aa r X i v : . [ m a t h . S G ] F e b A LORENTZIAN DISTANCE FUNCTION FOR POSITIVECONTACTOMORPHISMS
JAKOB HEDICKE
Abstract.
We define a Lorentzian distance function on the group of contac-tomorphisms of a closed contact manifold compatible with the relation givenby positivity. This distance function is continuous with respect to the Hofernorm on the group of contactomorphisms defined by Shelukhin ([She17]) andfinite if and only if the group of contactomorphisms is orderable. To provethis we show that intervals defined by the positivity relation are open withrespect to the topology induced by the Hofer norm. For orderable Legendrianisotopy classes we show that the Chekanov-type metric defined in [RZ18] isnon-degenerate. In this case similar results hold for a Lorentzian distancefunction on Legendrian isotopy classes. This leads to a natural class of metricsassociated to a globally hyperbolic Lorentzian manifold such that its Cauchyhypersurface has a unit co-tangent bundle with orderable isotopy class of thefibres. Introduction
Consider a closed co-oriented contact manifold ( M, ξ ) , i.e. an (2 n +1) -dimensionalsmooth manifold M with a hyperplane distribution ξ ⊂ T M that is the kernel of a -form α such that α ∧ dα n is nowhere vanishing. Denote by Cont ( M ) the identitycomponent of the group of contactomorphisms, i.e. the group of diffeomorphismspreserving ξ that are isotopic to id M through contactomorphisms. Fore details seee.g. [Gei08].In [EP00] Eliashberg and Polterovich introduced the concept of positivity on Cont ( M ) . An isotopy ϕ t of contactomorphisms is called positive if the contactvector field X ϕt ◦ ϕ t := dds | s = t ϕ s satisfies α ( X ϕt ) > for all t . Note that this definition only depends on the co-orientation defined by the contact form α . Similarly ϕ t is called non-negative if α ( X ϕt ) ≥ . This induces two relations on Cont ( M ) by ϕ Î ψ : ⇔ there exists a positive isotopy from ϕ to ψ and ϕ ψ : ⇔ there exists a non-negative isotopy from ϕ to ψ. The relations Î and turn Cont ( M ) into a causal space (see [KS18]).The properties of the relations on Cont ( M ) resemble properties of the chrono-logical and causal relation in Lorentzian geometry. Let ( N, g ) be a smooth time-oriented Lorentzian manifold, that is a manifold N with a smooth pseudo-Riemannianmetric of signature (1 , n ) and the choice of a vector field X such that g ( X, X ) < .A tangent vector v ∈ T N is called future pointing timelike if g ( v, v ) < and g ( v, X ) < and future pointing causal if g ( v, v ) ≤ and g ( v, X ) < . A smooth Date : February 26, 2021.This research is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebraand Dynamics”, funded by the Deutsche Forschungsgemeinschaft (Projektnummer 281071066 –TRR 191). curve γ ( t ) is called future pointing timelike (causal) if γ ′ ( t ) is future pointing time-like (causal) for all t . This induces two relations on N . The chronological relation p ≪ q : ⇔ there exists a future pointing timelike curve from p to q and the causal relation p ≤ q : ⇔ there exists a future pointing causal curve from p to q. Most causal properties of ( N, g ) can be phrased in terms of the chronological(causal) future/past of the points in N : I + ( p ) := { q ∈ N | p ≪ q } , I − ( p ) := { q ∈ N | q ≪ p } J + ( p ) := { q ∈ N | p ≤ q } , J − ( p ) := { q ∈ N | q ≤ p } . A natural topology on N related to g is the interval (Alexandrov) topology whosebasis is given by the open ’intervals’ I + ( p ) ∩ I − ( q ) of the chronological relation.The interval topology coincides with the manifold topology of N iff the Lorentzianmanifold is strongly causal , i.e. iff for every open U ⊂ N there exists a causallyconvex open V ⊂ U ([MS08]).Analogously to the length of a curve γ in a Riemannian manifold, one can definea Lorentzian length (eigentime) for causal curves by L g ( γ ) := b Z a p − g ( γ ′ ( t ) , γ ′ ( t )) dt. The
Lorentzian distance between two points is then defined by τ g : N × N → [0 , ∞ ]( p, q ) (cid:26) sup { L g ( γ ) } , if p ≤ q , otherwise . Here the supremum is taken over all future pointing causal curves connecting p and q . The Lorentzian distance satisfies (see [BEE96])(i) τ g ( p, q ) > if and only if p ≪ q .(ii) τ g is lower semi-continuous.(iii) τ g ( p, q ) ≥ τ g ( p, r ) + τ g ( r, q ) for p ≤ r ≤ q .If ( N, g ) is strongly causal the metric g can be recovered from τ g and the Lorentziandistance reflects many causal properties of ( N, g ) .A strongly causal time oriented Lorentzian manifold ( N, g ) is called globallyhyperbolic if J + ( p ) ∩ J − ( q ) is compact for all p, q ∈ N . In this case N containsa smooth Cauchy hypersurface Σ , i.e. a surface that is intersected in a uniquepoint by every inextendible causal curve. In particular N is diffeomorphic to R × Σ ([MS08]). Theorem ([BEE96, Corollary 4.7.]) . If ( N, g ) is globally hyperbolic, then τ g is finiteand continuous. Theorem ([BEE96, Theorem 4.30.]) . A strongly causal manifold ( N, g ) is globallyhyperbolic if and only if τ ˜ g is finite for all ˜ g in the conformal class of g , i.e. for all ˜ g with ˜ g = e f g for some smooth function f . As pointed out for instance in [KS18] Lorentzian distance functions can be usedto explore the causal properties of more general causal spaces. There the authorsconsider causal spaces and functions satisfying the properties (i)-(iii) (here lowersemi-continuity is with respect to some background metric). For closed ( M, ξ ) wewill define a Lorentzian distance function τ α for the relations Î and inspired ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 3 by Shelukhins definition of the Hofer norm on
Cont ( M ) . This Lorentzian dis-tance depends on the choice of a contact form α . It is not conjugation invariantbut continuous with respect to the Hofer-Shelukhin norm and shares some of itsproperties.Analogously to the relations on Cont ( M ) , positivity defines two relations onthe Legendrian isotopy classes of ( M, ξ ) (see e.g. [CN16], [CFP17] for details). InSection 6 we will define a Lorentzian distance function for isotopy classes of compactLegendrians. In the case of orderable isotopy classes this distance is continuous withrespect to a Chekanov-type metric defined by Rosen and Zhang in [RZ18].As observed in [Low06] a unit co-tangent bundle ST ∗ Σ is contactomorphic to thespace of null geodesics of globally hyperbolic manifolds with Cauchy hypersurface Σ . If the isotopy class of the fibre of the unit co-tangent bundle of the Cauchysurface is orderable, the points in the Lorentzian manifold can be identified withLegendrians isotopic to the fibres such that the identification respects the relationson the manifold and the Legendrian isotopy class (see [CN16, CN19]). Hence theLorentzian distance and the Chekanov-type metric on the Legendrian isotopy classinduce a Lorentzian distance and a metric on the Lorentzian manifold. Acknowledgements.
I am grateful to Alberto Abbondandolo, Stefan Nemirovski, Daniel Rosen andStefan Suhr for useful comments and discussions.2.
Main Results
Let ( M, ξ ) be a closed co-oriented contact manifold. Given a contact form α Shelukhin [She17] defined a non-conjugation invariant norm on
Cont ( M ) by | ϕ | α := inf Z max M | α ( X ϕt ) | dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ t isotopy with ϕ = id M , ϕ = ϕ . The norm | · | α has the following properties Theorem ([She17]) . The norm | · | α satisfies (i) | ϕ | α = 0 ⇔ ϕ = id M . (ii) | ϕψ | α ≤ | ϕ | α + | ψ | α . (iii) | ϕ − | α = | ϕ | α . (iv) | ψϕψ − | α = | ϕ | ψ ∗ α . Remark.
In the following we will work with the metric d α ( ϕ, ψ ) := | ψ − ϕ | α . Notethat the metrics induced by two different contact forms are equivalent ( [She17] ).Therefore they induce the same topology on Cont ( M ) . For a contactomorphism ϕ ∈ Cont ( M ) define the sets I + ( ϕ ) , I − ( ϕ ) , J + ( ϕ ) and J − ( ϕ ) with respect to the relations Î and analogously to the Lorentzian case. Anatural topology on Cont ( M ) introduced in [CN19] is then given by the intervaltopology, i.e. the topology induced by sets of the form I + ( ϕ ) ∩ I − ( ψ ) . Proposition 2.1.
The interval topology is coarser than the topology induced by d α ,in particular the sets I ± ( ϕ ) are open with respect to d α . Definition 2.2.
Given a contact form α on a closed contact manifold ( M, ξ ) define τ α : Cont ( M ) × Cont ( M ) → [0 , ∞ ] by τ α ( ϕ, ψ ) := sup (cid:26) R min M α ( X ϕt ) dt (cid:27) , if ϕ ψ , otherwise . Here the supremum is taken over all non-negative paths ϕ t with ϕ = ϕ and ϕ = ψ . JAKOB HEDICKE
Like the metric d α the function τ α fails to be left invariant. In [BIP +
08] it isshown that any conjugation invariant norm on
Cont ( M ) is discrete in the sensethat any contactomorphism that is not the identity has norm greater than somepositive constant. Conjecture.
Let τ : Cont ( M ) × Cont ( M ) → [0 , ∞ ] be a map satisfying τ ( ϕ, ψ ) > iff ϕ Î ψ and τ ( ϕ , ϕ ) ≥ τ ( ϕ , ψ ) + τ ( ψ, ϕ ) for ϕ ψ ϕ . Assume that τ is lower semi-continuous with respect to the interval topology and conjugationinvariant. Then τ ( ϕ, ψ ) = (cid:26) ∞ , if ϕ Î ψ , otherwise . In section 5 we will prove a version of this conjecture for Legendrian isotopyclasses.
Theorem 2.3.
The map τ α satisfies (i) τ α ( ϕ, ψ ) > ⇔ ϕ Î ψ . (ii) τ α is continuous with respect to the interval topology and the topology in-duced by d α . (iii) τ α ( ϕ , ϕ ) ≥ τ α ( ϕ , ψ ) + τ α ( ψ, ϕ ) for ϕ ψ ϕ . (iv) τ α ( ψϕ , ψϕ ) = τ ψ ∗ α ( ϕ , ϕ ) . Question 1.
Theorem 2.3 shows that (Cont ( M ) , d α , τ α ) is a Lorentzian pre-lengthspace in the sense of [KS18] . Are there contact manifolds such that (Cont ( M ) , d α , τ α ) has the structure of a Lorentzian length space? To answer the question one would need to show the local existence of non-negative paths maximizing the integral in Definition 2.2, i.e. the existence of pathsconnecting two d α -close contactomorphisms ϕ ψ so that their ’Lorentzian length’coincides with τ α ( ϕ, ψ ) .The following result is the key Lemma to prove Theorem 2.3 and many otherresults. Lemma 2.4.
Let ϕ t be a path of contactomorphisms with Z min M α ( X ϕt ) dt = ǫ. Then for any δ > there exists a path ψ t with ψ = ϕ and ψ = ϕ such that forall t ∈ [0 ,
1] min M α ( X ψt ) ∈ ( ǫ − δ, ǫ + δ ) . Call
Cont ( M ) orderable if defines a partial order on Cont ( M ) , i.e. if thereare no non-negative loops. By [EP00, Proposition 2.1.B] this is equivalent to thenon existence of a positive loop connecting id M to itself. Note that the proof in[EP00] is given for contractible non-negative loops but works analogously for noncontractible loops.The following Corollary of Lemma 2.4 answers [She17, Question 18] if Cont ( M ) is orderable. Corollary 2.5.
Suppose that ( M, ξ ) is closed and Cont ( M ) orderable. Denote by ϕ αt the Reeb flow of a contact form α . Then | ϕ αt | α = | t | , in particular sup ϕ ∈ Cont ( M ) | ϕ | α = ∞ . Similarly one can prove
ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 5
Corollary 2.6.
Let ( M, ξ ) be a closed contact manifold. The group Cont ( M ) isorderable if and only if τ α ( ϕ, ψ ) < ∞ for all ϕ, ψ ∈ Cont ( M ) . In this case, if ϕ αt denotes the Reeb-flow with respect to α , then τ α ( ϕ, ϕ αt ϕ ) = t for all t ≥ and all ϕ ∈ Cont ( M ) . Remark.
The fact that orderability is equivalent to the non existence of positiveloops and to the finiteness of τ α is a huge difference to Lorentzian geometry. In [MS08] there are given examples of manifolds that contain lightlike loops but notimelike loops. Moreover there are many examples of strongly causal spacetimessuch that the Lorentzian distance is not finite (see e.g. [BEE96, Theorem 4.30.] ). For < ǫ < ∞ consider the sets B + α ( ϕ, ǫ ) := { ψ ∈ Cont ( M ) | < τ α ( ϕ, ψ ) < ǫ } ,B − α ( ϕ, ǫ ) := { ψ ∈ Cont ( M ) | < τ α ( ψ, ϕ ) < ǫ } . Corollary 2.7.
The sets B + α ( ϕ, ǫ ) ∩ B − α ( ψ, ǫ ) form a basis of the interval topology.Proof. Theorem 2.3 implies that τ α is continuous with respect to the interval topol-ogy. Then the proof works analogous to the case of a strongly causal Lorentzianmanifold ( M, g ) such that τ g is continuous (see [BEE96, Proposition 4.31.]). (cid:3) Now consider the case when M is a spherical co-tangent bundle, i.e. M is thequotient bundle T ∗ N \ { } / R > for some smooth manifold N . Here R > actson T ∗ N \ { } by positive fibrewise homotheties. The spherical co-tangent bundlenaturally carries a contact structure ξ st : Consider the canonical Liouville form λ on T ∗ N . The R > -action induces a Liouville vector field Y on T ∗ N \ { } , i.e. avector field that satisfies dλ ( Y, · ) = λ . Since Y is tangent to the R > -action, thekernel of λ projects to a contact structure ξ st on the quotient (see [Gei08]).Using estimates for spectral invariants of Legendrians on jet-spaces ([Zap12]) wewill in section 4 show the following estimate for τ α : Theorem 2.8.
Assume that ( M, ξ ) ∼ = ( ST ∗ N, ξ st ) , where N is closed and smoothlycovered by an open subset of R n . Then there exists a constant C α depending on α such that for all ϕ, ψ ∈ Cont ( M ) τ α ( ϕ, ψ ) ≤ C α d α ( ϕ, ψ ) . Remark.
In general the interval topology is strictly coarser than the topology in-duced by d α . If Cont ( M ) is not orderable it is in general not even Hausdorff. Withthe assumptions of Theorem 2.8 (except for N being closed) [CN19] used similarmethods to show that the interval topology on Cont ( M ) is Hausdorff. Question 2.
Assume that
Cont ( M ) is orderable. Is Cont ( M ) then stronglycausal in the sense that the interval topology coincides with the topology inducedby d α ? In view of the results in [CN19] one might hope to use methods based on spectralinvariants to answer the question positively for manifolds satisfying the assumptionsof Theorem 2.8.
Question 3.
As pointed out above in Lorentzian geometry the strongly causal man-ifolds with a Lorentzian distance finite and continuous on the conformal class areglobally hyperbolic. Do Theorem 2.3 and Corollary 2.6 imply that if
Cont ( M ) is or-derable other properties of globally hyperbolic manifolds like the compactness of the’closed intervals’ of or the existence of Cauchy surfaces transfer to Cont ( M ) ? JAKOB HEDICKE Proofs
Proof of Lemma 2.4.
Let τ : [0 , → R be defined by τ (0) = 0 and τ ′ ( t ) = − min M α ( X ϕt ) + ǫ. Note that τ ′ is continuous since α ( X ϕt ) is smooth, i.e. τ ( t ) is C . Denote by ϕ αt theReeb-flow of the contact form α . Define ˜ ψ t := ϕ ατ ( t ) ◦ ϕ t . Then ˜ ψ t is a C path ofcontactomorphisms with ˜ ψ = ϕ and ˜ ψ = ϕ since τ (0) = 0 and τ (1) = − Z min M α ( X ϕt ) dt + ǫ = 0 . Moreover min M α ( X ˜ ψt ) = min M α ( X ϕt ) + τ ′ ( t ) = ǫ. Here we used that ϕ ατ ( t ) is a strict contactomorphism for all t , i.e. ϕ ατ ( t ) ∗ α = α .Approximating τ with smooth functions with fixed endpoints we get ψ t with thedesired properties. (cid:3) Proof of Proposition 2.1.
We show that I + ( ϕ ) is open with respect to d α . Theproof works analogously for I − ( ϕ ) . Let ψ ∈ I + ( ϕ ) and ϕ t be a positive path with ϕ = ϕ and ϕ = ψ . Choose ǫ > such that min M α ( X ϕt ) > ǫ . There exists a smoothfamily of positive functions ρ t such that ( ϕ t ◦ ψ − ) ∗ α = ρ t α . Define α := max [0 , × M ρ t α. Take ˜ ψ with d α ( ψ, ˜ ψ ) < ǫ . By Lemma 2.4 we can choose a path ψ t with ψ = ψ and ψ = ˜ ψ such that min M α ( X ψt ) > − ǫ . Define ˜ ϕ t := ϕ t ◦ ψ − ◦ ψ t . Then ˜ ϕ = ϕ and ˜ ϕ = ˜ ψ . Moreover min M α ( X ˜ ϕt ) ≥ min M ( ϕ t ψ − ) ∗ α ( X ψt ) + min M α ( X ϕt ) ≥ min(0 , min M α ( X ψt )) + min M α ( X ϕt ) > . Hence ˜ ϕ t is a positive path from ϕ to ˜ ψ , i.e. ˜ ψ ∈ I + ( ϕ ) . It follows that I + ( ϕ ) isopen with respect to d α and thus with respect to any d α since these metrics areequivalent. (cid:3) Proof of Theorem 2.3. (i) By definition ϕ Î ψ ⇒ τ α ( ϕ, ψ ) > . If τ α ( ϕ, ψ ) > there exists a path ϕ t between ϕ and ψ with ǫ := R min M α ( X ϕt ) dt > .Lemma 2.4 implies the existence of a positive path between ϕ and ψ bysetting e.g. δ = ǫ .(ii) We show that τ α is continuous with respect to the interval topology. Thenthe claim follows by Proposition 2.1. Let ( ϕ, ψ ) ∈ τ − α (( a, b )) , where ( a, b ) ⊂ [0 , ∞ ] is an open interval or ( a, b ) = [0 , b ) or ( a, b ) = ( a, ∞ ] . Denote by ϕ αt the Reeb flow of α . Claim 1:
There exists r > such that τ α ( ϕ αt ϕ, ϕ αs ψ ) ∈ ( a, b ) for all s, t ∈ [ − r, r ] .Suppose not.
1. Case:
Assume b < ∞ and for arbitrarily small r there exist t , s anda path ψ t with ψ = ϕ αt ϕ and ψ = ϕ αs ψ such that R min M α ( X ψt ) dt ≥ b . ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 7
Due to Lemma 2.4 one can assume that min M α ( X ψt ) > b − δ for δ arbi-trarily small. Let τ ( t ) := ( t − t − ts and ˜ ψ t := ϕ ατ ( t ) ψ t . Then ˜ ψ = ϕ , ˜ ψ = ψ and since ϕ ατ ( t ) is a strict contactomorphism min M α ( X ˜ ψt ) ≥ min M α ( X ψt ) + ( s − t ) > b − r − δ. Since we can choose r and δ arbitrarily small this contradicts ( ϕ, ψ ) ∈ τ − α (( a, b )) .
2. Case:
Assume a > and for arbitrarily small r there exist t , s with τ α ( ϕ αt ϕ, ϕ αs ψ ) ≤ a .There exists a path ψ t with ψ = ϕ , ψ = ψ and min M α ( X ψt ) > a for all t . Using the same argument like in Case 1 for small δ one can construct apath ˜ ψ t between ϕ αt ϕ and ϕ αs ψ with R min M α ( X ˜ ψt ) dt > a .One can assume that always one of the two cases holds. If b = ∞ every t, s satisfies τ α ( ϕ αt ϕ, ϕ αs ψ ) ≤ b . Then either a = 0 , i.e. τ − α (( a, b )) =Cont ( M ) × Cont ( M ) or the second case above holds. Similarly if a = 0 every t, s satisfies τ α ( ϕ αt ϕ, ψ αs ϕ ) ≥ a , Then either b = ∞ or the first caseabove holds.This shows Claim 1. Claim 2:
Choose r > like in Claim 1. Then ( I + ( ϕ α − r ϕ ) ∩ I − ( ϕ αr ϕ )) × ( I + ( ϕ α − r ψ ) ∩ I − ( ϕ αr ψ )) ⊂ τ − α ( a, b ) .W.l.o.g. assume a > and b < ∞ . Take ˜ ϕ ∈ I + ( ϕ α − r ϕ ) ∩ I − ( ϕ αr ϕ ) and ˜ ψ ∈ I + ( ϕ α − r ψ ) ∩ I − ( ϕ αr ψ ) . There exists a positive path ψ t with ψ = ˜ ϕ , ψ = ϕ αr ϕ , ψ = ϕ α − r ψ and ψ = ˜ ψ such that Z min M α ( X ψt ) dt > a. Here we used that R min M α ( X ψt ) dt is invariant under smooth orientationpreserving re-parametrisation of ψ t with respect to t , i.e. one can choose ψ t such that Z min M α ( X ψt ) dt = τ α ( ϕ αr ϕ, ϕ α − r ψ ) − δ > a for δ arbitrarily small. On the other hand, the existence of a path ψ t be-tween ˜ ϕ and ˜ ψ with R min M α ( X ψt ) dt ≥ b would imply that τ α ( ϕ α − r ϕ, ϕ αr ψ ) >b . This shows Claim 2, in particular, τ α is continuous with respect to theinterval topology.(iii) Assume id M ψ ϕ . Take non-negative paths ψ t from id M to ψ and ˜ ψ t from id M to ϕ . Following [She17] take smooth functions τ : [0 , → [0 , and τ : [0 , → [0 , with supp( τ ′ ) ⊂ [0 , ] , supp( τ ′ ) ⊂ [ , and τ ′ i ≥ such that τ (0) = 0 = τ (0) and τ (1) = 1 = τ (1) . Then ˆ ψ t := ψ τ ( t ) ˜ ψ τ ( t ) defines a smooth path between id M and ψϕ with Z min M α ( X ˆ ψ t t ) dt ≥ Z min M α ( X ψt ) dt + Z min M α ( X ˜ ψt ) dt. JAKOB HEDICKE
It follows that τ α ( id M , ψϕ ) ≥ τ α ( id M , ψ ) + τ α ( id M , ϕ ) . Then property (iii) follows from the right invariance of τ α .(iv) Analogous to property (iv) in [She17] . (cid:3) Proof of Corollary 2.5.
Assume that | ϕ α − t | α < t , where t > . Then there exists apath ψ s with ψ = id M and ψ = ϕ α − t such that R max M | α ( X ψs ) | ds < t . By Lemma2.4 we can assume that min M α ( X ψs ) > − t for all s . Define ˜ ψ s := ϕ αts ψ s . The path ˜ ψ s is a loop with ˜ ψ = id M = ˜ ψ . Moreover since ϕ αts is a strict contactomorphismone has min M α ( X ˜ ψs ) ≥ min M α ( X ψs ) + t > . This contradicts the orderability of ( M, ξ ) . The corollary follows from the fact that | ϕ αt | α = | ϕ α − t | α . (cid:3) Proof of Corollary 2.6. If Cont ( M ) is not orderable one can choose a positive loop ϕ t ([EP00, Proposition 2.1.B]). Iterating this loop, we get τ α ( ϕ , ϕ ) = ∞ . Let Cont ( M ) be orderable, i.e. there is no non-negative loop in Cont ( M ) . Take ϕ ψ . Assume that τ α ( ϕ, ψ ) = ∞ . Hence for any c > there exists a positivepath ϕ t from ϕ to ψ with R min M α ( X ϕt ) dt > c . Due to Lemma 2.4 one can assumethat min M α ( X ϕt ) > c for all t . Fix a path ψ t from id M to ϕ ◦ ψ − . Define ˜ ϕ t := ψ t ◦ ϕ t . Then ˜ ϕ = ϕ = ˜ ϕ . Let ρ t be the family of positive functions with ψ ∗ t α = ρ t α and α := ρ min α , where ρ min := min [0 , × M ρ t . Then min M α ( X ˜ ϕt ) ≥ min M ψ ∗ t α ( X ϕt ) + min M α ( X ψt ) ≥ min M α ( X ϕt ) + min M α ( X ψt ) . Since τ α ( ϕ, ψ ) = ∞ implies that τ α ( ϕ, ψ ) = ∞ and c was arbitrarily big, one canchoose ϕ t such that min M α ( X ˜ ϕt ) > . Then ˜ ϕ is a positive loop. This contradicts the orderability of ( M, ξ ) .Similarly if Cont ( M ) is orderable, consider the Reeb flow ϕ αt of α . Assume that τ α ( ϕ, ϕ αt ϕ ) > t , where t ≥ . Take a path ψ s from ϕ to ϕ αt ϕ with min M α ( X ϕs ) > t for all s ∈ [0 , . Then ϕ α − ts ψ s defines a positive loop contradicting the orderabilityof ( M, ξ ) . (cid:3) Spherical co-tangent bundles and spectral invariants
To prove Theorem 2.8 we will use spectral invariants on jet-spaces as defined forinstance in [CFP17], [CN10a] or [CN19]. We will follow the conventions in [Zap12].Let N be a closed manifold. Its -jet space is given by J N := T ∗ N × R . The space J N canonically carries the contact structure defined by the -form α := λ + dt , where λ denotes the canonical Liouville-form on T ∗ M and t the ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 9 projection to the R -component. An important example of Legendrian submanifoldsare given by -jets of functions. For f : N → R define its -jet by j f := { ( p, − df p , f ( p )) ∈ J N | p ∈ N } . In particular the -section of J N as the -jet of the -function is a Legendriansubmanifold. Note that all -jets are Legendrian isotopic to the -section. On theother hand not all Legendrians in this isotopy class are -jets of some function. Aneasy way to describe them is given by generating functions:For some m ∈ N consider a smooth function S : N × R m → R . Look at theset Σ S := { ( p, e ) ∈ N × R m | d e S = 0 } , where d e S denotes the fibre differential inthe R m direction. If is a regular value of d e S , Σ S is a submanifold of N × R m .Moreover the map i S ( p, e ) := ( p, − d p S ( p, e ) , S ( p, e )) immerses Σ S into J ( N ) as aLegendrian submanifold. Here d p S denotes the differential in the N direction whichis well defined (independent of the choice of a horizontal distribution) on Σ S . Thefunction S is called a generating function for the Legendrian i S (Σ S ) . We say that S is quadratic at infinity if S is of the form S ( p, e ) = f ( p, e ) + Q ( e ) , where f is compactly supported and Q is a non-degenerate quadratic form. ForLegendrians isotopic to the -section one can show Theorem ([Che96]) . Let L ⊂ J N be a Legendrian submanifold Legendrian iso-topic to the -section. Then there exists a quadratic at infinity generating function S generating L . Moreover if L t is a smooth isotopy of Legendrians isotopic to ,there exists a smooth family of quadratic at infinity generating functions S t suchthat S t generates L t . Generating functions can be used to define spectral invariants for Legendriansubmanifolds. Let S be a quadratic at infinity generating function for a Legendriansubmanifold L . Since S = f + Q and the function f has compact support, thereexist a , b ∈ R such that H ∗ ( { S < b } , { S < a } , Z ) ∼ = H ∗ ( { S < b } , { S < a } , Z ) for all a ≤ a < b ≤ b . Let q be the (negative) index of Q . There exists a naturalgraded isomorphism i : H ∗ ( { S < b } , { S < a } , Z ) → H ∗− q ( N, Z ) that is independent of a ≤ a and b ≥ b (see [Zap12]). Thus for a ≤ a and any b > a there is a natural inclusion i b : H ∗ ( { S < b } , { S < a } , Z ) → H ∗− q ( N, Z ) .For A ∈ H ∗ ( N, Z ) define l ( L, A ) := inf { b ∈ R | A ∈ im( i b ) } . Due to the Viterbo-Theret uniqueness theorem [Vit92],[Thé99] the spectral invari-ant l ( L, A ) is independent of the choice of a generating function. We will use thefollowing Lemma from Zapolsky, for further properties of the spectral invariantssee e.g. [Zap12]. Lemma 4.1 ([Zap12]) . Let L ⊂ J N be a Legendrian submanifold isotopic to and ϕ t a path of contactomorphisms such that ϕ = id M and ϕ (0) = L . Then forany A ∈ H ∗ ( N ) Z min ϕ t (0) α ( X ϕt ) dt ≤ l ( L, A ) ≤ Z max ϕ t (0) α ( X ϕt ) dt. Proof of Theorem 2.8.
Assume that M = ST ∗ N , where N is smoothly coveredby an open subset of R n . W.l.o.g this open subset contains the origin. Considera positive path of contactomorphisms ϕ t in Cont ( M ) . Note that M is covered by ST ∗ R n such that the projection map is a local contactomorphism. Hence ϕ t lifts to a positive path ˜ ϕ t on ST ∗ R n . Since τ α and d α are right invariant wecan assume that ϕ = id M . As in [CN10a] we use the hodograph transform toget a contactomorphism from ST ∗ R n to J ( S n − ) such that the fibre over theorigin is mapped to the -section in J ( S n − ) . In particular the spectral invariantscan be defined for Legendrians in ST ∗ R n isotopic to the fibres. Denote by ˜ α thecontact form on M induced by the standard contact form on J ( S n − ) and by L aLegendrian that lifts to the fibre over the origin. Then using Lemma 4.1 we get Z min M ˜ α ( X ϕt ) dt ≤ Z min ϕ t ( L ) ˜ α ( X ϕt ) dt ≤ l ( ϕ ( L ) , A ) ≤ Z max ϕ t ( L ) ˜ α ( X ϕt ) dt ≤ Z max M | ˜ α ( X ϕt ) | dt. Note that the spectral invariant l ( ϕ ( L ) , A ) is independent of the choice of pathbetween id M and ϕ . Taking the supremum over all paths from id M to ϕ on theleft hand side and the infimum over all paths on the right hand side shows τ ˜ α ( ϕ , ϕ ) ≤ d ˜ α ( ϕ , ϕ ) . For α = ρ ˜ α one gets τ α ( ϕ , ϕ ) ≤ max ρ min ρ d α ( ϕ , ϕ ) . (cid:3) A Lorentzian distance function for Legendrian isotopy classes
Let ( M, ξ ) be any co-orientable contact manifold (not necessarily closed). For aclosed Legendrian denote by L its Legendrian isotopy class. Given a parametrisa-tion l t : L → L t of a Legendrian isotopy denote by X lt the section of T M | L t definedby X lt ( l t ( p )) := dds | s = t l s ( p ) . A Legendrian isotopy L t is called positive (non-negative) if there is a parametri-sation l t : L → L t such that α ( X lt ) > ( ≥ )0 . Given a positive (non-negative) isotopy L t of closed Legendrians, there exists acompactly supported non-negative path of contactomorphisms ϕ t such that L t = ϕ f ( t ) ( L ) for a non-decreasing function f (see [CN16, Proposition 4.1]). Analogousto the relations on Cont ( M ) positivity defines relations Î and on L . The isotopyclass L is called orderable if is a partial order.There also holds a version of Lemma 2.4 for Legendrian isotopy classes. Lemma 5.1.
Let l t : L → L t be a Legendrian isotopy with Z min L t α ( X lt ) dt = ǫ. Assume that l t = ϕ t | L for some compactly supported path of contactomorphisms ϕ t . Then for all δ > there exists ˜ l t : L → ˜ L t with ˜ L = L such that min L t α ( X ˜ lt ) ∈ ( ǫ − δ, ǫ + δ ) . ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 11
Proof.
Take a compactly supported path of contactomorphisms such that ϕ t | L = l t . Let τ : [0 , → R with τ (0) = 0 and τ ′ ( t ) := − min L t α ( X lt ) + ǫ. Then ψ t := ϕ ατ ( t ) ϕ t satisfies ψ ( L ) = L and ψ ( L ) = L . Moreover since ϕ ατ ( t ) isa strict contactomorphism min ψ t ( L ) α ( X ψt ) = min L t α ( X lt ) + τ ′ ( t ) = ǫ. Approximating τ by smooth functions with fixed endpoints finally gives ˜ l t with thedesired properties. (cid:3) For closed M Rosen and Zhang in [RZ18] defined a Chekanov-type metric onthe orbit space of a subset M under the action of Cont ( M ) . For Legendrians L , L ∈ L it is given by d α ( L , L ) := inf {| ϕ | α | ϕ ( L ) = L } . The map d α has the following properties: Theorem ([RZ18]) . The map d α satisfies (i) d α ( L, L ) = 0 (ii) d α ( L , L ) = d α ( L , L ) (iii) d α ( L , L ) ≤ d α ( L , L ) + d α ( L , L ) (iv) For ϕ ∈ Cont ( M ) generated by a contact Hamiltonian F there exist con-stants C − ( ϕ, F ) and C + ( ϕ, F ) such that C − ( ϕ, F ) d α ( L , L ) ≤ d α ( ϕ ( L ) , ϕ ( L )) ≤ C + ( ϕ, F ) d α ( L , L ) . Remark.
Due to the non-conjugation invariance of the Hofer-Shelukhin norm thismetric fails to be left invariant, i.e. in general d α ( L , L ) = d α ( ϕ ( L ) , ϕ ( L )) . For closed Legendrians [RZ18, Conjecture 1.10] states that d α is always non-degenerate. The conjecture was proven in [Ush20, Corollary 3.5] for hypertightLegendrians. A simple proof of [RZ18, Conjecture 1.10] can be given for orderableLegendrian isotopy classes. Theorem 5.2.
Let L be a closed Legendrian such that L is orderable. Then d α isnon-degenerate.Proof. By [RZ18, Theorem 1.9] d α is either non-degenerate or vanishes identically.Thus it suffices to show that d α ( L , L ) > for two Legendrians L , L ∈ L . Lookat the inverse of the time- map of the Reeb-flow ϕ α − . Suppose d α ( L, ϕ α − ( L )) = 0 for some L ∈ L . Then for any ǫ > there exists a compactly supported ϕ t with ϕ = id M and ϕ ( L ) = ϕ α − ( L ) such that Z max M | α ( X ϕt ) | dt < ǫ. Look at ψ t := ϕ αt ϕ t . Then ψ ( L ) = ψ ( L ) = L . Moreover since ϕ αt is a strictcontactomorphism Z min M α ( X ψt ) dt ≥ Z min M α ( X ϕt ) dt ≥ − ǫ > for ǫ < . Lemma 5.1 implies that there exists a positive loop in L . (cid:3) Remark. In [CCDR19, Example 1.12] the authors give an example of a hyper-tight Legendrian contained in a (non-contractible) positive loop, i.e. there are non-orderable hypertight Legendrians. On the other hand let M be a closed manifoldsuch that its universal cover is not compact and π k ( M ) = 0 for some k > . Let i : M → Λ M be the map that maps a point p in M to the constant loop through p in the component of contractible loops of the free loop space Λ M . In this casethe induced map in singular homology is not surjective. It follows from the proof of [Vit99, Theorem 4.1] that ST ∗ M does not admit a contact form without contractibleReeb orbits (see also [AS06] for the isomorphism between the Floer homology of co-tangent bundles and the singular homology of the free loop space). Thus ST ∗ M isnot hypertight but as shown in [CN10b] the isotopy class of the fibre is orderable.This shows that Theorem 5.2 as well as [Ush20, Corollary 3.5] are not covered bythe other result. Remark.
Since M is closed, the metrics induced by different contact forms areequivalent. Corollary 5.3.
Let L be a closed Legendrian such that L is orderable. Then theinterval topology on L is coarser than the topology induced by d α , in particular, thesets I ± ( L ) are open in this topology.Proof. The proof works analogous to the one of Proposition 2.1. (cid:3)
Remark.
Using compactly supported contactomorphisms it is possible to define d α for non closed contact manifolds. The proofs of Theorem 5.2 and Corollary 5.3should also work in this case. However it is not clear if the metrics defined bydifferent contact forms are equivalent or induce the same topology on L . For Legendrian isotopy classes it is possible to prove a version of Conjecture 2.Thus any reasonable Lorentzian distance function is not invariant under the actionof
Cont ( M ) . Proposition 5.4.
Let τ : L × L → [0 , ∞ ] be a map satisfying τ ( L , L ) > iff L Î L and τ ( L , L ) ≥ τ ( L , L ) + τ ( L , L ) for L L L . Assume that τ ( ϕL , ϕL ) = τ ( L , L ) for any ϕ ∈ Cont ( M ) . Then τ ( L , L ) = (cid:26) ∞ , if L Î L , otherwise . Proof.
Let L be a closed Legendrian and assume there exists such a function τ on its Legendrian isotopy class L . Let L ∈ L with L Î L . Choose a positivepath ϕ t with ϕ = L and ϕ = L . Then for < t < t < small enough and L := ϕ t ( L ) , L := ϕ t ( L ) one can assume that L ∩ L = L ∩ L = L ∩ L = ∅ . Claim:
There exists ψ ∈ Cont ( M ) with ψ ( L ) = L and ψ ( L ) = L .Take an open neighbourhood U of L such that ϕ t ( L ) ∩ U = ∅ for all t ∈ [ t , t ] .Let V be an open neighbourhood of L with V ⊂ U . Let F : [ t , t ] × M → R be a smooth function with F ( t, p ) = α ( X ϕϕ − t t )( p ) for p ∈ M \ U and F ( t, p ) = 0 for p ∈ V . Define ψ to be the flow of the time-dependent contact Hamiltonianvector field of F at time t . Then ψ = id M on V and since for t ∈ [ t , t ] we have ϕ t ( L ) ∩ U = ∅ , ψ coincides with ϕ t ϕ − t around L .It follows τ ( L, L ) = τ ( ψ ( L ) , ψ ( L )) = τ ( L, L ) ≥ τ ( L, L ) + τ ( L , L ) . Since τ ( L , L ) > this implies τ ( L, L ) = ∞ . Due to the reverse triangle inequal-ity τ ( L, · ) is strictly increasing along positive paths. Hence τ ( L, L ) = ∞ . (cid:3) We define a Lorentzian distance function on L that is not Cont ( M ) -invariantby ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 13
Definition 5.5.
Define τ α ( L , L ) := sup (cid:26) R min L t α ( X lt ) dt (cid:27) , if L L , otherwiseHere the supremum is taken over all non-negative Legendrian isotopies l t with l ( L ) = L and l ( L ) = L such that l t = ϕ t | L for some compactly supportednon-negative path of contactomorphisms ϕ t . Using Lemma 5.1 one can analogously to the case of the Lorentzian distance on
Cont ( M ) prove the following. Theorem 5.6.
The map τ α satisfies (i) τ α ( L , L ) > ⇔ L Î L . (ii) τ α is continuous with respect to the interval topology. (iii) τ α ( L , L ) ≥ τ α ( L , L ) + τ α ( L , L ) for L L L .If M is closed and L orderable then (iv) τ α is continuous with respect to the topology induced by d α . Theorem 5.7.
The Legendrian isotopy class L is orderable if and only if τ α ( L , L ) < ∞ for all L , L ∈ L . In this case for t ≥ τ α ( L, ϕ αt ( L )) = t. Since Lemma 4.1 is formulated in terms of Legendrians, one also has
Theorem 5.8.
Assume that ( M, ξ ) ∼ = ( ST ∗ N, ξ st ) , where N is smoothly coveredby an open subset of R n . Then there exists a constant C α depending on α such thatfor all L , L isotopic to the fibres one has τ α ( L , L ) ≤ C α d α ( L , L ) . A metric for globally hyperbolic spacetimes
In [Low06] Low constructed the space of null geodesics N g of a Lorentzian man-ifold ( N, g ) . He observed that for globally hyperbolic ( N, g ) the space of nullgeodesics naturally carries the structure of a smooth contact manifold. Moreovergiven a Cauchy hypersurface Σ he constructed a contactomorphism ρ Σ from N g to ST ∗ Σ equipped with its standard contact structure. Given a point p ∈ N its sky S ( p ) is the set of null geodesics through the point p . Denote by S ( N ) the set of allskies. The set S ( p ) is always a Legendrian submanifold of N g and is mapped by ρ N to the isotopy class of the fibres in ST ∗ N ([Low06]).Chernov and Nemirovski [CN19, Proposition 4.5] proved that if the isotopy classof the fibre is orderable, then the map p S ( p ) bijectively maps N to S ( N ) sothat the natural orders on both sets coincide. Hence the maps τ α and d α restrictto S ( N ) and induce a Lorentzian distance and a metric on N . Theorem 6.1.
In the case described above the metric d Nα := d α | S ( N ) × S ( N ) inducesthe manifold topology on N .Proof. Since the isotopy class of the fibres in ST ∗ Σ is orderable Corollary 5.3 impliesthat the interval topology on S ( N ) is open with respect to the topology induced by d α . Thus [CN19, Corollary 4.6] implies that the manifold topology on N is coarserthan the topology induced by d Nα .Due to the Bernal-Sánchez theorem (see e.g. [MS08]) one can assume that N = R × Σ , where { t } × Σ is a Cauchy hypersurface for every t . Let ( t , p ) ∈ N and ǫ > . W.l.o.g. assume t = 0 . Since ( N, g ) is strongly causal it suffices to show thatthere exists δ > such that d α ( S ((0 , p )) , S (( t, q ))) < ǫ for any ( t, q ) ∈ I + (( − δ, p )) ∩ I − (( δ, p )) . Let ρ t : N g → ST ∗ ( { t } × Σ) be the natural contactomorphism describedin [Low06]. Define ϕ t := ρ ◦ ρ − t . Using the natural identification one has ϕ t ∈ Cont ( ST ∗ ( { } × Σ)) . Denote by F q the fibre over the point q ∈ Σ . Then bydefinition S ( t, q ) = ϕ t ( F q ) . In particular for a curve of the form ( f ( t ) , γ ( t )) onehas S ( f ( t ) , γ ( t )) = ϕ f ( t ) ( F γ ( t ) ) . Choose a parametrisation l t : S n → F γ ( t ) . Thenfor w ∈ S (( f ( t ) , γ ( t )) and u ∈ S n with w = ϕ f ( t ) ( l t ( u )) α w (cid:18) ddt ϕ f ( t ) ( l t ( u )) (cid:19) = α w (cid:18) dϕ f ( t ) (cid:18) ddt l t ( u ) (cid:19) + f ′ ( t ) X ϕf ( t ) ( w ) (cid:19) = ( ϕ ∗ f ( t ) α ) l t ( u ) (ˆ γ ′ ( t )) + f ′ ( t ) α w ( X ϕf ( t ) ( w )) . Here ˆ γ ′ ( t ) denotes any vector v ∈ T l t ( u ) ST ∗ Σ with dπv = γ ′ ( t ) .Define C := max t ∈ [0 , max ϕ t ( F p ) | α ( X ϕt ) | . It follows that for < δ < d α ( S ((0 , p )) , S (( δ, p ))) ≤ δ Z max ϕ t ( F p ) | α ( X ϕt ) | dt ≤ Cδ and d α ( S ((0 , p )) , S (( − δ, p ))) ≤ Z − δ max ϕ t ( F p ) | α ( X ϕt ) | dt ≤ Cδ.
Let ( s, q ) ∈ I + (( − δ, p )) ∩ I − (( δ, p )) . Since the metrics induced by different contactforms are equivalent one can assume that α is induced by a Riemannian metric h ,i.e. α [ v ] ( w ) = v ( dπ ( w )) , where v ∈ [ v ] ∈ ST ∗ Σ with h ∗ ( v, v ) = 1 and π : ST ∗ Σ → Σ denotes the projection. There exist smooth positive functions ρ t with ϕ ∗ t α = ρ t α .Define ˜ C := max [0 , × ST ∗ Σ ρ t . Then for < δ < one has ϕ ∗ t α ≤ ˜ Cα for all t ∈ [ − δ, δ ] .Moreover since N is globally hyperbolic one can choose ˆ C such that I + (( − δ, p )) ∩ I − (( δ, p )) ⊂ [ − δ, δ ] × B ˆ Cδ ( p ) . Here B ˆ Cδ ( p ) denotes the ball of radius ˆ Cδ around p with respect to h . Let ( f ( t ) , γ ( t )) be a causal curve from ( − δ, p ) to ( s, q ) . Then d α ( S ((0 , p )) , S (( s, q ))) ≤ d α ( S ((0 , p )) , S (( − δ, p ))) + d α ( S (( − δ, p )) , S (( s, q ))) ≤ Z max ϕ t ( F p ) | ϕ ∗ t α (ˆ γ ′ ( t )) | dt + Cδ ≤ ( ˜ C ˆ C + C ) δ. (cid:3) Contrary to the Riemannian case there is no canonical way to associate a metricto a Lorentzian manifold. So it is not surprising that our construction depends onthe choice of a contact form α on N g . Question 4.
Are there further relations of d Xα and τ α | S ( X ) × S ( X ) to the causalstructure of N and to the Lorentzian metric g for specific choices of the contactform α ? References [AS06] Alberto Abbondandolo and Matthias Schwarz. On the floer homology of cotangentbundles.
Communications on Pure and Applied Mathematics: A Journal Issued bythe Courant Institute of Mathematical Sciences , 59(2):254–316, 2006.[BEE96] John K. Beem, Paul E. Ehrlich, and Kevin L. Easley.
Global Lorentzian geometry ,volume 202 of
Monographs and Textbooks in Pure and Applied Mathematics . MarcelDekker, Inc., New York, second edition, 1996.
ORENTZIAN DISTANCE FOR POSITIVE CONTACTOMORPHISMS 15 [BIP +
08] Dmitri Burago, Sergei Ivanov, Leonid Polterovich, et al. Conjugation-invariant normson groups of geometric origin. In
Groups of Diffeomorphisms: In honor of ShigeyukiMorita on the occasion of his 60th birthday , pages 221–250. Mathematical Society ofJapan, 2008.[CCDR19] Baptiste Chantraine, Vincent Colin, and Georgios Dimitroglou Rizell. Positive legen-drian isotopies and floer theory. In
Annales de l’Institut Fourier , volume 69, pages1679–1737, 2019.[CFP17] Vincent Colin, Emmanuel Ferrand, and Petya Pushkar. Positive isotopies of legen-drian submanifolds and applications.
International Mathematics Research Notices ,2017(20):6231–6254, 2017.[Che96] Yu. V. Chekanov. Critical points of quasifunctions, and generating families of Legen-drian manifolds.
Funktsional. Anal. i Prilozhen. , 30(2):56–69, 96, 1996.[CN10a] Vladimir Chernov and Stefan Nemirovski. Legendrian links, causality, and the Lowconjecture.
Geometric and Functional Analysis , 19(5):1320–1333, 2010.[CN10b] Vladimir Chernov and Stefan Nemirovski. Non-negative Legendrian isotopy in ST ∗ M . Geometry & Topology , 14(1):611–626, 2010.[CN16] Vladimir Chernov and Stefan Nemirovski. Universal orderability of Legendrian isotopyclasses.
Journal of Symplectic Geometry , 14(1):149–170, 2016.[CN19] Vladimir Chernov and Stefan Nemirovski. Interval topology in contact geometry.
Com-munications in Contemporary Mathematics , 22(05):1950042, May 2019.[EP00] Yakov Eliashberg and Leonid Polterovich. Partially ordered groups and geometry ofcontact transformations.
Geometric & Functional Analysis GAFA , 10(6):1448–1476,2000.[Gei08] Hansjörg Geiges.
An introduction to contact topology , volume 109 of
Cambridge Studiesin Advanced Mathematics . Cambridge University Press, Cambridge, 2008.[KS18] Michael Kunzinger and Clemens Sämann. Lorentzian length spaces.
Ann. Global Anal.Geom. , 54(3):399–447, 2018.[Low06] Robert J. Low. The space of null geodesics (and a new causal boundary). In
Analyticaland numerical approaches to mathematical relativity , volume 692 of
Lecture Notes inPhys. , pages 35–50. Springer, Berlin, 2006.[MS08] Ettore Minguzzi and Miguel Sánchez. The causal hierarchy of spacetimes.
Recent de-velopments in pseudo-Riemannian geometry, ESI Lect. Math. Phys , pages 299–358,2008.[RZ18] Daniel Rosen and Jun Zhang. Chekanov’s dichotomy in contact topology.
PreprintarXiv:1808.08459 , 2018.[She17] Egor Shelukhin. The Hofer norm of a contactomorphism.
J. Symplectic Geom. ,15(4):1173–1208, 2017.[Thé99] David Théret. A complete proof of viterbo’s uniqueness theorem on generating func-tions.
Topology and its Applications , 96(3):249–266, 1999.[Ush20] Michael Usher. Local rigidity, contact homeomorphisms, and conformal factors.
Preprint arXiv:2001.08729 , 2020.[Vit92] Claude Viterbo. Symplectic topology as the geometry of generating functions.
Math-ematische Annalen , 292:685–710, 03 1992.[Vit99] Claude Viterbo. Functors and computations in floer homology with applications, i.
Geometric & Functional Analysis GAFA , 9(5):985–1033, 1999.[Zap12] Frol Zapolsky. Geometric structures on contactomorphism groups and contact rigidityin jet spaces.
Preprint arXiv:1202.5691 , 2012.
Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstraße 150, 44801Bochum, Germany
Email address ::