (R×D,g_L) iscausally simple (⇔(D,F)is convex)

2 (∂D;F) is convex

3 (R×∂D;gL) is light-convex.

4 Any point w ∈R×D and any line l_q,q ∈D connected in R×D by a future–pointinglightlike geodesicminimizing the (future) arrival time T (⇔ idem for past)

In this case,if D is not contractible,infinitely many connecting lightlike geodesicswith diverging arrival times exist

(59)

Results: lightlike geodesics

1 (R×D,g_L) iscausally simple (⇔(D,F) is convex)

2 (∂D;F) is convex

(60)

Results: lightlike geodesics

2 (∂D;F) isconvex

(61)

Results: lightlike geodesics

2 (∂D;F) isconvex

(62)

Results: lightlike geodesics

2 (∂D;F) isconvex

4 Any point w ∈R×D and any line lq,q ∈D connected in R×D by a future–pointinglightlike geodesicminimizing the (future) arrival time T (⇔ idem for past)

(63)

Results: lightlike geodesics

2 (∂D;F) isconvex

4 Any point w ∈R×D and any line lq,q ∈D connected in R×D by a future–pointinglightlike geodesicminimizing the (future) arrival time T (⇔ idem for past)

(64)

Results: timelike geodesics

R×M standard stationary,non-necessarily normalized (−Λdt²), product metric forRu ≡(R,du²) and (R×M,g_L), Π_u projection on the first factor, in addition to Π_M. Fermat F_Λ on R×M:

F_Λ= s

Π^∗_Mh+ Π^∗_udu²

Λ◦Π_M + Π^∗_Mω=p

h_Λ+ω₁.

(extra dimension plus non-conformal invariance)

(65)

Results: timelike geodesics

Theorem

Assume that allB¯_s^D^¯(p,r) are compact(which happens, in particular, whenB¯_s(p,r)∩D are compact¯ ). Then:

(Ru×∂D;F_Λ) is convex⇔

for anylength l >0,each point w ∈R×D and line lq are joined by a future (and a past) pointing timelike geodesic in R×D,with length l minimizing the arrival time (among causal curves of length l ).

In this case, if D is not contractible,a sequence of such connecting geodesics with diverging arrival times exists.

(66)

Further results: asympt. flat spacetimes

Asymptotically flat stationary spacetime:

(M,g) complete, outside a compact subset C, is

diffeomorphic to Rⁿ\B(0,R0), elementsg, ω,Λ turning Euclidean with large radial coordinater.

Model isolated systems -gravity outside a star.

Makes sense to speak on (stationary) large balls B(0,R)×R⊂R×M and spheres.

Application: Large spheres in asympt. flat spacetimes are always light-convex(and all the previous results are applicable) but, typically,(including reasonable matter, when gravity attracts) they are not time-convex

(67)

Further results: asympt. flat spacetimes

Asymptotically flat stationary spacetime:

(M,g) complete, outside a compact subset C, is

diffeomorphic to Rⁿ\B(0,R0), elementsg, ω,Λ turning Euclidean with large radial coordinater.

Model isolated systems -gravity outside a star.

Makes sense to speak on (stationary) large balls B(0,R)×R⊂R×M and spheres.

Application: Large spheres in asympt. flat spacetimes are always light-convex(and all the previous results are applicable) but, typically,(including reasonable matter, when gravity attracts) they are not time-convex

(68)

Riemannian boundaries Finslerian boundaries C-boundary on stationary s-t

THIRD PART

3. CAUSAL BOUNDARY

(69)

Introduction

Causal boundary∂V of a spacetimeV:

Involved structure: conformal structure (Causality).

Intrinsic alternative to common Penrose conformal boundary applicable to any strongly causal spacetime

Purpose: attach a boundary endpoint P ∈∂V to any inextensible future or past directed timelike curve γ

(70)

Introduction

Basic idea: the boundary point would be represented by P =I⁻(γ) or F =I⁺(γ) or, more precisely, a pair (P,F).

(71)

Introduction

Long story from Geroch, Kronheimer & Penrose ’72 until its recent redefinition Flores, Herrera & — ’11 (with contributions by many authors: Budic & Sachs ’74, Szabados ’88, ’89, Harris ’97-’07, Marolf & Ross ’03...):

As a point set,the completion V is composed by(S related) pairs (P,F) (“(IP,IF)”, for example

V 3p≡(I⁺(p),I⁻(p))∈V)

Chronological relation: (P,F)(P⁰,F⁰)⇔F ∩P⁰6=∅

(72)

Introduction

Subtle topology... non always Hausdorff

(73)

Introduction

When computed for standard stationary spacetimes, relations with Cauchy boundary

Gromov boundary

Busemann-type boundary for Randers manifolds:

previous study for any Riemannian and Finslerian manifold with interest by itself

(74)

Introduction

When computed for standard stationary spacetimes, relations with Cauchy boundary

Gromov boundary

Busemann-type boundary for Randers manifolds:

previous study for any Riemannian and Finslerian manifold with interest by itself

(75)

Cauchy boundary for Riemannian manifolds

Remark: it may be non-locally compact

(76)

Gromov boundary for Riemannian manifolds

Classical Gromov’s compactification forcompleteRiemannian manifolds(Gromov ’81)

L₁(M,g) 1-Lipschitz functions (pointwise topology)

x ∈M can be seen in L₁(M,g) as d_x :y 7→d(x,y) and also dx+C for anyC ∈R

f ∼f⁰ ⇐⇒ f −f⁰=constant (quotient topology)

eachx ∈M is represented in L₁(M,g)/∼as the class of −d_x MG = closure ofM in L₁(M,g)/∼