generalized fermat principle and zermelo navigation: a...

OverviewFinslerian and spacetime viewpoints

Applications: Fermat and Zermelo

IntroductionBackground: Riemann-Finsler, LorentzFermat vs Zermelo

Generalized Fermat principle and Zermelonavigation: a link between

Lorentzian and Generalized Finslerian Geometries

Miguel Sanchez

Universidad de Granada, IEMath-GR

8th Int. Meeting on Lorentzian Geom. (Malaga, 23/09/2016)

M. Sanchez Generalized Fermat and Zermelo




General aims

Show a correspondence between problems in:

1 Lorentzian Geometry

2 Finslerian and generalized (singular) Finslerian Geometry





General aims

Applications:

1 For Lorentz: appropriate description of relativistic notions inFinslerian terms

2 For Finsler: new problems and results by using Lorentzianviewpoint

3 Dynamical systems/optimal control:Non singular description of apparently singular problems

Emphasis in the most general viewpoint! extended and singular Finslerand (non-singular) relativistic interpretations





General aims

Applications:




Emphasis in the most general viewpoint! extended and singular Finsler

and (non-singular) relativistic interpretations





General aims

Applications:




Emphasis in the most general viewpoint! extended and singular Finslerand (non-singular) relativistic interpretations





General aims

Focus on a pair of variational goals:

Generalization of relativistic Fermat principle

Solution to generalized Zermelo problem(navigation in arbitrary wind)





Joint work with E Caponio and MA Javaloyes

Main reference:

Caponio, Javaloyes, S. arxiv 1407.5494 [CJS]

Previous work

Caponio, Javaloyes, Masiello’11 [CJM]

Caponio, Javaloyes, Sanchez’11 [CJS11]

(+ others)





Background: Riemannian-Finsler, Lorentz





Finslerian elements

Finsler metric F : TM → R: generalization of Riemann’s replace pointwise Euclidean scalar products by norms Fp

1 Smooth: F smooth outside zero section or, equally:

1 Smooth indicatrix (set of unit spheres)Σ := F−1(1)(⊂ TM)

2 Transversality Σ t TpM for all p ∈ M

2 Strong convexity of pointwise indicatrices (unit spheres)Σp = F−1

p (1) ovaloids (II > 0 in particular strictly convex)bound the unit (open) ball Bp = F−1

p ([0, 1))

3 No reversibility assumed: Fp(λvp) = λFp(v) just for λ ≥ 0(and even 0p ∈ Bp no barycenter) non-symmetric distance





Finslerian geodesics

Energy functional: E (γ) = (1/2)∫F 2(γ′(s))ds

Geodesics:

Critical points of E (for length pregeodesics)Locally minimize: energy, non-symmetric distance

An example for interpretations: mobile

Σp: maximum velocity depending on p (Riemann. case)and direction (properly Finsler)Length of (unit) curves ≡ arrival time at maximum speed[non-reversible](Pre)geodesics ≡ locally fastest paths [non-reversible]Some cases:

hill (Matsumoto),mild wind (Zermelo, Shen et al.)





Zermelo Navigation

• Plane/Zeppelin in the air with a (stationary) wind• Submarine in the sea dragged by a (stationary) current Zermelo problem: find fastest path between two points





Zermelo navigation

Riemannian metric 〈·, ·〉: unit spheresSp maximum speed zeppelin/air

Vector field W : velocity of the wind respect to Earth

Finsler model indicatrix Σp = Sp + Wp (Randers) metric Z

Z -geodesics solve Zermelo’s...

under mild wind, 〈W ,W 〉 < 1





Zermelo navigation

Riemannian metric 〈·, ·〉: unit spheresSp maximum speed zeppelin/air

Vector field W : velocity of the wind respect to Earth

Finsler model indicatrix Σp = Sp + Wp (Randers) metric Z

Z -geodesics solve Zermelo’s... under mild wind, 〈W ,W 〉 < 1M. Sanchez Generalized Fermat and Zermelo




Zermelo navigation

Note: if the wind is not mild... f. ex. critical 〈Wp,Wp〉 = 1

“Singular” Finsler metric: 0p ∈ Σp (“Kropina metric”)

Forbidden directions unreachable regions





Zermelo navigation

Strong wind 〈Wp,Wp〉 > 1

No Finsler metric but Σ ⊂ TM still makes sense“Wind Riemannian/ Finslerian structure”





Lorentzian manifolds and spacetimes

Lorentz metric g , (−,+, · · ·+)

Cone structure (conformal class): vp ∈ TpM \ {0}timelike g(vp, vp) < 0, lightlike g(vp, vp) = 0; (≤ 0, causal)spacelike g(vp, vp) > 0

Spacetime: g + time orientation





Remark on projection of cones

Lorentzian is richer (rather than a generalization) of Riemannian

Choose a spacelike hyperplane Π and a transversal vector K atsome p ∈ M:

K ⊥ Π (and unit): Euclidean indicatrix of Π from the cone






Lorentzian is richer (rather than a generalization) of RiemannianChoose a spacelike hyperplane Π and a transversal vector K atsome p ∈ M:

K ⊥ Π (and unit): Euclidean indicatrix of Π from the cone






Choose a spacelike hyperplane Π and a transversal direction K atsome p ∈ M:

K timelike but non orthogonal: “Finslerian” indicatrix fromthe cone







K lightlike: Koprina/ “critical wind” indicatrix







K spacelike: “strong wind” indicatrix (+ cone on Π)





Lorentzian geodesics

Geodesics:

Critical points of E (γ) = (1/2)∫g(γ′(s), γ′(s))ds

Euler-Lagrange equation in terms of Levi-Civita ∇g(γ′, γ′) constant: timelike, lightlike, spacelike

Local maximization properties only for causal (timelike orlightlike)

Interpretations

f-d timelike (unit) curves ≡ observers

f-d lightlike geod. ≡ light rays

Fermat principle ≡ light arrives fastest/critical





Fermat principle

Classical relativistic Fermat principle (Kovner ’90, Perlick ’90):

Point p ∈ M (event), observer α : I ⊂ R→ M

Among lightlike curves from p to α:pregeodesics are critical curvesfor the arrival time t ∈ I at α (parameter of α) in particular, first arriving (minima) are pregedesics

Existence of lightlike geodesics, multiplicity, Morse relations:Existence of critical points: Fortunato, Giannoni, Masiello ’95, etc.





Link Fermat/Zermelo

Start with Zermelo on M and represent graphs of curves adding acoordinate “time” as a dimension more





Link Fermat/Zermelo

Maximum velocities: add a “unit of time” to all theindicatrices cone structure compatible with a (conformalclass of) Lorentz g (independent of t)





Link classical Fermat/Zermelo

Now, for mild wind, let x , y ∈ M:

“Vertical” lines R× {x}, R× {y} are timelike observers

Connecting Z -unit curve c : [0,T ]→ M ⇐⇒g -lightlike curve γ(t) = (t, c(t)) on R×Mfrom (0, x) to (T , y) ∈ R× {y}c unit Z -geodesic (critical for length) ⇐⇒γ = (t, c(t)) a lightlike g -pregeodesic ⇐⇒γ Fermat critical curve from p to the observerαy (s) = (s, y) ∈ R× {y}





Generalized Fermat and Zermelo

What about if the wind is not mild?

Arrival vertical curve (observer?) R× {y} non-timelike

Goal However, there is still a Fermat principle

No Zermelo (Finsler) metric but a wind Riemann. st.

Goal

wind Riemm./ Finslerian st. admit a notion of geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem

Overall goal basics on wind Finslerian, spacetimes andFinsler/Lorentz correspondence (including Randers/stationaryspacetimes)










Goal

wind Riemm./ Finslerian st. admit a notion of geodesicThe relation with spacetimes holds

Fermat principle solves Zermelo problem











Goal

wind Riemm./ Finslerian st. admit a notion of geodesicThe relation with spacetimes holds Fermat principle solves Zermelo problem






Plan

General wind Finslerian structures+ Spacetime viewpoint

Applications: generalized Fermat and Zermelo (and more)




General Wind Finslerian structuresSpacetimes and Wind RiemannianConclusion on geodeics

Notion of wind Finslerian structure

Definition

For a vector space V :—Wind Minkowskian structure: Compact strongly convexsmooth hypersurface ΣV embedded in V—Unit ball B Bounded open domain B enclosed by ΣV

—Conic domain A : region determined half lines from 0 to B.

0 ∈ B ⇒ A = V

0 ∈ ΣV ⇒ A = half space

0 6∈ B ⇒ properly conic A






Definition

For a manifold M:— Wind Finslerian str.: smooth hypersurface Σ ↪→ TM:Σp = Σ ∩ TpM is wind Minkowski in TpM (+transversality)— Ball at p: Bp ⊂ TpM ( Ap)— Conic domain A := ∪pAp

— Region of strong wind: Ml := {p ∈ M : 0 /∈ Bp}— Properly conic domain: Al := Σ ∩ π−1(Ml)






Proposition

Any Σ is the displacement of the indicatrix of Finsler metric F0

along some vector field W :

F0

(v

Z (v)−W

)= 1,

(v ∈ Σ⇐⇒ Z (v) is a solution)

— Uniqueness if 0p is required to be the barycentre of each Fp— Wind Riemannian: displacement of F0 =

√gR (ellipsoids)






Proposition

Any Σ determines two “conic” pseudo-Finsler metrics:

(i) F : A→ [0,+∞) conic Finsler metric on all M,

(ii) Fl : Al → [0,+∞) Fl is a Lorentz-Finsler metric in the regionMl of strong wind with F < Fl . Moreover, a cone structureappears






Cone structure on Ml :

Limit region F = Fl : Cone ∪A: “Σ-admissible vectors”it characterizes accessibility from x0 to x1 (x0 ≺ x1)For wind Riemannian, associated to a Lorentzian metricCurvatures for F and Fl are computable [JV]





Balls and geodesics for wind Finsler

No “distance” dF for Σ

redefinitions of balls and geodesics for any wind Finsler

Σ admissible γ from x0 to x : γ′ in a closure of A(⊃ Al).(Forward/backwards) wind balls [mild wind: usual open balls]

B+Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : `F (γ) < r < `Fl

(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) < r < `Fl

(γ)}.

Wind c-balls:

B+Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x0 to x : `F (γ) ≤ r ≤ `Fl

(γ)},B−Σ (x0, r) = {x ∈ M | ∃ γ Σ-a. x to x0 : `F (γ) ≤ r ≤ `Fl

(γ)}.

Closed balls: (usual closures) B+Σ (x0, r), B−Σ (x0, r)

B+Σ (x0, r) ⊂ B+

Σ (x0, r) ⊂ B+Σ (x0, r)






w-convexity: c-balls are closed (extend usual convexity)






w-convexity: c-balls are closed (extend usual convexity)M. Sanchez Generalized Fermat and Zermelo





Geodesic parametrized by arc length: Σ-admissible curve s.t.γ(t + ε) ∈ B+

Σ (γ(t), ε) \ B+Σ (γ(t), ε) (locally, i.e., for small ε > 0)

Proposition

When γ(t) ∈ A (open):γ geodesic of (M,Σ) (parametrized by arc length) ⇔γ (unit) geodesic for either F or Fl .





Going further

What about when γ is Σ-admissible but belongs to ∂A?

In principle, one could follow but there are technicaldifficulties (“abnormal” geodesics) Focus on wind Riemannian (but generalizable to Finslerian)

Develop in a “non-singular” way through the spacetimeviewpoint





Definition of SSTK spacetime

SSTK s.t: standard with a space-transverse Killing v.f. (K = ∂t)(R×M, g), g = −Λdt2 + 2ωdt + g0

≡ −(Λ ◦ π)dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g0

for Λ (function), ω (1-form), g0 (Riemannian) on M withΛ > −‖ω‖2

0 (Lorentz restriction)

Cases:

ω = 0,Λ ≡ 1: Product st :R×M, g = −dt2 + π∗g0 ≡ −dt2 + g0

ω = 0,Λ > 0 Static st :R×M, g = −Λdt2 + g0 = Λ(−dt2 + g0/Λ) Conformal to product





Product/ static case

K = ∂t induces a Riemannian metric g0(≡ g0/Λ) on M





SSTK spacetimes

SSTK (R×M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −‖ω‖20)

Cases:

Λ ≡ 1, arbitrary ωNormalized (standard) stationary s.t. :R×M, g = −1dt2 + 2ωdt + g0

Λ > 0, arbitrary ωStationary s.t. : R×M, g = −Λdt2 + 2ωdt + g0

Conformal to normalized Λ(−dt2 + 2(ω/Λ) + (g0/Λ)

)





Stationary case

K = ∂t induces the indicatrix of a Finslerian metric on MM. Sanchez Generalized Fermat and Zermelo




Stationary case

Induces a (pair of) Finslerian metric on MM. Sanchez Generalized Fermat and Zermelo




Appearance of Finsler

For each v ∈ TxM:

Future-d. lightlike vector (F+(v), v)

Past-d. lightlike vector (−F−(v), v)

where F± : TM → R, for normalized Λ ≡ 1:

F±(v) =√g0(v , v) + ω(v)2 ± ω(v)

F±: Finsler metrics of Randers type, “Fermat metrics”

F−(v) = F+(−v), F− “reversed metric” of F+ (≡ F ).





Expression with wind

F : Randers metric with indicatrix

Σ = SR + W

W : vector field (wind):g0(W , ·) = −ωSR : Riemannian metric indicatrix (sphere bundle) ofgR = g0/(1 + |W |20)

Necessarily gR(W ,W ) < 1 (mild wind)





General SSTK spacetime

SSTK (R×M, g), g = −Λdt2 + 2ωdt + g0 (with Λ > −‖ω‖20)

General case:

K := ∂t Killing and

timelike Λ > 0lightlike Λ = 0spacelike Λ < 0

The projection t : R×M → R time function[for v causal dt(v) > 0 defines the future direction]





Interpretation of K = ∂t





Interpretation of K = ∂t

K = ∂t induces a wind-Riemannian structure





SSTK ←→ Wind Riemannian

Proposition

Σ = {v ∈ TM : (1, v) is (future-p.) lightlike in T (R×M)}is a wind Riemannian structure on M (Fermat structure of theconformal class of the SSTK) with;

Σ computable from:

Wind vector W : g0(·,W ) = −ωRiemannian metric gR = g0/(Λ + g0(W ,W ))

Moreover, cone structure on Ml computable from the sign.changing metric h (Lorentzian (+,−, . . . ,−) on Ml)

h(v , v) = Λg0(v , v) + ω(v)2

Conversely, each wind Riemannian structure selects a uniqueconformal class of SSTK





Unified viewpoint

Merging SSTK and Wind Riemannian for geodesics:

Theorem

For associated SSTK ↔ Σ, these classes of curves coincide:

1 Projections on M of the future-d. lightlike pregeodesics forSSTK R×M

2 Pregeodesics for wind Riemannian Σ on M

3 The set of all the pregeodesics for

F (locally minimizing F -distance, including critical/Kropinaand strong wind regions)Fl (on strong wind region Ml , locally maximizing) andlightlike for −h (Lorentzian metric on Ml)




Fermat and ZermeloGeneralized Fermat: Sketch of proof... and some of the applications to Lorentz

General Fermat principle

Theorem (CJS)

Let (L, g) be any spacetime and any (smooth embedded)arbitrary arrival curve α.For any piecewise smooth future-directed lightlike curve γ fromp0 to α, such that γ is not orthogonal to α (at its arrival):

γ : [a, b]→ L is a pregeodesic ⇐⇒it is a critical point of the arrival functional (parameter of α)

Includes classical one (Kovner [Ko], Perlick [Pe]): α timelike

Based on a sharp characterization of which vector fields on γcome from a variation by lightlike curves from p0 to α





Sharpest Fermat for SSTK

Theorem

Let (R×M, g) SSTK, x0, x1 ∈ M, x0 6= x1, p0 = (t0, x0) andγ(s) = (ζ(s), x(s)) lightlike from p0 to R× {x1}.a) γ critical point of the arrival time T =⇒ pregeodesic.b) γ pregeodesic ⇐⇒ (Cγ = g(∂t , γ) constant and:)

(i) Cγ < 0, x lies in A, x pregeodesic of F parametrized withh(x , x) = const., γ is a critical point of T (locally min.)

(ii) Cγ > 0, Λ < 0 on all x , x a pregeodesic of Fl parametrizedwith h(x , x) = const., γ critical point of T (locally max.)

(iii) Cγ = 0, Λ ≤ 0 on all x : whenever Λ < 0, x lightlike geodesicof h/Λ on M; Λ vanishes on x only at isolated points where xvanishes.





Characterization for Zermelo

For arbitrary gR ,W (generalizing Shen’s et al. [Sh], [BRS]):

Solutions x(s) of Zermelo’s connecting x0, x1 are(pre)geodesics of Σ and they lie in exactly one of the threeprevious cases.

If solution exists if:

1 An admissible curve exists from x0 to x1

(⇐⇒ x0 ≺ x1 for −h on M , whereh(u, v) := (1− gR(W ,W ))gR(u, v) + gR(u,W )(W , v))

2 and Σ is w-convex(⇐⇒ associated SSTK causally simple)





General Fermat principle: precise statement

Theorem (CJS)

(L, g) any spacetime, α any arrival curve (smooth, embedded)Np0,α := {γ : [a, b]→ L|γ piece. smooth f.-d. light. from p0 to α}Arrival functional: T (γ) = α−1(γ(b)), ∀γ ∈ Np0,α

γ ∈ Np0,α with γ(b) 6⊥ α:pregeodesic ⇐⇒ critical point of T





Lemma 1: critical in terms of admissible v.f. Z

Z v.f. on γ admissible: variational v.f. by means of longitudinalcurves γw ∈ Np0,α

Lemma 1. γ critical for T ⇔ Z (b) = 0, ∀Z admissible

Proof.

Z (b) =d

dwγw (b) |w=0=

d

dwα(T (γw )) |w=0

=

(d

dwT (γw ) |w=0

)α(T (γ))





Lemma 2: characterization of admissible Z

Lemma 2. Z v.f. on γ, Z (a) = 0, with Z (b) ‖ α:Z admissible ⇐⇒ Z ′ ⊥ γ

(⇒ trivial)

• Note: (⇐) Typical results(i) no lightlike longit. or(ii) geodesic γ ⊥ α non-lightlike






Lemma 2. Z v.f. on γ, Z (a) = 0, with Z (b) ‖ α:Z admissible ⇐⇒ Z ′ ⊥ γ

Sketch (⇐):

Neighborhood of γ covered by a finite number of coordinateswhich looks like a t-dependent SSTK and:(a) γ nowhere orthogonal to ∂t(b) α ‖ ∂t at γ(b).






Lemma 2. Z v.f. on γ, Z (a) = 0, with Z (b) ‖ α:Z admissible ⇐= Z ′ ⊥ γ

Neighborhood of γ covered by a finite number of coordinateswhich looks like a t-dependent SSTK and:(a) γ nowhere orthogonal to ∂t(b) α ‖ ∂t at γ(b).

Put Z = (Y ,W ) in each local splitting R× SW : fixed endpoint variation for x(s) = ΠS(γ(s))

Lift this variation imposing longitudinal curves in Np0,α

diff. eqn. for t coordinate

Check: (i) consistency eqn. from Z ′ ⊥ γ, Z (b) ‖ α and (b)(ii) non-degeneracy eqn. (uniqueness) because of (a) constructed admissible v.f. agrees Z





Lemma 3: construction of admissible Z

Lemma 3. Explicit construction of such admissible Z :— Choose U along γ at no point orthogonal— For each W along γ with W (a) = W (b) = 0, put:

ZW (s) = W (s) + fW (s)U(s), where

fW (s) = −e−ρ(s)

∫ s

a

g(W ′, γ)

g(U, γ)eρdµ with ρ(s) =

∫ s

a

g(U ′, γ)

g(U, γ)dµ

Sketch of proof.ZW is admissible: check g(Z ′W , γ) = 0 (eqn for fW )






Lemma 3. Explicit construction of admissible Z :— Choose U along γ at no point orthogonal— For each W along γ with W (a) = W (b) = 0, put:ZW (s) = W (s) + fW (s)U(s)

Sketch. Any admissible Z is some ZW :

1 Define W (s) = Z (s)− (c(s − a)/(b − a))U(s)with c s.t. Z (b) = cU(b).








1 Define W (s) = Z (s)− (c(s − a)/(b − a))Uwith c s.t. Z (b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible...








1 Define W (s) = Z (s)− (c(s − a)/(b − a))Uwith c s.t. Z (b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible ...

3 ... but Z − ZW = (fW (s)− c(s − a)/(b − a))U =: p(s)U








1 Define W (s) = Z (s)− (c(s − a)/(b − a))Uwith c : Z (b) = cU(b).

2 Z and ZW admissible ⇒ Z − ZW admissible ...

3 ... but Z − ZW = (fW (s)− c(s − a)/(b − a))U =: pU

4 As 0 = g((pU)′, γ) = pg(U, γ) + pg(U ′, γ) and p(a) = 0⇒ p ≡ 0





General Fermat principle: proof

Sketch proof of theorem

Lemma 1: γ critical for T⇔ Z (b) = 0 for all admissible Z







Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0, as W (b) = 0 (= W (a))







Lemma 3 (chosen U): Z = ZW = W + fWU⇔ fW (b) = 0 (as W (b) = 0 = W (a))

Using the explicit formula for fW :

⇔∫ ba

g(W ′,γ)g(U,γ) e

ρdµ = 0.









⇔∫ ba


ρdµ = 0.

Integrating by parts (with smooth W vanishing at breaks)

⇔∫ ba g(W , (ϕγ)′)dµ = 0, for some function ϕ









⇔∫ ba


ρdµ = 0.

Integrating by parts (with smooth W vanishing at breaks)

⇔∫ ba g(W , (ϕγ)′)dµ = 0, for some function ϕ

Using standard variational arguments:⇔ (ϕγ)′ = 0 (well-known characterization of pregeodesics)





Characterization of the causal ladder

SSTK are always stably continuous (t time function)

Causal continuity characterizable in terms of the associatedwind Finslerian structure

Causal simplicity and global hyperbolicity especially interesting





Causal simplicity of SSTK

Causal simplicity (for SSTK spacetimes, J±(p) closed)⇐⇒ w-convexity (c-balls are closed)

Variational methods type Fortunato et al. [FGM], becomeapplicable providing results on existence and multiplicityApplications even for stationary s.t.:

Extension of previous results

Applications to gravitational lensing [CGS]

...now extensible to SSTK







Variational methods type Fortunato et al. [FGM], becomeapplicable providing results on existence and multiplicity

Applications even for stationary s.t.:










Variational methods type Fortunato et al. [FGM], becomeapplicable providing results on existence and multiplicityApplications even for stationary s.t.:








Chracterization of global hyperbolicity

Global hyperbolicity (J+(p) ∩ J−(q) compact)equivalent to any of

1 All intersections B+Σ (x0, r1) ∩ B−

Σ (x1, r2) compact

2 All intersections B+Σ (x0, r1) ∩ B−

Σ (x1, r2)3 In the case of K timelike (stationary/Randers):

Compactness of B+s (p, r)

Spacelike slices St = {(t, x) : x ∈ R×M} Cauchy hypers.(crossed exactly once by any inextendible causal curve)equivalent to any of:

1 All closed B+Σ (x , r), B−

Σ (x , r) compact

2 All c-balls B+Σ (x , r), B−

Σ (x , r) compact3 Σ (forward and backward) geodesically complete





An unexpected application for Riemann, Finsler & Lorentz

Application to Riemann/Finsler/wind Finsler Geom. [FHS]

Relativistic notion of causal boundary New notion of boundary extending classical Cauchy,Gromov and Busemann for Riemannian and FinslerianGeometries, now extensible to wind Finsler

Application to Lorentz Geom. [FHS]:description of the c-boundary of static/ stationary/ SSTK s.t. interms of Riemannian [FHa]/ Finslerian/ wind Finslerian elements





...and some other applications to Finsler [CJS11]

1 To weaken completeness by compactness of balls B+s (p, r)

(Heine-Borel) in classical Finsler theorems such as Myers

2 Characterization of the differentiability of the distance from asubset d(C , ·) with applications to Hamilton Jacobi equation(extended by Tanaka & Sabau [TS])

3 Properties of completeness in classes of projectively relatedmetrics (extended by Matveev ’12)

4 Properties of the Hausdorff dimension for the cut locus,extending a previous result of Lee & Nirenberg ’06 [LN]

5 Appropriate description of Randers manifolds of constant flagcurvature [CJS14] and Javaloyes & Vitorio [JV]





References

[CJS] Caponio, Javaloyes, Sanchez: arxiv 1407.5494

Lorentz-Finsler

[CJS11] Caponio, Javaloyes, Sanchez: Rev. Mat. Iberoam (2011)[CJM] Caponio, Javaloyes, Masiello: Math. Ann. (2011)+ [FHS] Flores, Herrera, Sanchez: Memoirs AMS (2013)

Fermat’s principle, visibility and lensing

[Ko] Kovner: Astroph. J. (1990)[Pe] Perlick: Class. Quant. Grav (1990)[FGM] Fortunato, Giannoni, Masiello, J. Geom. Phys. (1995)[CGS] Caponio, Germinario, Sanchez, J. Geom. Anal. (2016)





References

Zermelo’s navigation

[Sh] Shen: Canadian J. Math. (2003)[BRS] Bao, Robles, Shen: J. Diff. Geom. (2004)[YS] Yoshikawa, Sabau: Geom. Dedicata (2014)[JV] Javaloyes, Vitorio, arXiv:1412.0465.

Related Finslerian problems

[FHa] Flores, Harris: Class. Quant. Grav. (2007)[JV] Javaloyes, Vitorio, in progress[LN] Li, Nirenberg: Comm. Pure Appl. Math. (2005)[Ma] Matveev: Springer Proc. Math. & Stat. 26 (2013)[TS] Tanaka, Sabau: arXiv:1207.0918





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