Geometric structures in the study of the geodesic …gunther/MSRI_IP...Concepts from Finsler-geometry Geodesic ray transform Geometric structures in the study of the geodesic ray transform

Concepts from Finsler-geometryGeodesic ray transform

Geometric structures in the study of the geodesicray transform

Juha-Matti Perkkio

Mathematical Sciences Research InstituteBerkeley, CA

Inverse Problems and Applications

Juha-Matti Perkkio Geometric structures in the study of the geodesic ray transform


Outline

1 Concepts from Finsler-geometryWhat are Finsler-spaces?Sprays and nonlinear connectionsHorizontal-vertical calculus

2 Geodesic ray transformNon-trapping Finsler-spacesFirst results on geodesic ray transformUltrahyperbolicity and ray transform



Outline

1 Concepts from Finsler-geometryWhat are Finsler-spaces?Sprays and nonlinear connectionsHorizontal-vertical calculus

2 Geodesic ray transformNon-trapping Finsler-spacesFirst results on geodesic ray transformUltrahyperbolicity and ray transform



What are Finsler-spaces?Sprays and nonlinear connectionsHorizontal-vertical calculus

Finsler-spaces I

Definition

Let M be a smooth manifold. Then d : M ×M → R is acompatible quasi-metric on M if

1 d(x , x) = 0 for every x ∈ M.

2 d(x , y) > 0 whenever x 6= y.

3 d(x , y) ≤ d(x , z) + d(z , y) for every x , y , z ∈ M.

4 For any smooth chart (U, ϕ) and compact K ⊂ U

1C ‖ϕ(x)− ϕ(y)‖ ≤ d(x , y) ≤ C‖ϕ(x)− ϕ(y)‖, x , y ∈ K .

The Finsler-function Fd : TxM → R of a compatiblequasi-metric d on M is well-defined at almost every TxM by

Fd(ξ) := limt→0+

d(γ(0), γ(t))

t, γ(0) = ξ ∈ TxM.




Finsler-spaces I

Definition

Let M be a smooth manifold. Then d : M ×M → R is acompatible quasi-metric on M if

1 d(x , x) = 0 for every x ∈ M.

2 d(x , y) > 0 whenever x 6= y.

3 d(x , y) ≤ d(x , z) + d(z , y) for every x , y , z ∈ M.

4 For any smooth chart (U, ϕ) and compact K ⊂ U

1C ‖ϕ(x)− ϕ(y)‖ ≤ d(x , y) ≤ C‖ϕ(x)− ϕ(y)‖, x , y ∈ K .

The Finsler-function Fd : TxM → R of a compatiblequasi-metric d on M is well-defined at almost every TxM by

Fd(ξ) := limt→0+

d(γ(0), γ(t))

t, γ(0) = ξ ∈ TxM.




Finsler-spaces II

Let d : M ×M → R be a compatible quasi-metric. Then

Fd(λξ) = λFd(ξ) for every ξ ∈ TxM and λ > 0.Fd(ξ + η) ≤ Fd(ξ) + Fd(η) for every ξ, η ∈ TxM.For any smooth chart (U, ϕ) of M and K ⊂ U

1C ‖ϕ∗X‖ ≤ F (X ) ≤ C‖ϕ∗X‖, X ∈ TxM, x ∈ K .

Let us make the additional regularity assumptions

d : M ×M → R is smooth outside the diagonal.SxM := ξ ∈ TxM | Fd(ξ) = 1 are strictly convex.

Then Fd ∈ C∞(TM \ 0), and the fundamental tensor

gξ(X ,Y ) :=1

2

d2

dsdt

∣∣∣s=t=0

F 2d (ξ + sX + tY )

of Fd is positive definite at every ξ ∈ TM \ 0.




Finsler-spaces II



1C ‖ϕ∗X‖ ≤ F (X ) ≤ C‖ϕ∗X‖, X ∈ TxM, x ∈ K .




gξ(X ,Y ) :=1

2

d2

dsdt

∣∣∣s=t=0

F 2d (ξ + sX + tY )





Finsler-spaces II



1C ‖ϕ∗X‖ ≤ F (X ) ≤ C‖ϕ∗X‖, X ∈ TxM, x ∈ K .




gξ(X ,Y ) :=1

2

d2

dsdt

∣∣∣s=t=0

F 2d (ξ + sX + tY )





Finsler-spaces III

Definition

Let M be a smooth manifold and F ∈ C∞(TM \ 0). Then (M,F )is a Finsler-space, if

1 F (λξ) = λF (ξ) for every λ > 0.

2 The fundamental tensor gξ of F is positive definite.

Due to the homogeneity of F the length functional

L(γ) :=

∫ b

aF (γ(t))dt, γ : [a, b]→ M

of (M,F ) is invariant in positive reparametrizations.If (M,F ) is a Finsler-space, the function

d(x , y) := inf L(γ) | γ(a) = x , γ(b) = y is a compatible quasi-metric on M with Fd = F .




Finsler-spaces III

Definition





L(γ) :=

∫ b

aF (γ(t))dt, γ : [a, b]→ M






Finsler-spaces III

Definition





L(γ) :=

∫ b

aF (γ(t))dt, γ : [a, b]→ M






Geodesics

The extremal paths γ : [a, b]→ M of the length functionalL(γ) are geodesics of the Finsler-space (M,F ).

The constant speed geodesics of (M,F ) coincide with theextremal paths of the energy functional

E(γ) :=1

2

∫ b

aF (γ(t))2dt.

Define the Hilbert-form α ∈ Ω1(TM \ 0) by

αξ(X ) :=1

2

d

dt

∣∣∣s=0

F 2(ξ + t(πTM)∗X

).

Then dα ∈ Ω2(TM \ 0) is a symplectic form on TM \ 0.

Let H be the Hamiltonian vector field for E := 12 F 2, that is,

the unique vector field on TM \ 0 such that ιHdα = −dE .

Then the Euler-Lagrange equation for E(γ) is γ(t) = Hγ(t).




Geodesics



E(γ) :=1

2

∫ b

aF (γ(t))2dt.


αξ(X ) :=1

2

d

dt

∣∣∣s=0


).








Geodesics



E(γ) :=1

2

∫ b

aF (γ(t))2dt.


αξ(X ) :=1

2

d

dt

∣∣∣s=0


).








Geodesics



E(γ) :=1

2

∫ b

aF (γ(t))2dt.


αξ(X ) :=1

2

d

dt

∣∣∣s=0


).








Geodesics



E(γ) :=1

2

∫ b

aF (γ(t))2dt.


αξ(X ) :=1

2

d

dt

∣∣∣s=0


).








Geodesics



E(γ) :=1

2

∫ b

aF (γ(t))2dt.


αξ(X ) :=1

2

d

dt

∣∣∣s=0


).








Spray structures

The canonical vector field and tangent structure on TM are

Vξ[f ] :=d

dt

∣∣∣t=0

f (etξ), (JξX )[f ] :=d

dt

∣∣∣t=0

f (ξ+t(πTM)∗X ).

Definition

A smooth vector field H on TM \ 0 is a spray, if

JH = V and [V ,H] = H.

Consider a smooth family γξξ∈TM\0 of regular curvesγξ : R→ M such that γξ(0) = ξ and γλξ(t) = γξ(λt).

Then (t, ξ) 7→ γξ(t) is one-parameter group action whosegenerating vector field H such that γξ(t) = Hγξ(t) is a spray.

Any complete spray H defines a smooth family γξξ∈TM\0 ofregular curves as above by γξ(t) := (πTM Φt

H)(ξ).




Spray structures


Vξ[f ] :=d

dt

∣∣∣t=0

f (etξ), (JξX )[f ] :=d

dt

∣∣∣t=0


Definition






H)(ξ).




Spray structures


Vξ[f ] :=d

dt

∣∣∣t=0

f (etξ), (JξX )[f ] :=d

dt

∣∣∣t=0


Definition






H)(ξ).




Spray structures


Vξ[f ] :=d

dt

∣∣∣t=0

f (etξ), (JξX )[f ] :=d

dt

∣∣∣t=0


Definition






H)(ξ).




Spray structures


Vξ[f ] :=d

dt

∣∣∣t=0

f (etξ), (JξX )[f ] :=d

dt

∣∣∣t=0


Definition






H)(ξ).




Natural vector subbundles

Define the pullback vector bundles

π∗TMTM := (ξ,X ) | ξ,X ∈ TxM, x ∈ M ,π∗TMT ∗M := (ξ, ω) | ξ ∈ TxM, ω ∈ T ∗x M, x ∈ M .

Sections of their tensor products are semibasic tensor fields.

Vertical tangent bundle and horizontal cotangent bundle are

VTM := Ker(πTM)∗, H∗TM := Ran(π∗TM).

Vertical lift is the natural vector bundle isomorphism

vl : π∗TMTM → VTM; (vl ξX )[f ] :=d

dt

∣∣t=0

f (ξ + tX ).

The pullback of πTM is a natural vector bundle isomorphism

(πTM)∗ : π∗TMT ∗M → H∗TM; (π∗TMω)(X ) = ω((πTM)∗X ).












dt

∣∣t=0

f (ξ + tX ).














dt

∣∣t=0

f (ξ + tX ).














dt

∣∣t=0

f (ξ + tX ).














dt

∣∣t=0

f (ξ + tX ).






Nonlinear connection

The spray H induces a connection on (TM \ 0, πTM ,M) by

hpr := 12

(I − LHJ

), vpr := 1

2

(I + LHJ

).

The vector bundle mappings hl : π∗TM\0TM → H(TM \ 0)

and κ : T (TM \ 0)→ TM are defined by the diagram

T (TM \ 0)

hpr

uu

vpr

))H(TM \ 0)

(πTM\0)∗))

π∗TM\0TMhl

oovl

//

fib.id.

V (TM \ 0)

κuu

TM

The curvature of the connection is the semibasic tensor field

Rξ ∈ TxM∧TxM⊗TxM; vl ξR(X ,Y ) = vpr [hl X , hl Y ]ξ.






hpr := 12

(I − LHJ

), vpr := 1

2

(I + LHJ

).



T (TM \ 0)

hpr

uu

vpr

))H(TM \ 0)

(πTM\0)∗))

π∗TM\0TMhl

oovl

//

fib.id.

V (TM \ 0)

κuu

TM








hpr := 12

(I − LHJ

), vpr := 1

2

(I + LHJ

).



T (TM \ 0)

hpr

uu

vpr

))H(TM \ 0)

(πTM\0)∗))

π∗TM\0TMhl

oovl

//

fib.id.

V (TM \ 0)

κuu

TM






Jacobi-fields

Define the actions of H and R on semibasic vector fields by

(HX )ξ := (πTM)∗[H, hl X ]ξ, (RX )ξ := Rξ(ξ,X ).

Then one can verify the formulae

[H, hl X ] = hl HX + vl RX , [H, vl Y ] = vl HY − hl Y .

A vector field Xγ along a geodesic γ is a Jacobi-field, ifXγ(t) = (πTM)∗Zγ(t) for some Z on TM \ 0 with [H,Z ] = 0.Let Z = hl X + vl Y . Then [H,Z ] = 0 if and only if

HHX + RX = 0 and HX = Y .

Hξ = hl ξξ and Vξ = vl ξξ induce the trivial Jacobi-fields

γξ(t) = (πTM ΦtH)∗Hξ, tγξ(t) = (πTM Φt

H)∗Vξ.

If H is the Hamiltonian vector field on (M,F ), then

H[g(X ,Y )] = g(HX ,Y )+g(X ,HY ), g(X ,RY ) = g(RX ,Y ).Juha-Matti Perkkio Geometric structures in the study of the geodesic ray transform



Jacobi-fields









H)∗Vξ.





Jacobi-fields









H)∗Vξ.





Jacobi-fields









H)∗Vξ.





Jacobi-fields









H)∗Vξ.





Jacobi-fields









H)∗Vξ.





Sasaki-metric

For a Finsler-space (M,F ) the fundamental tensor ξ 7→ gξdefines a vector bundle metric on π∗TM\0TM.

The Sasaki-metric on TM \ 0 is the Riemannian metric

gTM(hl X + vl Y , hl U + vl W ) := g(X ,U) + g(Y ,W ).

Let gradTM f denote the gradient, ωTM the volume form anddivTM the divergence with respect to gTM .The horizontal and vertical gradient of f ∈ C∞(TM \ 0) arethe semibasic vector fields defined by

hlh∇ f + vl

v∇ f = gradTM f .

The horizontal and vertical divergence of X ∈ Γ(π∗TM\0TM)

are the smooth functions on TM \ 0 defined by

hdiv X := divTM(hl X ),

vdiv X := divTM(vl X ).




Sasaki-metric





hlh∇ f + vl

v∇ f = gradTM f .








Sasaki-metric





hlh∇ f + vl

v∇ f = gradTM f .








Sasaki-metric





hlh∇ f + vl

v∇ f = gradTM f .








Sasaki-metric





hlh∇ f + vl

v∇ f = gradTM f .








Commutator relations

The symplectic form dα ∈ Ω2(TM \ 0) can be written as

dα(hl X + vl Y , hl U + vl W ) = g(Y ,U)− g(X ,W ).

The symplectic volume and the symplectic gradient are

ωdα = ωTM , graddα u = hlv∇ u − vl

h∇ u.

The identity divdα graddα u = 0 reads as

hdiv

v∇ f =

vdiv

h∇ f .

The identity [H, graddα u] = graddα Hu reads as

Hh∇ u =

h∇Hu + R

v∇ u, H

v∇ u =

v∇Hu −

h∇ u.

The identity H[divdα Z ] = divdα[H,Z ] reads as

Hh

div X =h

div HX +v

div RX , Hv

div Y =v

div HY −h

div Y .









h∇ u.


hdiv

v∇ f =

vdiv

h∇ f .


Hh∇ u =

h∇Hu + R

v∇ u, H

v∇ u =

v∇Hu −

h∇ u.


Hh

div X =h

div HX +v

div RX , Hv

div Y =v

div HY −h

div Y .









h∇ u.


hdiv

v∇ f =

vdiv

h∇ f .


Hh∇ u =

h∇Hu + R

v∇ u, H

v∇ u =

v∇Hu −

h∇ u.


Hh

div X =h

div HX +v

div RX , Hv

div Y =v

div HY −h

div Y .









h∇ u.


hdiv

v∇ f =

vdiv

h∇ f .


Hh∇ u =

h∇Hu + R

v∇ u, H

v∇ u =

v∇Hu −

h∇ u.


Hh

div X =h

div HX +v

div RX , Hv

div Y =v

div HY −h

div Y .









h∇ u.


hdiv

v∇ f =

vdiv

h∇ f .


Hh∇ u =

h∇Hu + R

v∇ u, H

v∇ u =

v∇Hu −

h∇ u.


Hh

div X =h

div HX +v

div RX , Hv

div Y =v

div HY −h

div Y .




Restriction to the unit sphere bundle I

The Sasaki-metric gTM restricts to a Riemannian metric onSM := ξ ∈ TM | F (ξ) = 1 with V as the unit normal.SM inherits the connection TSM = HSM ⊕ VSM.The semibasic tensor fields restrict in sections of π∗SMTM and

hl ξX ∈ HξSM, vl ξY ∈ VξSM iff gξ(ξ,Y ) = 0.

The gradient gradSM with respect to gSM reads as

gradSM u = hlh∇ u + vl

vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.

For any X ∈ Γ(π∗SMTM) it holds that

divSM(hl X ) = divTM(hl X ) =h

div X .

For any Y ∈ Γ(π∗SMTM) with gξ(ξ,Y ) = 0 we define

vsdiv Y := divSM(vl Y ).









vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.



div X .











vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.



div X .











vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.



div X .











vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.



div X .











vs∇ u,

vs∇ u :=

v∇ u − (Vu)ξ.



div X .






Restriction to the unit sphere bundle II

For convenience we define the horizontal spherical gradient as

hs∇ u ∈ Γ(π∗SMTM);

hs∇ u :=

h∇ u − (Hu)ξ.

Then the commutator relations can be rewritten as

hdiv

vs∇ u =

vsdiv

hs∇ u + (n − 1)Hu,

Hhs∇ u =

hs∇Hu + R

vs∇ u,

Hvs∇ u =

vs∇Hu −

hs∇ u,

Hh

div X =h

div HX +vs

div RX ,

Hvs

div Y =vs

div HY −h

div Y ,

where gξ(ξ,Y ) = 0 if and only if gξ(ξ,HY ) = 0.




Restriction to the unit sphere bundle II

For convenience we define the horizontal spherical gradient as

hs∇ u ∈ Γ(π∗SMTM);

hs∇ u :=

h∇ u − (Hu)ξ.

Then the commutator relations can be rewritten as

hdiv

vs∇ u =

vsdiv

hs∇ u + (n − 1)Hu,

Hhs∇ u =

hs∇Hu + R

vs∇ u,

Hvs∇ u =

vs∇Hu −

hs∇ u,

Hh

div X =h

div HX +vs

div RX ,

Hvs

div Y =vs

div HY −h

div Y ,

where gξ(ξ,Y ) = 0 if and only if gξ(ξ,HY ) = 0.



Non-trapping Finsler-spacesFirst results on geodesic ray transformUltrahyperbolicity and ray transform

Non-trapping Finsler-spaces

The tangent bundle of SM can be identified with

TSM := Y ∈ TξTM | F (ξ) = 1, dF (Y ) = 0 .

Since H[E ] = 0, also H[F ] = 0, so the geodesic flow restrictsinto a Lie-group action ΦH : R× SM → SM and the geodesicspray H restricts to a smooth vector field on SM.

Definition

A Finsler-space (M,F ) without a boundary is non-trapping, if forevery compact K ⊂ M and ξ ∈ SM there exists tK ,ξ > 0 such that

(πSM ΦtH)(ξ) ∈ M \ K whenever |t| ≥ tK ,ξ.

So (M,F ) is non-trapping, if it is complete and if everygeodesic finally stays away from any compact set K ⊂ M.






TSM := Y ∈ TξTM | F (ξ) = 1, dF (Y ) = 0 .


Definition









TSM := Y ∈ TξTM | F (ξ) = 1, dF (Y ) = 0 .


Definition









TSM := Y ∈ TξTM | F (ξ) = 1, dF (Y ) = 0 .


Definition







Manifold of geodesics I

Lemma

Let (M,F ) be a complete Finsler-space without a boundary. Thenthe Lie-group action ΦH : R × SM → SM is free and proper if andonly if (M,F ) is non-trapping.

Free proper Lie-group actions give rise to smooth quotients.

Definition

Let (M,F ) be a non-trapping Finsler-space. Then the manifold ofgeodesics on M is GM := SM/ΦH

.

ξ ∼ η if and only if η = ΦtH(ξ) for some t ∈ R.

The elements [ξ]∼ ∈ GM correspond to the unit speedgeodesics γ on (M,F ) modulo reparametrizations t 7→ t + s.





Lemma



Definition


.







Lemma



Definition


.







Lemma



Definition


.







Lemma



Definition


.






Manifold of geodesics II

f ∈ C∞(SM) descends into f ∈ C∞(GM) given byf ([ξ∼]) = f (ξ) if and only if H[f ] = 0.

Vector field X of SM descends to a vector field X on GMgiven by X [f ] = X [f ] if and only if [H,X ] = 0.

Any vector field X on GM is induced by some X ∈ SM with

[H,X ] = 0 and α(X ) = 0.

Therefore the tangent vectors at [ξ]∼ correspond toequivalence classes [Xγξ ]∼ of normal Jacobi-fields alongγξ(t) = (πSM Φt

H)(ξ), where Yγη ∼ Xγξ if and only ifYγη(t) = Xγξ(t+s) for some s ∈ R.

dα descends into a symplectic form on GM and

ωGM := ιHωSM ∈ Ω2n−2(SM)

descends into its symplectic volume.








[H,X ] = 0 and α(X ) = 0.













[H,X ] = 0 and α(X ) = 0.













[H,X ] = 0 and α(X ) = 0.













[H,X ] = 0 and α(X ) = 0.













[H,X ] = 0 and α(X ) = 0.









Geodesic ray transform

Definition

The geodesic ray transform in a non-trapping (M,F ) is

I : C∞c (SM)→ C∞c (GM) ; If (γ) :=

∫ ∞−∞

f (γ(t))dt,

where γ(t) = ΦtH(ξ) for any ξ ∈ SM such that γ = [ξ]ΦH

.

I(H[f ]) = 0, so If ∈ C∞c (GM) is well-defined.

The Santalo-formula takes the form∫SM

f (ξ)ωSM(ξ) =

∫GMIf (γ)ωGM(γ) , f ∈ C∞c (SM).





Definition


I : C∞c (SM)→ C∞c (GM) ; If (γ) :=

∫ ∞−∞

f (γ(t))dt,


.



f (ξ)ωSM(ξ) =






Definition


I : C∞c (SM)→ C∞c (GM) ; If (γ) :=

∫ ∞−∞

f (γ(t))dt,


.



f (ξ)ωSM(ξ) =





First main theorem

Lemma

Let (M,F ) be a non-trapping Finsler-space. Then

Ker(I : C∞c (SM)→ C∞c (GM)

)= Ran

(H : C∞c (SM)→ C∞c (SM)

)It is not known in the general case, if there are non-zerof ∈ C∞c (M) and u ∈ C∞c (SM) such that f πSM = Hu.

Theorem

Let (M,F ) be a non-trapping Finsler-space without conjugatepoints. If f ∈ C∞c (M) satisfies f πSM = Hu for someu ∈ C∞c (SM), then f = 0.

So in this case f ∈ C∞c (M) is determined by I(f πSM).




First main theorem

Lemma



)= Ran

(H : C∞c (SM)→ C∞c (SM)


Theorem






First main theorem

Lemma



)= Ran

(H : C∞c (SM)→ C∞c (SM)


Theorem






First main theorem

Lemma



)= Ran

(H : C∞c (SM)→ C∞c (SM)


Theorem






Proof of first main theorem I

The commutator relations on SM yield

Hvs

divhs∇ u + (n − 1)HHu = H

hdiv

vs∇ u =

hdiv H

vs∇ u +

vsdiv R

vs∇ u

=h

divvs∇Hu −

hdiv

hs∇ u +

vsdiv R

v∇ u.

Multiplying by w ∈ C∞c (SM) and integrating over SM usingthe divergence theorem and the Santalo-formula we arrive atthe integrated Pestov identity∫

SM

(g(

vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(

hs∇w ,

hs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .




Proof of first main theorem I

The commutator relations on SM yield

Hvs

divhs∇ u + (n − 1)HHu = H

hdiv

vs∇ u =

hdiv H

vs∇ u +

vsdiv R

vs∇ u

=h

divvs∇Hu −

hdiv

hs∇ u +

vsdiv R

v∇ u.

Multiplying by w ∈ C∞c (SM) and integrating over SM usingthe divergence theorem and the Santalo-formula we arrive atthe integrated Pestov identity∫

SM

(g(

vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(

hs∇w ,

hs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .




Proof of first main theorem II

Using the commutator relations again we can write

g(hs∇w ,

hs∇ u) = g(

vs∇Hw − H

vs∇w ,

vs∇Hu − H

vs∇ u)

= g(vs∇Hw ,

vs∇Hu)− g(

vs∇Hw ,H

vs∇ u)

− g(Hvs∇w ,

vs∇Hu) + g(H

vs∇w ,H

vs∇ u)

= g(vs∇Hw ,

vs∇Hu)− g(

vs∇Hw ,

vs∇Hu −

hs∇ u)

− g(vs∇Hw −

hs∇w ,

vs∇Hu) + g(H

vs∇w ,H

vs∇ u).

= g(Hvs∇w ,H

vs∇ u)− g(

vs∇Hw ,

vs∇Hu)

+ g(vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu).




Proof of the Main Theorem III

Plugging this into the Pestov-identity we obtain

∫SM

(g(

vs∇Hw ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(H

vs∇w ,H

vs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .

Assume that Hu = f πSM for some f ∈ C∞c (M). Then thefirst term in the left hand side vanishes.

Let w = u. Since there are no conjugate points, the righthand side is non-negative by the index-lemma, so we have

−(n − 1)

∫SM

(Hu)2ωSM = 0.






∫SM

(g(

vs∇Hw ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(H

vs∇w ,H

vs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .



−(n − 1)

∫SM

(Hu)2ωSM = 0.






∫SM

(g(

vs∇Hw ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(H

vs∇w ,H

vs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .



−(n − 1)

∫SM

(Hu)2ωSM = 0.




Ultrahyperbolic equation

Recall the integrated Pestov-identity∫SM

(g(

vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(

hs∇w ,

hs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .

Let w = u and Hu = f for some f ∈ C∞c (SM) that satisfies

hdiv

vs∇ f = 0.

This ultrahyperbolic equation is equivalent to the condition∫SM

g(hs∇w ,

vs∇Hu)ωSM = 0 for every w ∈ C∞c (SM).






(g(

vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(

hs∇w ,

hs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .


hdiv

vs∇ f = 0.


g(hs∇w ,







(g(

vs∇Hw ,

hs∇ u) + g(

hs∇w ,

vs∇Hu)− (n − 1)(Hw)(Hu)

)ωSM

=

∫SM

(g(

hs∇w ,

hs∇ u)− g(

vs∇w ,R

vs∇ u)

)ωSM .


hdiv

vs∇ f = 0.


g(hs∇w ,





Second main theorem

This immediately gives the second main theorem.

Theorem

Let (M,F ) be a non-trapping Finsler-space with non-positiveflag-curvature. If f ∈ C∞c (M) satisfies

hdiv

vs∇ f = 0

and f = Hu for some u ∈ C∞c (SM), then f = 0.

So ultrahyperbolic functions f ∈ C∞c (SM) are uniquelydetermined by their geodesic ray transform If onnon-trapping Finsler-spaces with non-positive flag curvature.

It seems to be a reasonable hypothesis that this would be thecase on all non-trapping Finsler spaces.




Second main theorem


Theorem


hdiv

vs∇ f = 0







Second main theorem


Theorem


hdiv

vs∇ f = 0







Tensor tomography

Define d : Sk(M)→ Sk+1(M) for symmetric tensor fields oforder k on a Riemannian manifold (M, g) by

(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).

Let δ : Sk(M)→ Sk+1(M) be its formal adjoint.

A ∈ Sk+1(M) is potential, if A = dB for some A ∈ Sk(M).

A ∈ Sk(M) is solenoidal, if δA = 0.

For compactly supported A ∈ Sk(M) define fA ∈ C∞c (SM) by

fA(ξ) := A(ξ, . . . , ξ).

The geodesic ray transform of A ∈ Sk(M) is then simply IfA.

Our theorems extend the results in tensor tomography, since

H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.




Tensor tomography


(dA)(X1, . . . ,Xk+1) := 1(k+1)!

∑σ∈S(k+1)

(∇Xσ(k+1)A)(Xσ(1), . . . ,Xσ(k)).





fA(ξ) := A(ξ, . . . , ξ).



H[fA] = fdA andh

divvs∇ fA = fδA.


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