Wind Riemannian spaceforms and Randers metrics of constant flag curvature

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by  Miguel Angel Javaloyes

WIND RIEMANNIAN SPACEFORMS
AND RANDERS METRICS OF
CONSTANT FLAG CURVATURE
7
1
0 MIGUELANGELJAVALOYESANDMIGUELSA´NCHEZ
2

n
a Abstract. Recently, wind Riemannian structures (WRS) have been intro-
J duced as a generalization of Randers and Kropina metrics. They are con-
structed from the natural data for Zermelo navigation problem, namely, a
5
Riemannian metric gR and a vector field W (the wind), where, now, the re-
] strictionofmildwindgR(W,W)<1isdropped.
G Here,themodelsofWRSspaceformsofconstant flagcurvature aredeter-
mined. Indeed, the celebrated classification of Randers metrics of constant
D
flag curvature by Bao, Robles and Shen [2], extended to the Kropina case in
. the works by Yoshikawa, Okubo and Sabau [29, 30], can be used to obtain
h
the local classification. For the global one, a suitable result on completeness
t
a for WRS yields the complete simply connected models. In particular, any of
m the local models in the Randers classification does admit an extension to a
uniquemodelofwindRiemannianstructure,evenifitcannotbeextendedas
[
acompleteFinslerianmanifold.
1 Thus,WRS’semergeasthenaturalframeworkfortheanalysisofRanders
v spaceforms and, prospectively, wind Finslerian structures would become im-
3 portant for other global problems too. For the sake of completeness, a brief
7 overview aboutWRS(includingausefullinkwiththeconformalgeometry of
2 aclassofrelativisticspacetimes)isalsoprovided.
1
0
.
1
0 1. Introduction
7
1 Wind Riemmanian structures (WRS’s) are generalizations of the indicatrices of
: both, Randers and Kropina metrics on a manifold M, introduced in [7] for several
v
purposes. The simplestone is thatthey providea naturalframeworkfor modelling
i
X
Zermelonavigationproblem. Namely,considerthe motionofa zeppelin: its engine
r is able to develop a maximum speed with respect to the air, modelled by the
a
indicatrix of a Riemanian metric g , but the air is moving with respect to earth
R
witha(time-independent)velocitymodelledbyavectorfieldW. So,themaximum
speedofthezeppelinwithrespecttoearthismodelledbyΣ=S +W. Thisisthe
R
indicatrix of a Randers metric when g (W,W)<1, but when g (W,W)=1, it is
R R
the indicatrix of a Kropina metric, a singular Finsler metric in the sense that it is
notdefinedinallthedirections;thegeodesicsofsuchmetricssolveZermeloproblem
offinding the fastestpath betweentwoprescribedpoints. Inthe moregeneralcase
in that no restriction on g (W,W) is imposed, Σ is a WRS and, as proven in [7],
R
such a WRS admits a notion of geodesic which also solves Zermelo navigation.
Date:30.11.2016.
2010Mathematics Subject Classification: Primary53C60,53C22
Key words: Finslermetrics,RandersandKropinametrics,Zermelonavigation,windFinslerian
structure,constantflagcurvature,modelspace,Randersspaceforms,(α.β)-metric.
1
2 M.A.JAVALOYESANDM.SA´NCHEZ
However,there is a deeper motivationforstudying WRS’s becauseofthe existence
of a link between the geometry of such WRS’s and the conformal geometry of a
class of relativistic spacetimes, those which are standard with a space-transverse
Killing vector field, or SSTK spacetimes.
The existence of a fruitful correspondence between the geometry of Randers
manifolds and the conformal geometry of stationary spacetimes (a particular case
of SSTK ones) has already been pointed out and systematically exploited in [6, 7,
12,10,14]andothers. However,theimportanceofthecorrespondenceinthecaseof
generalWRS’s becomes especially usefulfor the Finslerianframework. Indeed, the
existenceofapparentlysingularFinslerianelementsforWRS’s(suchasregionswith
a Kropina metric) is reinterpreted from the Lorentzian viewpoint in a completely
non-singular way. In particular, geodesics for WRS’s can be seen as projections
of lightlike pregeodesics of SSTK spacetimes. Moreover, geodesic completeness of
WRS’s becomes equivalent to the global hyperbolicity of the spacetime. Notice
thatboth, lightlike pregeodesicsandglobalhyperbolicityare conformallyinvariant
elements in Lorentzian Geomety.
OurmainaimalongthepresentnotewillbetoshowanapplicationofWRS’s to
Randers metrics of constant flag curvature (CFC). The complete classification of
these manifolds was a landmark in Finslerian Geometry, obtained by Bao, Robles
andShen[2]. The solutionsadmitaneatdescription,whichcanbe summarizedby
saying that g must have constant curvature and W must be a homothetic vector
R
field. Froma globalviewpoint,however,thereis astrikingdifference withthe clas-
sical(simply connected)modelsofRiemannianmanifoldsofconstantcurvature: in
the Randers case, some of the models are necessarily incomplete. Namely, some
localmodelscannotbe extendedtoacompleteone,andonlyitsinextensibility can
be claimed. Nevertheless, we will see that this comes from the fact that g (W,W)
R
may not remain bounded by 1, and we will show that all the local models admit
a unique extension as a complete simply connected WRS. Moreover, taking into
accountthe classificationofKropinametrics ofCFC byYoshikawaandOkubo[29]
andYoshikawaand Sabau[30] we will provideboth, the localand globalclassifica-
tion of WRS’s.
This article is organized as follows. In Section 2, the motivations and necessary
results on WRS’s are briefly summarized. In Section 3, the classificationof WRS’s
of CFC is achieved. For this purpose, we summarize first the known results in the
Randers[2]andKropinacases[29,30](subsection3.1),andshowhowtheseresults
can be used for the local classification (subsection 3.2). The global classification
is obtained in subsection 3.3. Here, the key issue is completeness. First, we prove
a result of independent interest on completeness for WRS’s, Theorem 3.10; the
subtleties of this result are stressed in Example 3.11. The global classification is
reached in Theorem 3.12, by using the previous result on completeness, the local
result and a specific study of the case when W is properly homothetic. Finally, in
the appendix (Section 4), we summarize some known results on Killing and homo-
thetic vector fields. The results in subsection 4.1 are useful to understand when
Randers models can be naturally extended in WRS’s, and those in subsection 4.2
foraneatdescriptionoftheKropinacaseandtheregionoftransitionbetweenmild
and strong wind.
WIND RIEMANNIAN STRUCTURES 3
h(v,v) 0 and g (W,v)>0 region
R
≥
z
y g (W,v)>0 region
R
p
C
Σ
p
Wp x
Figure 1. The diagram on the left represents the domain of F
p
when Λ(p)<0. In this case, we consider g the Euclidean metric
R
in R3 and W = (0,0,2). The condition h(v,v) 0 determines
≥
that the region must lie in the two cones of h and the condition
g (W,v)>0thatitmustbe containedinthe half-spacez >0;so,
R
the cone is selected. The diagram on the right represents the
p
C
indicatrix and the domain of F , z >0, in the Kropina case when
p
Λ(p)=0 and W =(0,0,1).
2. A Finslerian overview on Wind Riemannian Structures
2.1. Generalizing Randers and Kropina metrics. Let (g ,W) be Zermelo
R
datain amanifoldM, namely, g is aRiemannianmetric andW, a vectorfield on
R
M. Assumethat W <1,where denotesthepointwiseg -norm,andconsider
R R R
| | |·|
the Randers metric F whose indicatrix Σ at p M is obtained by displacing the
∈
g -indicatrix S with the vector W , that is, Σ=S +W . This metric is the key
R R p R p
of Zermelo navigationproblem (the F-geodesics solve it), and it canbe written as:
1
F(v )= Λ(p)v 2 +g (v ,W )2 g (W ,v ) , whereΛ=1 W 2,
p Λ(p) | p|R R p p − R p p −| |R
(cid:18)q (cid:19)
for any v T M. Clearly, this expressioncrashes when W =1, as Λ vanishes.
p p p R
∈ | |
However,we can rewrite it by removing the root from the numerator and, then, Λ
from the denominator:
v 2 v 2
F(v )= | p|R = | p|R ,
p
g (W ,v )+ Λ(p)v 2 +g (v ,W )2 g (W ,v )+ h(v ,v )
R p p | p|R R p p R p p p p
(1)
p p
where
h(v ,v )=Λ(p)v 2 +g (v ,W )2. (2)
p p | p|R R p p
This expressionfor F(v ) makes sense for an arbitrarywind W even if Λ vanishes,
p
suggesting a possibility for the description of the case W 1. However, a
R
| | ≥
cautionshould be taken into account: both, F(v ) and the expressionfor h (which
p
liesinsidearoot)shouldbenonnegative. So,restrictthedomainofF byimposing:
h(v ,v ) 0, and
ifΛ(p) 0restrict to p p ≥ (3)
≤ gR(Wp,vp)>0.
(cid:26)
These restrictions have the following meaning (see Figure 1). Formula (2) shows
4 M.A.JAVALOYESANDM.SA´NCHEZ
Figure 2. The threepossibilities forindicatricesofthe WRSina
2-dimensional example.
that h is a signature-changing metric, which becomes Riemannian in the region
of mild wind W < 1, degenerate when the wind is critical W = 1, and
R R
| | | |
Lorentzian of signature (+, ,..., ) when the wind is strong W > 1. While
R
in the region W <1 one r−ecover−s a Randers metric with indic|atr|ix F−1(1)=Σ
R
| |
(= S +W), in the region W = 1 (i.e., Λ = 0) one has a Kropina metric α2/β
R R
| |
with α = , β = 2g (W, ). Indeed the restriction g (W ,v ) > 0 selects the
R R R p p
|·| ·
(pointwise) tangent open half-space A where F becomes positive, and one can
p
still regard Σ = S +W as the indicatrix of F (up to the “singular” vector 0 ),
R p
Figure 2. In the region W > 1, the Lorentzian metric h determines at every
R
| |
tangent space two (lightlike) cones, and the restrictions (3) have a neat meaning:
the allowedv must belong either to one ofthese two h-cones (the cone selected
p p
C
byg (W ,v )>0)ortotheinteriorregionA determinedbyit(seeFigure1). So,
R p p p
F can be regarded as a “conic” Finsler metric defined only on the tangent vectors
satisfying (3). Notice that, now, Σ = S +W includes the “indicatrix” F−1(1).
R
Indeed, Σ divides Σ into two pieces, and F−1(1) corresponds with one of
p p p
∩C
them; namely, the (strongly) convex one, when looking from infinity into the cone
region. Easily, one can check that the other piece is equal to F−1(1) where
l
v 2
F(v ):= | p|R . (4)
l p
g (W ,v ) h(v ,v )
R p p p p
−
RecallthatF−1(1)is(strongly)concaveand,sop,F willbe calledaLorentz-Finsler
l l
metric. Notice that F is univocally determined by1 F; consequently, most of our
l
computations will deal only with F. However, F has a nice interpretation for
l
Zermelo navigation under strong wind: the indicatrix of F provides the minimum
l
speed of the moving object at each allowed direction.
Let us summarize the previous approachand introduce suitable conventions:
(1) Given a (connected) manifold M, a wind Riemannian structure (WRS) is
anyhypersurfaceΣ TM whichcanbeexpressedasΣ=S +W forsome
R
⊂
vector field W and Riemannian metric g (both univocally determined)2.
R
At each p M, Σ encloses an open domain B which will be called the
p p
∈
unit ball at p.
1ThispropertydoesnotholdforthegeneralwindFinslerianstructuresstudiedin[8].
2In[8],generalwindFinslerianstructuresareintroducedinamoreabstractway,andwindRie-
mannianonesarethenregardedasaparticularcaseofthatdefinition. However,bothapproaches
areclearlyequivalent(see[8,Prop. 2.13]).
WIND RIEMANNIAN STRUCTURES 5
(2) Such a Σ determines the (possibly signature-changing) metric h in (2), as
wellasadomainA:= p∈MApincludedintheslittangentbundleTM 0 ,
∪ \{ }
defined by choosing each A as follows: A = T M 0 in the region of
p p p
\{ }
mild wind (Λ(p) > 0), A is the open half space determined by (3) in the
p
region of critical wind (Λ(p) = 0), and A is the interior of the solid cone
p
determined by (3) in the region M of strong wind (Λ(p)<0).
l
(3) Σ also determines:
(i) A conic Finsler metric F :A R.
(ii) A Lorentz-Finsler metric F :→A R, where A is the union of the
l l l
→
pointwise domains A in the region M of strong wind.
p l
Moreover,the following three extensions of the domains A,A will be used
l
when necessary:
(a) F and F are extended continuously on M to the h-cones T M
l l p p
C ⊂ \
0 ,
{ }
(b) F is extendedas outsidethe regionofstrongwind, thatis,inTM
l
∞ \
( 0 TM), and
l
{ }∪
(c) intheregionofcriticalwind(Λ(p)=0)),wedefineF(0 )=F (0 )=1
p l p
(even though, necessarily, such a choice is discontinuous, see below).
2.2. Wind curves and the appearance of Lorentzian geometry. In the
framework of Zermelo navigation, consider two points p,q M and a WRS Σ
∈
whichprovides the maximum velocities at eachdirection. If a moving object going
from p to q is represented by the curve γ : [t ,t ] M, where t ,t are, respec-
0 1 0 1
→
tively, the instant of departure from p and arrival to q, then its velocity at each
instant t must be an allowed one, that is:
F(γ˙(t)) 1 F (γ˙(t)), t [t ,t ]. (5)
l 0 1
≤ ≤ ∀ ∈
Any curve in M satisfiying these inequalities will be called a wind curve. Recall
that (5) assumes implicitly that γ˙(t) lies in the domains of F and F explained at
l
the end ofthe lastsubsection, so: (a)γ˙(t) is allowedto belong not only to the unit
ball B but also to its closure (in the slit tangent bundle) Σ , (b) as F
γ(t) γ(t) l
≡ ∞
outside M , the last inequality in (5) imposes no restriction when the wind is not
l
strong, and (c) the zero velocity 0 is excluded in both, the region of strong wind
p
(becauseitisnotanallowedvelocity)andthe regionofmildwind(byconvenience,
analogous to the restriction to regular curves in Riemannian Geometry); however,
thezero-velocityisallowedintheregionofcriticalwind,asithasaspecialmeaning
there,namely,itis the minimum allowedvelocityinthe directionofthe wind. The
arrivaltime ofthe movingobject t t is bounded by the F andF lengths ofthe
1 0 l
−
curves, that is:
t1 t1
ℓ (γ):= F(γ˙(t))dt t t ℓ (γ):= F (γ˙(t))dt.
F ≤ 1− 0 ≤ Fl l
Zt0 Zt0
Obviously, when γ is reparametrized so that F(γ˙(t)) 1, this corresponds to a
≡
movingobjectwhichusesthemaximumpossiblevelocityalongthetrajectoryofγso
thatitspendstheminimumpossibletraveltimealongthattrajectory;analogously,
in the case that γ lies entirely in the region of strong wind, a reparametrization
with F (γ˙(t)) 1 corresponds to minimum velocity and maximum travel time.
l
≡
A better insight is obtained by considering the graph (t,γ(t)),t [t ,t ]
1 2
R M of the wind curve. The allowed velocities for γ at{each instan∈t t R}an⊂d
× ∈
6 M.A.JAVALOYESANDM.SA´NCHEZ
each point p are represented by
(1,v ) T (R M):F(v ) 1 F(v ) . (6)
p (t,p) p l p
{ ∈ × ≤ ≤ }
Now, notice that the half lines in T (R M) which start at 0 and cross any
(t,p)
of these allowed velocities provide a solid co×ne on T (R M) (recall Figure 3).
(t,p)
Thus,theWRSyieldsnaturallyaconestructureonallR ×M. Thisconestructure
×
is invariantin the t-coordinate,as neither the wind W nor the metric g are time-
R
dependent. A simple computation shows that the cone structure is equal to the
(future) cone structure associated with a Lorentzian metric g on R M, namely:
×
g = (Λ π)dt2+π∗ω dt+dt π∗ω+π∗g (7)
0
− ◦ ⊗ ⊗
where π :R M M is the natural projection and:
× →
Λ=1 W 2, ω = g (W, ), g =g . (8)
−| |R − R · 0 R
Some comments on this Lorentzian metric are in order:
The metric g is Lorentzianwith signature ( ,+,...,+). At each (t,p) the
• non-zerotangentvectors(τ,v ) T (R M−)satisfyingg((τ,v ),(τ,v ))=
p (t,p) p p
∈ ×
0 (resp. <0, >0) are called lightlike (resp. timelike, spacelike). The light-
like vectors at each (t,p) are distributed in two cones. One of them, the
future-directedlightlike cone,containstangentvectorswithτ >0;theother
one is called the past-directed lightlike cone. The future-directed lightlike
cone plus the corresponding future-directed timelike vectors (those inside
the solid cone) contain the set (6) determined by the wind curves.
The natural vector field K = ∂ is Killing for g. When Λ > 0 (resp.
t
•
= 0,< 0), K is future-directed timelike (resp. future-directed lightlike,
spacelike), see Figure 3. The natural projection t : R M R is a time
× →
function because it is strictly increasing on any future-directed timelike
or lightlike curve; its slices t =constant are spacelike hypersurfaces (they
inheritaRiemanninmetric). Accordingto[8],thesespacesarecalledstan-
dard with a spacelike-transverse Killing vector field, or just SSTK space-
times. They include important families of relativistic spacetimes, as the
standard stationary ones(thosewithΛ>0),whichcorrespondtothe Ran-
ders case under our approach.
TheSSTKspacetime(R M,g)willbecanonicallyassociatedwithaWRS.
• ×
However,theonlyrelevantpropertiesofthespacetimeforourpurposeswill
betheconformallyinvariantones3. Indeed,anyconformalmetricg∗ =Ω g,
·
Ω > 0, will have the same lightlike cones as g. Moreover, it is well-known
thatiftwoLorentzianmetricsg,g∗ sharethesametimelikeconesthenthey
are (pointwise) conformally related through some function Ω>0 [3, 21].
The correspondence between the WRS and the SSTK spacetime is very fruitful
for both, our Finslerian problem and the geometry of relativistic spacetimes. In-
deed, this happens even in the particular case Λ > 0, where the WRS is just a
Randers metric and the SSTK spacetime, a standard stationary spacetime (see
the detailed study in [7] and further developments such as [12, 14]). However,
the correspondence becomes crucial for general WRS. Indeed, the study of WRS
3So,sometimesadifferentrepresentativeoftheconformalclassoftheSSTKmaybepreferred.
For example, the normalization Λ ≡ 1 was chosen in the case of Randers metrics and standard
stationaryspacetimes studied in[7]. (Thisisthereasonwhywepreferredtowritethe metricg0
in(7)evenifitistakeng0=gR later.)
WIND RIEMANNIAN STRUCTURES 7
t
Figure 3. The three possibilities for the cones of the associ-
ated SSTK spacetime. At the tangent space of each p, the in-
teresection of the (n+1)-dimensional cone with the hyperplane
dt =1projectsintotheindicatrixofaRanders,Kropinaorwind-
p
Riemannian metric.
through more classical Finslerian elements such as the metrics F, F presents the
l
important drawback of having “singular” elements (as the 0 vector in the Kropina
region)ordiscontinuouselements (asthe suddenjump inthe structureof A when
p
p varies from the region of mild wind to the non-mild one). This reason underlies
the difficulties to develop general Zermelo navigation in spite of attempts such as
[9]. Nevertheless,thespacetimeviewpointallowsboth: (i)adescriptionintermsof
completelyregularandnon-singularelements(the metric g)and(ii)the possibility
to derive results for WRS by using results or techniques in the well established
conformaltheoryofspacetimes. Indeed,thegeneralproblemofZermelonavigation
canbe describedandsolvedsatisfactorilybyusingthis correspondence[8]. Wewill
not go deeper here in this correspondence and refer to the exhaustive study in [8];
however, we would like to point out that some of the results below were obtained
by using it.
2.3. Balls,geodesicsandcompleteness. LetΣbeaWRSand,foranyp,q M,
∈
letCΣ denotethesetofallthewindcurvesfromptoq. Theforwardandbackward
p,q
wind balls ofcenter p M andradiusr >0associatedwith the WRS Σ are,resp:
0
∈
B+(p ,r)= x M : γ CΣ , s.t. r =b a andℓ (γ)<r <ℓ (γ) ,
Σ 0 { ∈ ∃ ∈ p0,x γ − γ F Fl }
B−(p ,r)= x M : γ CΣ , s.t. r =b a andℓ (γ)<r <ℓ (γ) .
Σ 0 { ∈ ∃ ∈ x,p0 γ − γ F Fl }
8 M.A.JAVALOYESANDM.SA´NCHEZ
These balls are open [8, Remark 5.2] and their closures are called (forward, back-
ward) closed wind balls, denoted B¯±(p ,r). Between these two types of balls, the
Σ 0
forward and backward c-balls are defined, resp., by:
Bˆ+(p ,r)= x M : γ CΣ , s.t. r =b a (so,ℓ (γ) r ℓ (γ)) ,
Σ 0 { ∈ ∃ ∈ p0,x γ − γ F ≤ ≤ Fl }
Bˆ−(p ,r)= x M : γ CΣ , s.t. r =b a (so,ℓ (γ) r ℓ (γ))
Σ 0 { ∈ ∃ ∈ x,p0 γ − γ F ≤ ≤ Fl }
for r > 0; for r = 0, by convention Bˆ±(p ,0) = p (so that, consistently with our
Σ 0 0
conventions, if 0 Σ then p Bˆ±(p ,r) for all r 0). If Σ comes from
p0 ∈ p0± 0 ∈ Σ 0 ≥
a Randers metric, then B (p ,r) coincides with the usual (forward or backward)
Σ 0
±
open balls. However, even in the Riemannian case, one may have B (p ,r) (
Σ 0
Bˆ±(p ,r)(B¯±(p ,r) (put M =R2 (1,0) , p =(0,0), r =2).
Σ 0 Σ 0 \{ } 0
Starting at these notions of balls, geodesics can be defined as follows. A wind
curve γ :I =[a,b] M, a<b, is called a unit extremizing geodesic if
→
γ(b) Bˆ+(γ(a),b a) B+(γ(a),b a). (9)
∈ Σ − \ Σ −
Then,acurveisanextremizinggeodesicifitisanaffinereparametrizationofaunit
extremizing geodesic, and it is a geodesic if it is locally an extremizing geodesic.
ThegeodesicsofaWRScoincide,uptoareparametrization,withtheprojection
on M of the future-directed lightlike geodesics of the associated SSTK spacetime
(R M,g). This allows one to prove that a curve γ : I M is a geodesic if an
× →
only if it lies in one of the following cases:
(1) γ isageodesicoftheconicFinslermetricF. Inthiscase,γ˙(t)liesalwaysin
A (it cannot belong to its boundary) and γ may lie in the regions of mild,
critical or strong wind, eventually crossing them; moreover, γ minimizes
locally the F-length in a natural sense.
(2) γ isageodesicoftheLorentz-FinslermetricF . Inthiscase,γ˙(t)liesalways
l
in A (it cannot belong to its boundary) and γ is entirely contained in the
l
regionofstrong wind M ; moreover,γ maximizes locally the F -lengthin a
l l
natural sense.
(3) γ is an exceptional geodesic, that is, γ is constantly equal to some point p
0
with Λ(p )=0 and dΛ(v )=0 for all v satisfying g (W ,v )=0.
0 p0 p0 R p0 p0
(4) γ is included in the closure of M and it satisfies: (i) whenever γ re-
l
mains in M , it is a lightlike pregeodesic of the Lorentzian metric h in
l
(2), reparametrizedsothatF(γ˙) F (γ˙)isaconstantc>0(thatis, γ isa
l
≡
boundary geodesic in M , with velocity in the boundary of each solid cone
l
A¯ ,p M ), and (ii) γ can reach the boundary ∂M (which is included in
p l l
∈
the critical region Λ=0) only at isolated points s I,j =1,2,..., where4
j
∈
γ˙(s ) = 0, dΛ does not vanish on all the g -orthogonal to W and the
j R γ(sj)
second derivative of γ (in one and then in any coordinates) is continuous
and does not vanish at s .
j
The WRS is (geodesically) complete when its inextendible geodesics are defined
on all R; it is called forward (resp. backward) complete when only the upper (resp.
lower)unboundednessoftheirintervalsofdefinitionisrequired. GivenΣ,itsreverse
WRS is Σ = S W. Clearly, the reverse parametrization of a geodesic for Σ
R
− −
becomesageodesicfor ΣandΣisforwardcompleteiff Σisbackwardcomplete.
− −
4Recall that the equality F(γ˙) = Fl(γ˙) = c > 0 close to sj is compatible with γ˙(sj) = 0
becauseoftheKropinacharacter ofΣatγ(sj).
WIND RIEMANNIAN STRUCTURES 9
y axis
3
1
0.5
6 3 0 3 6 x axis f(x)
− − 6 4 2 2 4 6
− − −
0.5
−
3 1
−
−
Figure 4. Randers-Kropina metric F in the Euclidean space R2
with W = f(x)∂ and f 1 as in the graph. The regions
(x,y) x
| | ≤
x 3 and x 3 are disconnected (non-connectable by wind
≤ − ≥
curves),butthemetriciscompletesincetheforwardandbackward
ballsofradiusrarecontainedinthecorrespondingEuclideanballs
with radius 2r because F(v) 2v for every v R2. Moreover,
≤ | | ∈
the disconnected regions x 6 and x 6 are endowed with the
≤ − ≥
Euclidean metric.
From now on, completeness will mean forward and backward completeness. We
have the following characterizationextracted from [8, Prop. 6.4].
Theorem 2.1. Let (M,Σ) be a WRS. The following properties are equivalent:
(i) Σ is geodesically complete,
(ii) B+(x,r) and B−(x,r) are precompact for every x M and r >0.
Σ Σ ∈
(iii) Bˆ+(x,r) and Bˆ−(x,r) are compact for every x M and r >0.
Σ Σ ∈
In particular, if M is compact then Σ is geodesically complete.
The following result of completeness will be used throughout the text.
Theorem 2.2. Let (M,Σ) be a complete WRS and W a complete homothetic field
oftheassociatedconicpseudo-Finsler metricsF andF . Then, Σ+W isacomplete
l
WRS.
Proof. It is a straightforwardconsequence of [15, Theorem 1.2]. (cid:3)
Theequalitybetweenc-ballsandclosedballsina(connected)Finslerianmanifold
becomes equivalent to its convexity (in the sense that any p,q M can be joined
∈
by a geodesicof minimum length). In general,a WRS is calledw-convex whenthis
equality between balls hold. In a w-convex WRS, an F-extremizing wind geodesic
from p to q will exist whenever there exists a wind curve starting from p and
ending at q. A complete WRS is always w-convex, and even in this case some of
their points might be non-connectable through wind curves (see Figure 4). These
possibilitiesareveryappealingfromtheSSTKviewpoint,astheyarerelatedtothe
existenceofKilling horizonsin globallyhyperbolic spacetimesandother notionsin
the relativistic fauna.
3. Wind Riemannian Structures of constant flag curvature
3.1. Randers and Kropina solutions. Randers manifolds of constant flag cur-
vature have been completely classified by Bao, Robles and Shen [2, Th. 3.1]. This
10 M.A.JAVALOYESANDM.SA´NCHEZ
result has a local nature, even though it can be used to obtain models of inex-
tendible Randers spaceforms (some of them necessarily incomplete as Finslerian
manifolds). Their result can be summarized as follows.
Theorem 3.1. A Randers metric F has constant flag curvature κ if and only if
its Zermelo data (g ,W) satisfy the following:
R
(i) the metric g has constant curvature κ+ 1µ2 for some constant µ,
R 4
(ii) the wind W satisfies g (W,W) < 1 and it is µ-homothetic for g , namely,
R R
g =2µg , where is the Lie derivative.
W R R
L L
Moreover, the only complete simply connected Randers spaceforms are:
(0) if κ = 0, those with Zermelo data given by the Euclidean metric and a
parallel vector field with norm less than 1,
(-) if κ<0, the hyperbolic space of constant curvature κ,
(+) if κ>0, those with Zermelo data given by the round sphere of radius 1/√κ
and a Killing vector field with norm less than 1.
Observe that given a vector field in the sphere, there is a multiple of it with
norm less than 1. However, completeness excludes the possibility of being prop-
erly homothetic for W, as well as most of the cases of Randers metrics that are
not Riemannian for κ 0. About the Kropina case with Zermelo data (g ,W),
R
≤
Yoshikawa and Okubo [29, Th. 4] solution becomes a natural extension of Bao et
al.’s: g must have constant curvature but now W must be a homothetic vector
R
field satisfying W 1 (in particular, W is then necessarily Killing, see below).
R
| | ≡
The correspondingexplicit cases were obtained by Yoshikawa, and Sabau[30]. For
the convenience of the reader, we include some results on homothetic and Killing
vectorfieldsintheAppendixsothatonecanrewritetheconcreteKropinasolutions
[29, 30] as follows.
Theorem3.2. AKropinamanifold(M,F)ofconstantflagcurvatureisdetermined
by Zermelo data (g ,W) which lie in one of the following two cases:
R
(i) g is flat and W is parallel and unit,
R
(ii) g is locally isometric to an odd round sphere S2m+1(r) and W is a unit
R
Killing vector field, i.e. a unit Hopf vector field.
Proof. By Proposition 4.3, W must be a Killing field. Moreover, Proposition 4.6
determines the two possibilities of the theorem. (cid:3)
Again the result has a local nature, but all the local examples of Theorem 3.2
can be extended to complete simply connected models.
Theorem 3.3. The complete simply connected Kropina manifolds (M,F) of con-
stant flag curvature lie in one of the following two cases, up to isometries:
(i) (M,g )=Rn is flat and W is a parallel, unit vector field of Rn.
R
(ii) (M,g )=Rn =S2m+1(r) and W is a unit Hopf vector field.
R
In particular, W is also a complete geodesic vector field.
Remark 3.4. As emphasized in subsection 2.3, completeness always means “for-
ward and backward completeness”. Otherwise, further possibilities appear; for
example, in the case (i) the half space x > 0 of Rn with W = ∂ is forward
n xn
complete.