pseudo-riemannian manifolds all of whose geodesics of one ...gelosp2013/files/suhr.pdfquestion: what...

Pseudo-Riemannian manifolds all of whosegeodesics of one causal type are closed

Stefan Suhr (Hamburg University)

July 23, 2013

Stefan Suhr (Hamburg University) Semi-Riemannian manifolds all of whose geodesics are closed

Question: What do we know about pseudo-Riemannian manifoldsall of whose geodesics of one causal type are closed?

Causal type ∼= timelike, spacelike or lightlike.

Answer: Next to nothing! Focus on the topological question.

Riemannian case: A large theory contained in the first book by A.Besse. E.g.

Theorem (Bott, Samelson)

Let (M, g) be a Riemannian manifold such that all geodesics aresimply closed. Then H∗(M,Z) ∼= H∗(CROSS,Z) where CROSS∈ {Sn,RPn,CPn,HPn,CaP2}.Proof with Morse theory not directly applicable inpseudo-Riemannian geometry (future development).




Answer: Next to nothing!

Focus on the topological question.








Riemannian case:

A large theory contained in the first book by A.Besse. E.g.







Riemannian case: A large theory contained in the first book by A.Besse.

E.g.









Let (M, g) be a Riemannian manifold such that all geodesics aresimply closed. Then H∗(M,Z) ∼= H∗(CROSS,Z) where CROSS∈ {Sn,RPn,CPn,HPn,CaP2}.

Proof with Morse theory not directly applicable inpseudo-Riemannian geometry (future development).







Let (M, g) be a Riemannian manifold such that all geodesics aresimply closed. Then H∗(M,Z) ∼= H∗(CROSS,Z) where CROSS∈ {Sn,RPn,CPn,HPn,CaP2}.Proof with Morse theory

not directly applicable inpseudo-Riemannian geometry (future development).







Let (M, g) be a Riemannian manifold such that all geodesics aresimply closed. Then H∗(M,Z) ∼= H∗(CROSS,Z) where CROSS∈ {Sn,RPn,CPn,HPn,CaP2}.Proof with Morse theory not directly applicable inpseudo-Riemannian geometry

(future development).


Known so far:

Proposition (Guillemin)

Let (M, g) be a compact pseudo-Riemannian 2-manifold such thatall lightlike geodesics are closed. Then (M, g) is finitely covered by(T 2, g) which is globally conformal to (R2/Z2, dxdy).

I Easily extended to non-compact 2-manifolds.

I For (Sn × S1, gcan − λdθ2) with λ ∈ Q all lightlike geodesicsare closed.

I Similar problem known for refocussing spacetimes.

What about examples with all spacelike/timelike geodesics closed?

Example

ConsiderSnν (r) = {x ∈ Rn+1| 〈x , x〉ν = r2}.

Then all spacelike geodesics of the induced metric are closed. Bychange of sign obtain examples with all timelike geodesics closed.


Known so far:







Example


Then all spacelike geodesics of the induced metric are closed.

Bychange of sign obtain examples with all timelike geodesics closed.


Known so far:







Example


Then all spacelike geodesics of the induced metric are closed. Bychange of sign obtain examples with all timelike geodesics closed.


Topological classification for 2-dimensional spacetimes.

Theorem (Mounoud/–)

Let (M, g) be a pseudo-Riemannian and non-Riemannian2-manifold all of whose timelike/spacelike geodesics are closed.Then (M, g) is finitely covered by (S1 × R, g) such that alltimelike/spacelike g-geodesics are simply closed (timelike/spacelikeZoll).

I The result is optimal, due to the previous examples.

I Zoll surfaces are Riemannian 2-manifolds all of whosegeodesics are simply closed, i.e. metrics on S2 and RP2.

I For 2-manifolds: If all timelike/spacelike geodesics are closedthen all non-timelike/non-spacelike geodesics are non-closed.Due to the theorem and the Poincaré-Bendixson theorem.

Corollary (Mounoud/–)

There does not exist a 2-dimensional pseudo-Riemannian andnon-Riemannian manifold all of whose geodesics are closed.







I For 2-manifolds: If all timelike/spacelike geodesics are closedthen all non-timelike/non-spacelike geodesics are non-closed.

Due to the theorem and the Poincaré-Bendixson theorem.









I For 2-manifolds: If all timelike/spacelike geodesics are closedthen all non-timelike/non-spacelike geodesics are non-closed.Due to the theorem and the Poincaré-Bendixson theorem.




Connection to geodesic foliations

Proposition

Let (M, g) be a pseudo-Riemannian manifold. Then there exists apseudo-Riemannian metric G on TM such that the tangent curvesof g-geodesics are G-geodesics of the same causal type.

Sketch of proof.

Consider the connection map Kg of the Levi-Civita connection ofg .

Tv TM ∼= ker(πTM)∗ ⊕ ker Kg ∼= TpM ⊕ TpM (v ∈ TpM)

Define G such that this isomorphism induces an isometry withg ⊕ g . Then πTM : TM → M becomes an pseudo-Riemanniansubmersion. Note that the tangent curves of geodesics are parallellifts.

RemarkIf all geodesics of one causal type (say timelike) are closed then{v ∈ TM| g(v , v) < 0} is foliated by closed geodesics.



Proposition


Sketch of proof.



Define G such that this isomorphism induces an isometry withg ⊕ g .

Then πTM : TM → M becomes an pseudo-Riemanniansubmersion. Note that the tangent curves of geodesics are parallellifts.




Proposition


Sketch of proof.



Define G such that this isomorphism induces an isometry withg ⊕ g . Then πTM : TM → M becomes an pseudo-Riemanniansubmersion. Note that the tangent curves of geodesics are parallellifts.



Theorem (Wadsley, Mounoud/–)

Let F be a smooth foliation by circles of M. The followingconditions are equivalent:

(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the samecausal character, i.e. the leaves of F are either timelike orspacelike geodesics.

(2) For any compact subset K of M, the circles meeting K havebounded length with respect to some (hence every)Riemannian metric.

(3) Let M be the double cover of M obtained by taking the twodifferent possible local orientations of the leaves. There is asmooth action of the orthogonal group O(2) on M and thenon-trivial deck transformation σ : M → M is an element ofthe non-trivial component of O(2). Each orbit under theO(2)-action consists of two components and each componentis mapped diffeomorphically onto a leaf of F by the coveringprojection.


Remark

I (2) ⇔ (3) is surprising, especially on non-compact manifolds(period could jump unboundedly).

I Epstein: If M3 is compact every foliation by circles has locallybounded length of the leafs.

I (1) cannot be extended to possibly lightlike geodesics. Thurston-Sullivan examples


Sketch of proof of the pseudo-Riemannian Wadsley theorem.

The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.

The claim follows from Koszul’s formula. h is well definedindependent of orientability of F .


Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:

(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.

(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.

This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.

(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation.

Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Sketch of proof of the pseudo-Riemannian Wadsley theorem.The following are equivalent:(1) There is a smooth pseudo-Riemannian metric rendering F ageodesic foliation by non-degenerate geodesics of the same causalcharacter, i.e. the leaves of F are either timelike or spacelike.(1’) There is a smooth Riemannian metric rendering F a geodesicfoliation.This is the condition in the known formulation of Wadsley’stheorem.(1)⇒ (1’): Let X be a locally defined unit-tangent field to thefoliation. Choose any Riemannian metric h⊥ on the orthogonalcomplement X⊥ and define the Riemannian metric

h = h⊥ + X [ ⊗ X [.



Theorem (”Signature-rigidity-theorem”, Mounoud/–)

A pseudo-Riemannian manifold having a geodesic flow that can beperiodically reparametrized is Riemannian or anti-Riemannian.

Proposition

Let F be an oriented 1-dimensional geodesic foliation on apseudo-Riemannian manifold (M, g). If the leaves of F are circleswith locally bounded Riemannian(!) length then they all have thesame type.

Remark

I If examples exist of pseudo-Riemannian manifolds with allgeodesics closed, then their geodesics flow is complicated.

I The problem lies on the lightcones.

I There exist examples of foliations by circles such that thelength of the leafs are not locally bounded (Thurston-Sullivanexamples).


Thurston-Sullivan examples:

Consider H/Γ× S1 × S1, where His the 3-dimensional Heisenberg group and Γ is the lattice ofinteger matrices in H. Denote with (x , y , z , t, u) coordinates onH × R× R. Set

X = sin(2u)(− sin(t)∂x + cos(t)∂y )+ (x sin(2u) cos(t)− cos2(u))∂z + 2 sin2(u)∂t .

X descends to the quotient H/Γ× S1 × S1 (S1 = R/2πZ). Notethat X is tangent to H × R and therefore the projection is tangentto H/Γ× S1. The flowlines of X are all closed and of unboundedlength (period 2π

sin2(u)for u 6= 0, π and 2π for u = 0, π).


Thurston-Sullivan examples: Consider H/Γ× S1 × S1, where His the 3-dimensional Heisenberg group and Γ is the lattice ofinteger matrices in H.

Denote with (x , y , z , t, u) coordinates onH × R× R. Set





Thurston-Sullivan examples: Consider H/Γ× S1 × S1, where His the 3-dimensional Heisenberg group and Γ is the lattice ofinteger matrices in H. Denote with (x , y , z , t, u) coordinates onH × R× R. Set







X descends to the quotient H/Γ× S1 × S1 (S1 = R/2πZ).

Notethat X is tangent to H × R and therefore the projection is tangentto H/Γ× S1. The flowlines of X are all closed and of unboundedlength (period 2π





X descends to the quotient H/Γ× S1 × S1 (S1 = R/2πZ). Notethat X is tangent to H × R and therefore the projection is tangentto H/Γ× S1.

The flowlines of X are all closed and of unboundedlength (period 2π



Turning the flowlines of X into a geodesic foliation:

Consider the frame (X , ∂u,V ,W , 2∂t + ∂z) with

V = cos(t)∂x + sin(t)(∂y + x∂z)

andW = − sin(t)∂x + cos(t)(∂y + x∂z).

Note that the frame descends to the quotient. Define theLorentzian metric g to be lightlike on X and ∂u and unitRiemannian on the other vector fields. Clearly the flowlines of Xform a g -geodesic foliations by lightlike geodesics.

RemarkThis construction works for type-changing foliations as well. Butnothing is known for geodesic foliations on tangent bundles.


Turning the flowlines of X into a geodesic foliation:Consider the frame (X , ∂u,V ,W , 2∂t + ∂z) with









Note that the frame descends to the quotient.

Define theLorentzian metric g to be lightlike on X and ∂u and unitRiemannian on the other vector fields. Clearly the flowlines of Xform a g -geodesic foliations by lightlike geodesics.






Note that the frame descends to the quotient. Define theLorentzian metric g to be lightlike on X and ∂u and unitRiemannian on the other vector fields.

Clearly the flowlines of Xform a g -geodesic foliations by lightlike geodesics.







RemarkThis construction works for type-changing foliations as well.

Butnothing is known for geodesic foliations on tangent bundles.


Idea of the topological classification:

(1) Non-compact case: pseudo-Riemannian Wadsley Thefundamental class of the closed geodesics lies in the center ofπ1(M). π1(M) is a free group and no pseudo-Riemannian2-manifold contains contractible non-spacelike/non-timelikeloops. π1(M) ∼= Z M is covered by S1 × R. The Zollproperty follows since the geodesic flow on the unit tangentbundle is induced by an S1-action.

(2) Compact case: The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle, i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics. The unit tangents are unbounded and thegeodesic flow is continuous. There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle. Contradiction toWadsley’s theorem.



(1) Non-compact case:

pseudo-Riemannian Wadsley Thefundamental class of the closed geodesics lies in the center ofπ1(M). π1(M) is a free group and no pseudo-Riemannian2-manifold contains contractible non-spacelike/non-timelikeloops. π1(M) ∼= Z M is covered by S1 × R. The Zollproperty follows since the geodesic flow on the unit tangentbundle is induced by an S1-action.




(1) Non-compact case: pseudo-Riemannian Wadsley Thefundamental class of the closed geodesics lies in the center ofπ1(M).

π1(M) is a free group and no pseudo-Riemannian2-manifold contains contractible non-spacelike/non-timelikeloops. π1(M) ∼= Z M is covered by S1 × R. The Zollproperty follows since the geodesic flow on the unit tangentbundle is induced by an S1-action.




(1) Non-compact case: pseudo-Riemannian Wadsley Thefundamental class of the closed geodesics lies in the center ofπ1(M). π1(M) is a free group and no pseudo-Riemannian2-manifold contains contractible non-spacelike/non-timelikeloops.

π1(M) ∼= Z M is covered by S1 × R. The Zollproperty follows since the geodesic flow on the unit tangentbundle is induced by an S1-action.




(1) Non-compact case: pseudo-Riemannian Wadsley Thefundamental class of the closed geodesics lies in the center ofπ1(M). π1(M) is a free group and no pseudo-Riemannian2-manifold contains contractible non-spacelike/non-timelikeloops. π1(M) ∼= Z M is covered by S1 × R.

The Zollproperty follows since the geodesic flow on the unit tangentbundle is induced by an S1-action.





(2) Compact case:

The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle, i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics. The unit tangents are unbounded and thegeodesic flow is continuous. There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle. Contradiction toWadsley’s theorem.




(2) Compact case: The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle,

i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics. The unit tangents are unbounded and thegeodesic flow is continuous. There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle. Contradiction toWadsley’s theorem.




(2) Compact case: The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle, i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics.

The unit tangents are unbounded and thegeodesic flow is continuous. There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle. Contradiction toWadsley’s theorem.




(2) Compact case: The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle, i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics. The unit tangents are unbounded and thegeodesic flow is continuous.

There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle. Contradiction toWadsley’s theorem.




(2) Compact case: The closed unit-speed geodesics all intersecta fixed compact subset of the tangent bundle, i.e. the unittangents to a timelike/spacelike foliation with the samerotation number (image under the Hurewicz homomorphism)as the geodesics. The unit tangents are unbounded and thegeodesic flow is continuous. There exist arbitrary long(Riemannian sense) closed geodesics whose tangents meet acompact subset of the tangent bundle.

Contradiction toWadsley’s theorem.


3-dimensional spacetimes with all lightlike geodesics closed.

Definition (Guillemin)

A compact 3-dimensional pseudo-Riemannian manifold (M, g) isZollfrei, if the geodesic flow on the lightlike vectors induces afibration by circles.

Remark

I Zollfrei is inspired by notion of Zoll surfaces.

I (S2 × S1, gcan − λdθ2) is Zollfrei iff λ ∈ Q.I Connection to Low’s notion of refocussing spacetimes.

Theorem (Tollefson)

The only diffeomorphism types of compact manifolds covered byS2 × S1 are S2 × S1 itself, RP2 × S1, RP3]RP3 and the uniquenon-orientable 2-sphere bundle over S1.





Remark


I (S2 × S1, gcan − λdθ2) is Zollfrei iff λ ∈ Q.

I Connection to Low’s notion of refocussing spacetimes.

Theorem (Tollefson)






Remark


I (S2 × S1, gcan − λdθ2) is Zollfrei iff λ ∈ Q.I Connection to Low’s notion of refocussing spacetimes.

Theorem (Tollefson)



Remark (Guillemin)

The metrics gcan − λdθ2 descend to all quotients. They are Zollfreiiff λ ∈ Q and are called the standard examples.

Conjecture (Guillemin)

Every Zollfrei manifold has the diffeomorphism type of one of thestandard examples.

TheoremEvery non-trivial orientable circle bundle over a closed andorientable surface admits a Zollfrei metric.

Corollary

Guillemin’s conjecture is wrong.

By the Gysin sequence all diffeomorphism types in the theorem aredifferent and none is one of the standard examples.


Guillemin gives a weaker version of his conjecture assumingcausality of the universal cover.

But we have:

TheoremIf (M, g) is Zollfrei and the universal cover is globally hyperbolic,then M has the diffeomorphism type of one of the standardexamples.

Follows from a result of Low on refocussing spacetimes. Here(M, g) Zollfrei and globally hyperbolicity of the universal coverimply that the universal cover is refocussing.

Theorem (Low, Chernov/Rudyak)

Let (M, g) be a globally hyperbolic spacetime with non-compactCauchy hypersurfaces. Then (M, g) is not refocussing.

For Zollfrei spacetimes this implies that the Cauchy hypersurfacesof the universal cover are compact and simply connected, i.e.2-spheres. Then the Zollfrei spacetimes are covered by S2 × S1.


Guillemin gives a weaker version of his conjecture assumingcausality of the universal cover. But we have:

TheoremIf (M, g) is Zollfrei and the universal cover is globally hyperbolic,then M has the diffeomorphism type of one of the standardexamples.

Follows from a result of Low on refocussing spacetimes. Here(M, g) Zollfrei and globally hyperbolicity of the universal coverimply that the universal cover is refocussing.

Theorem (Low, Chernov/Rudyak)

Let (M, g) be a globally hyperbolic spacetime with non-compactCauchy hypersurfaces. Then (M, g) is not refocussing.

For Zollfrei spacetimes this implies that the Cauchy hypersurfacesof the universal cover are compact and simply connected, i.e.2-spheres. Then the Zollfrei spacetimes are covered by S2 × S1.


Construction of the metrics:

I (B2, g) closed oriented surface with constant curvature, suchthat volg (B) ∈ 2πZ

I S1 ↪→ M → B principal bundle with Euler class[−dvolg2π

]∈ H2(B,Z), R tangent field to the fibres

I α connection 1-form on M, such that LR(α) = 0, α(R) = 1and curvature dvolg (contact form)

I For φ ∈(0, π2

)set (π : M → B)

hφ = π∗g − cot2(φ)α⊗ α

I opening angles of the lightcones is π2 − φI hφ is stationary with timelike Killing vector field R





]∈ H2(B,Z),

R tangent field to the fibresI α connection 1-form on M, such that LR(α) = 0, α(R) = 1

and curvature dvolg (contact form)

I For φ ∈(0, π2

)set (π : M → B)







]∈ H2(B,Z), R tangent field to the fibres

I α connection 1-form on M, such that LR(α) = 0, α(R) = 1and curvature dvolg (contact form)

I For φ ∈(0, π2

)set (π : M → B)




Example

For (B, g) = (CP1, gFS) (K = 4) we have M ∼= S3 and

hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg . In the present case the construction is therefore not global.(ii) Global description is not well defined on curves, but on ”films”(Taimanov). A smooth map f : S → B is called a film, where S isa compact surface with boundary. The charged particlesfunctional cpφ for films is

cpφ(f ) = Lg (f |∂S)− cot2(φ)

∫S

f ∗dvolg .

If the ”magnetic” term −dvolg is exact the arrival time functionalis retained.


Example


hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg .

In the present case the construction is therefore not global.(ii) Global description is not well defined on curves, but on ”films”(Taimanov). A smooth map f : S → B is called a film, where S isa compact surface with boundary. The charged particlesfunctional cpφ for films is


∫S

f ∗dvolg .



Example


hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg . In the present case the construction is therefore not global.

(ii) Global description is not well defined on curves, but on ”films”(Taimanov). A smooth map f : S → B is called a film, where S isa compact surface with boundary. The charged particlesfunctional cpφ for films is


∫S

f ∗dvolg .



Example


hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg . In the present case the construction is therefore not global.(ii) Global description is not well defined on curves, but on ”films”(Taimanov).

A smooth map f : S → B is called a film, where S isa compact surface with boundary. The charged particlesfunctional cpφ for films is


∫S

f ∗dvolg .



Example


hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg . In the present case the construction is therefore not global.(ii) Global description is not well defined on curves, but on ”films”(Taimanov). A smooth map f : S → B is called a film, where S isa compact surface with boundary.

The charged particlesfunctional cpφ for films is


∫S

f ∗dvolg .



Example


hφ = J∗ ⊗ J∗ + K ∗ ⊗ K ∗ − cot2(φ)I ∗ ⊗ I ∗.

(i) The lightlike geodesics are locally given by the arrivial timefunctional of a Finsler metric defined via g and a local primitive ofdvolg . In the present case the construction is therefore not global.(ii) Global description is not well defined on curves, but on ”films”(Taimanov). A smooth map f : S → B is called a film, where S isa compact surface with boundary. The charged particlesfunctional cpφ for films is


∫S

f ∗dvolg .



(iii) The critical points of cpφ have closed “magnetic geodesics” asboundary. A curve γ is a magnetic geodesic if

∇γ̇ γ̇ = cot2(φ) dvolg (γ̇, .)].

(The charged particle functional on films induces anEuler-Lagrange flow.)

(iv) cpφ is invariant under isometries of (B, g). All magneticgeodesics are simply closed and induce a fibration of T 1B bycircles for secg ≥ 0. If secg < 0 there exist a φ0 such that for all0 < φ ≤ φ0 the same holds.

Proposition

(M, hφ) is Zollfrei iff cpφ ∈ 2πQ on its critical points.

For B = CP1 we have cpφ = 4 tan(φ)−1√1+4 tan2 φ

2π + 2πZ on the criticalpoints



∇γ̇ γ̇ = cot2(φ) dvolg (γ̇, .)].

(The charged particle functional on films induces anEuler-Lagrange flow.)(iv) cpφ is invariant under isometries of (B, g).

All magneticgeodesics are simply closed and induce a fibration of T 1B bycircles for secg ≥ 0. If secg < 0 there exist a φ0 such that for all0 < φ ≤ φ0 the same holds.

Proposition






∇γ̇ γ̇ = cot2(φ) dvolg (γ̇, .)].

(The charged particle functional on films induces anEuler-Lagrange flow.)(iv) cpφ is invariant under isometries of (B, g). All magneticgeodesics are simply closed and induce a fibration of T 1B bycircles for secg ≥ 0.

If secg < 0 there exist a φ0 such that for all0 < φ ≤ φ0 the same holds.

Proposition





(iii) The c

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