Sprays metrizable by Finsler functions of constant flag curvature
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Di e en ial Geome y and i s Applica ions ••• (••••)•••–•••
Con en s lis s a ailable a SciVe se ScienceDi ec
Di e en ial Geome y and i s Applica ions
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Sp ays me izable by Finsle unc ions o cons an flag
cu a u e
Ioan Buca a ua,∗,Zol án Muzsnayb
aFacul y o Ma hema ics, Alexand u Ioan Cuza Uni e si y, Ia¸si, Romania
bIns i u e o Ma hema ics, Uni e si y o Deb ecen, Deb ecen, Hunga y
a icle in o abs ac
A icle his o y:
Recei ed 6 Decembe 2012
A ailable online xxxx
Communica ed by Z. Shen
MSC:
53C60
58B20
49N45
58E30
Keywo ds:
Iso opic sp ays
Finsle me izabili y
Flag cu a u e
In his pape we cha ac e ize sp ays ha a e me izable by Finsle unc ions o cons an
flag cu a u e. By sol ing a pa icula case o he Finsle me izabili y p oblem, we p o ide
he necessa y and sufficien condi ions ha can be used o decide whe he o no a
gi en homogeneous sys em o second o de o dina y di e en ial equa ions ep esen s he
geodesic equa ions o a Finsle unc ion o cons an flag cu a u e. The condi ions we
p o ide a e enso ial equa ions on he Jacobi endomo phism. We iden i y he class o
homogeneous SODE whe e he Finsle me izabili y is equi alen wi h he me izabili y
by a Finsle unc ion o cons an flag cu a u e.
©2013 Published by Else ie B.V.
1. In oduc ion
The in e se p oblem o Lag angian mechanics can be o mula ed as ollows: decide whe he o no a gi en sys em o
second o de o dina y di e en ial equa ions (SODE) coincides wi h he Eule –Lag ange equa ions o some Lag angian, [2,6,
11,15,17,19]. When he gi en sys em o SODE is homogeneous and he Lag angian o sea ch o is he squa e o a Finsle
unc ion, he p oblem is known as he Finsle me izabili y p oblem, [8,14,18,23]. I he sough a e Lag angian is a Finsle
unc ion, he p oblem is known as he p ojec i e me izabili y p oblem, o as he Finsle ian e sion o Hilbe ’s ou h
p oblem, [1,7–10,24].
In his pape we add ess he special case o he Finsle me izabili y p oblem, whe e he Finsle unc ion we seek o has
cons an cu a u e. When he sp ay has ze o cons an cu a u e, hen he e is no obs uc ion o he exis ence o a locally
defined Finsle s uc u e ha me icizes he gi en sp ay, [7,9,18]. The e o e, in his wo k we will ocus on he case when
he cu a u e is non-ze o.
In Theo em 4.1, we sol e he abo e men ioned p oblem by p o iding a se o equa ions, which con ains an algeb aic
equa ion A) and wo enso ial di e en ial equa ions D1) and D2)in(4.1), which ha e o be sa isfied by he Jacobi en-
domo phism. One o hese wo enso ial equa ions es ic s he class o homogeneous SODE (sp ays), which we discuss,
o he class o iso opic sp ays. The e o e, we ocus ou a en ion on iso opic sp ays and hei ela ion wi h he Finsle
*Co esponding au ho .
E-mail add ess: [email p o ec ed] (I. Buca a u).
URLs: h p://www.ma h.uaic. o/~buca a u/ (I. Buca a u), h p://www.ma h.kl e.hu/~muzsnay/ (Z. Muzsnay).
0926-2245/$ – see on ma e ©2013 Published by Else ie B.V.
h p://dx.doi.o g/10.1016/j.di geo.2013.02.001
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me izabili y p oblem. In Theo em 4.2, we cha ac e ize he class o iso opic sp ays S o which he ollowing condi ions a e
equi alen :
•Sis Finsle me izable,
•Sis me izable by a Finsle me ic o scala flag cu a u e;
•Sis Finsle me izable by an Eins ein me ic;
•Sis me izable by a Finsle me ic o cons an flag cu a u e;
•Sis Ricci cons an .
I he Finsle unc ion is educible o a Riemannian me ic, he equi alence o he abo e condi ions, on mani olds o di-
mension g a e han o equal o h ee, is always ue o affine iso opic sp ays, and i is no ue in he gene al Finsle ian
con ex . The e o e, Theo em 4.2 iden ifies he class o sp ays whe e his equi alence is s ill ue. The las condi ion is a
echnical one, can be checked e y easily, and i is common o bo h Theo ems 4.1 and 4.2.Theo em 4.1 has he ad an age
o being ue o sp ay spaces o dimension g a e han o equal o wo, and i has he disad an age o dis ega ding Finsle
me izable sp ays o non-cons an flag cu a u e. The main ad an age o Theo em 4.2 is ha i ea s sp ays o which he
Finsle me izabili y is equi alen wi h he me izabili y by a Finsle me ic o cons an non-ze o flag cu a u e. The key
ing edien in p o ing Theo em 4.2 is he Finsle ian e sion o Schu ’s Lemma, [4, Lemma 3.10.2], which is ue only o
sp ay spaces o dimension g a e han o equal o h ee.
Since any sp ay on a wo-dimensional mani old is iso opic, one o he wo equa ions (4.1) in Theo em 4.1 simpli y. In
Theo em 4.3 we p o ide necessa y and sufficien condi ions o he me izabili y o a wo-dimensional sp ay space by a
Finsle unc ion o cons an (Gaussian) cu a u e.
To suppo ou esul s, in he las sec ion, we conside a ious examples o iso opic sp ays ha sa is y, o no , one o
mo e o he necessa y and sufficien condi ions, which we p o ide, o Finsle me izabili y.
2. Sp ays and hei geome ic se ing
The na u al geome ic amewo k o s udying sys ems o second o de o dina y di e en ial equa ions is he angen
bundle o some configu a ion mani old.
In his wo k, Mdeno es a C∞-smoo h, eal, and n-dimensional mani old. We will deno e by TM i s angen bundle and
by T0M=TM {0} he angen bundle wi h he ze o sec ion emo ed. Local coo dina e cha s (U,(xi)) on Minduce local
coo dina e cha s (π−1(U), (xi,yi)) on TM, whe e π:TM→Mis he canonical subme sion. We will assume ha Mis a
connec ed mani old o dimension n⩾2. The e o e, TM and T0Ma e 2n-dimensional connec ed mani olds.
In his sec ion we discuss he na u al geome ic se ing de e mined by a sp ay S, which includes canonical nonlinea
connec ion, dynamical co a ian de i a i e and cu a u e enso s. This se ing, as well as he p oo s o ou esul s in he
nex sec ions, a e based on he F öliche –Nijenhuis heo y and he co esponding di e en ial calculus ha can be de eloped
on TM,[11,13,22]. The e a e wo canonical s uc u es on TM, which we will use o de elop ou se ing. One is he angen
s uc u e, J, and he o he one is he Liou ille ec o field, C, locally gi en by
J=∂
∂yi⊗dxi,C=yi∂
∂yi.
A sys em o homogeneous second o de o dina y di e en ial equa ions on a mani old M, whose coefficien s do no
depend explici ly on ime, can be iden ified wi h a special ec o field on T0M ha is called a sp ay. A ec o field S∈
X(T0M)is called a sp ay i JS=Cand [C,S]=S. Locally, a sp ay Sis gi en by
S=yi∂
∂xi−2Gi(x,y)∂
∂yi,(2.1)
whe e unc ions Gi(x,y)a e smoo h unc ions on domains o induced coo dina es on T0Mand 2-homogeneous wi h espec
o he y- a iable.
I is well known ha a sp ay induces a nonlinea connec ion, wi h he co esponding p ojec o s hand gi en by
h=1
2(Id−LSJ), =1
2(Id+LSJ).
Fo a sp ay S, we conside he map ∇:X(T0M)→X(T0M),gi enby[6]
∇=LS+h◦LSh+ ◦LS .(2.2)
We equi e ha he ac ion o ∇on scala unc ions is gi en by ∇ =S( ), o ∈C∞(T0M). Fu he equi emen s ha ∇
sa isfies he Leibni z ule and commu es wi h con ac ions allow us o ex end i s ac ion o a bi a y enso fields on T0M.
We will e e o ∇as o he dynamical co a ian de i a i e induced by he sp ay S. I s ac ion on semi-basic o ms was called
he semi-basic de i a ion and s udied, in connec ion wi h he in e se p oblem o he calculus o a ia ion, in [11].
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An impo an geome ic s uc u e induced by a sp ay is he Jacobi endomo phism, which is he ec o alued semi-basic
1- o m gi en by
Φ= ◦LSh=LSh◦h.
Locally, he Jacobi endomo phism, is gi en by
Φ=Rij
∂
∂yi⊗dxj=2∂Gi
∂xj−S∂Gi
∂yj−∂Gi
∂y
∂G
∂yj∂
∂yi⊗dxj.(2.3)
We conside also he Ricci cu a u e,Ric, and he Ricci scala ,ρ,[5],[21, Defini ion 8.1.7], which a e gi en by
Ric =(n−1)ρ=Ri
i=T (Φ). (2.4)
Defini ion 2.1. Asp aySis called
i) Ricci-cons an i dhρ=0,
ii) weakly Ricci-cons an i S(ρ)=0.
We say ha a sp ay Sis iso opic i i s Jacobi endomo phism has he o m,
Φ=ρJ−α⊗C,(2.5)
whe e αis a semi-basic 1- o m on T0M.
I is easy o see ha i a sp ay Sis Ricci cons an , hen i is also weakly Ricci cons an . Indeed, o a Ricci cons an
sp ay S, i ollows ha he Ricci scala sa isfies dhρ=0. Using he co esponding commu a ion o mula, we ob ain
0=iSdhρ=−dhiSρ+LhSρ+i[h,S]ρ=S(ρ),
and hence he sp ay Sis weakly Ricci cons an .
Fo an iso opic sp ay S, due o he homogenei y condi ion, i ollows ha 0 =Φ(S)=(ρ−iSα)Con T0Mand hence
ρ=iSα. Iso opic sp ays can be cha ac e ized using he Weyl cu a u e, see P oposi ion 13.4.1 in [21]. The semi-basic ec o
alued 1- o m Φis 2-homogeneous, which means Φ=[C,Φ]. Fo an iso opic sp ay S,weha e
Φ=[C,Φ]=[C,ρJ]−[C,α⊗C]=C(ρ)−ρJ−LCα⊗C,
which implies C(ρ)=2ρand LCα=α. The e o e, he Ricci cu a u e Ric and he Ricci scala ρa e 2-homogeneous, while
he 1- o m αis 1-homogeneous.
Lemma 2.2. Conside S an iso opic sp ay, whose Jacobi endomo phism is gi en by o mula (2.5). The ollowing wo condi ions a e
equi alen
i) dJα=0,
ii) dJρ=2α.
P oo . Using he ac ha ρ=iSαand he commu a ion ules, i ollows
dJρ=dJiSα=−iSdJα+LJSα+i[J,S]α=−iSdJα+2α.(2.6)
In he abo e equa ions, we ha e used ha [J,S]=h− and since αis a semi-basic 1- o m, i ollows ha i[J,S]α=ihα=α.
The e o e, he assump ion dJα=0 implies ha dJρ=2α. The o he implica ion is s aigh o wa d, using he ac ha he
angen s uc u e Jis in eg able, and hence d2
J=0. 2
I is known ha he p ojec i e de o ma ions o a sp ay p ese es he class o iso opic sp ays. These de o ma ions may
p ese e o no he condi ion dJα=0. In Sec ion 5.1 we p o ide examples o iso opic sp ays such ha dJα= 0. See o mula
(5.1) o a con enien choice o a 1-homogeneous unc ion P∈C∞(T0M).
3. Finsle me izable sp ays
In his sec ion we ecall he no ion o a Finsle space and i s geodesic sp ay. We will ocus ou discussions on Finsle
spaces o scala flag cu a u e.
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Defini ion 3.1. ByaFinsle unc ion we mean a con inuous unc ion F:TM→Rsa is ying he ollowing condi ions:
i) Fis smoo h and s ic ly posi i e on T0M;
ii) Fis posi i ely homogeneous o o de 1, which means ha F(x,λy)=λF(x,y), o allλ⩾0 and (x,y)∈TM;
iii) The me ic enso wi h componen s
gij(x,y)=1
2
∂2F2
∂yi∂yjhas ank non T0M.(3.1)
The abo e condi ions o Defini ion 3.1 imply ha he me ic enso gij o a Finsle unc ion is posi i e defini e on T0M,
see [16]. Some elaxa ions o he abo e condi ions, which lead o he no ion o conic pseudo-Finsle me ic,we ep oposed
in [12]. See also [3, §1.1.2, §1.2.1] o mo e discussions abou he egula i y condi ions and hei elaxa ion o a Finsle
unc ion. In Subsec ion 5.4 we will discuss an example o a sp ay me izable by such a conic pseudo-Finsle unc ion.
A Finsle unc ion is educible o a Riemannian me ic i he me ic enso gij in o mula (3.1) does no depend on he fibe
coo dina es y.I gij(x)is a Riemannian me ic on M, henF:TM→R,F(x,y)=gij(x)yiyjis a Finsle unc ion.
The egula i y condi ion iii) o Defini ion 3.1 is equi alen o he ac ha he Poinca é–Ca an 2- o m o F2,ωF2=
−dd JF2, is non-degene a e and hence i is a symplec ic s uc u e. The e o e, he equa ion
iSdd JF2=−dF2(3.2)
uniquely de e mine a ec o field Son T0M, which is called he geodesic sp ay o he Finsle unc ion. In his wo k we will
usemo e equen ly, heequa ionLSdJF2=dF2, which is equi alen o Eq. (3.2).
Defini ion 3.2. Asp ayS∈X(T0M)is called Finsle me izable i he e exis s a Finsle unc ion F ha sa isfies Eq. (3.2).
Necessa y and sufficien c i e ia o he Finsle me izabili y p oblem o a sp ay Swe e o mula ed in [18] using he
holonomy dis ibu ion HS. Also, such necessa y and sufficien condi ions we e o mula ed in e ms o a semi-basic 1- o m
in [6]. We will use hese condi ions in he nex sec ion o discuss he Finsle me izabili y p oblem o a pa icula class o
iso opic sp ays.
Defini ion 3.3. Conside Fa Finsle unc ion and Φ he Jacobi endomo phism o i s geodesic sp ay S.
i) Fis said o be o scala (cons an ) flag cu a u e i he e exis s a scala unc ion (cons an ) κon T0M, such ha
Φ=κF2J−FdJF⊗C.(3.3)
ii) Fis called an Eins ein me ic i he e exis s a unc ion λ∈C∞(M)such ha he Ricci scala sa isfies ρ(x,y)=
λ(x)F2(x,y).
The no ion o flag cu a u e ex ends o he Finsle ian se ing he concep o sec ional cu a u e om he Riemannian
se ing.
Rema k 3.4. I a Finsle unc ion Fis educible o a Riemannian me ic gon a mani old o dimension g a e han o equal
o h ee, and Sis i s geodesic sp ay, hen he ollowing condi ions a e equi alen , see [21, §13.4]:
i) Sis iso opic;
ii) gis o scala cu a u e;
iii) gis o cons an cu a u e.
In he gene al Finsle ian con ex , condi ions ii) and iii) abo e a e no equi alen anymo e. In he nex sec ion, we p o ide
he necessa y and sufficien condi ion o an iso opic geodesic sp ay such ha his equi alence emains ue. See he
equi alence o he condi ions o Theo em 4.2.
4. Finsle me izable iso opic sp ays
In his sec ion we use he necessa y and sufficien condi ions, exp essed in e ms o a semi-basic 1- o m, which we e
o mula ed in [6, Theo em 5.4], o discuss he Finsle me izabili y p oblem o some classes o iso opic sp ays.
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4.1. Sp ays me izable by Finsle unc ions o cons an cu a u e
In he nex heo em we p o ide he necessa y and sufficien condi ions one has o check i we wan o decide i a sp ay
is me izable by a Finsle unc ion o non-ze o cons an cu a u e.
Theo em 4.1. Conside S a sp ay wi h non- anishing Ricci cu a u e. The sp ay S is me izable by a Finsle unc ion o non-ze o
cons an flag cu a u e i and only i i s Jacobi endomo phism sa isfies he ollowing equa ions:
A) ankdd J(T Φ) =2n;
D1)2(n−1)Φ −2(T Φ) J+dJ(T Φ) ⊗C=0;
D2)dh(T Φ) =0.(4.1)
P oo . Conside Sa sp ay wi h non- anishing Ricci cu a u e. We assume ha i s Jacobi endomo phism, Φ, sa isfies he
algeb aic assump ion A) as well as he wo enso ial equa ions (4.1).SinceΦsa isfies D1), i ollows ha he Jacobi en-
domo phism is gi en by o mula (2.5), whe e 2(n−1)α=(n−1)dJρ=dJ(T Φ). The e o e he sp ay Sis iso opic and
sa isfies he condi ion dJα=0.
Due o condi ion D2), we ha e ha Sis Ricci cons an and, as we ha e seen al eady, i ollows ha he sp ay Sis weakly
Ricci cons an .
Using he ac ha 2α=dJρwe ob ain
2LSα=LSdJρ=d[S,J]ρ+dJLSρ=d ρ=dρ.(4.2)
Wi hin he assump ion ha he Ricci cu a u e does no anish on T0M, we may conside he unc ion F>0 such ha
F2=sign(ρ)ρ>0onT0M.Sincedd J(T Φ) =(n−1)dd Jρ=(n−1)dd JF2, he assump ion A) assu es ha Fis a Finsle
unc ion. The condi ion 2α=dJρ eads now 2α=dJF2and using o mula (4.2) we ob ain LSdJF2=dF2, which means
ha Sis he geodesic sp ay o he Finsle unc ion F.
We eplace T Φ=(n−1)F2=(n−1)ρ=(n−1)iSαand dJ(T Φ) =2(n−1)FdJF=2(n−1)α=(n−1)dJρin he
exp ession o Φand ob ain o mula (3.3) o κ=sign(ρ). I ollows ha sp ay Sis Finsle me izable by he Finsle
unc ion Fo cons an cu a u e κ=sign(ρ).
Con e sely, i he sp ay Sis Finsle me izable by a Finsle unc ion o non-ze o cons an flag cu a u e hen i s Jacobi
endomo phism is gi en by o mula (3.3). I is a s aigh o wa d compu a ion o see ha Φsa isfies all h ee condi ions A),
D1) and D2)in(4.1).2
Some o he condi ions o Theo em 4.1 a e ela ed o he condi ions o Theo em 7.2 in [11] as ollows. Bo h heo ems
use he assump ion o non- anishing Ricci cu a u e ρ=iSα. Condi ion D1), which implies ha he sp ay Sis iso opic and
dJα=0, is s onge hen condi ion 2. o Theo em 7.2 in [11]. Also condi ion D2) implies ha ∇α=0 and he e o e his
implies condi ion 3. o Theo em 7.2 in [11]. No e ha ∇α=αis he semi-basic de i a ion in [11]. Howe e , he conclusion
in Theo em 4.1 is s onge , o a gi en sp ay, we seek o he me izabili y by a Finsle unc ion o cons an flag cu a u e.
Theo em 7.2 in [11] cha ac e izes local a ia ional non-fla iso opic ypical sp ays. The di e en iabili y assump ions a e
di e en , while we use smoo hness, in [11] he analy ici y o all geome ic s uc u es is assumed.
The algeb aic condi ion A) assu es he egula i y condi ion o he sough a e Finsle unc ion. Condi ion D1)isequi a-
len o he ac ha he sp ay Sis iso opic, i s Jacobi endomo phism is gi en by o mula (2.5), whe e 2(n−1)α=dJ(T Φ)
and hence sa isfies he condi ion dJα=0. Condi ion D2) says ha he sp ay Sis Ricci cons an .
Condi ions D1) and D2)in(4.1) a e e y use ul o decide whe he o no a gi en sp ay is me izable by a Finsle
unc ion o non-ze o cons an cu a u e. Howe e , he e a e me izable sp ays by Finsle unc ions o non-cons an flag
cu a u e, and hence whe e condi ions D1) and D2) canno be used. See he example in Sec ion 5.2.InTheo em 4.2 we will
s eng hen he esul o Theo em 4.1 by limi ing ou discussion o he case whe e Finsle me izabili y is equi alen o he
me izabili y by a Finsle unc ion o cons an cu a u e. In his discussion, we use he Finsle ian e sion o Schu ’s Lemma,
[4, Lemma 3.10.2]. The e o e, we will ha e o limi ou conside a ions o he case whe e dim M⩾3. The case dim M=2
will be ea ed sepa a ely.
4.2. dim M ⩾3
Conside San iso opic sp ay, whose Jacobi endomo phism is gi en by o mula (2.5). In he nex heo em we will see
ha he condi ion dJα=0 is he bes we can equi e o make su e ha he h ee equi alen condi ions in Rema k 3.4 a e
also ue in he Finsle ian con ex . We add wo mo e condi ions, he las one is condi ion D2)inTheo em 4.1.
Theo em 4.2. Conside S a sp ay o non- anishing Ricci cu a u e. Then, he sp ay is iso opic, sa isfies he algeb aic condi ion A),and
he condi ion d Jα=0, i and only i he ollowing fi e condi ions a e equi alen :
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i) S is Finsle me izable;
ii) S is me izable by a Finsle me ic o non- anishing scala flag cu a u e;
iii) S is Finsle me izable by an Eins ein me ic;
i ) S is me izable by a Finsle me ic o non-ze o cons an flag cu a u e;
) S is Ricci cons an .
P oo . We assume ha he sp ay is iso opic, sa isfies he algeb aic condi ion A), and he condi ion dJα=0. Wi hin hese
assump ions, we will p o e he ollowing implica ions i) ⇒ii) ⇒iii) ⇒i ) ⇒ ) ⇒i). Fo some o hese implica ions we will
no need he condi ion dJα=0. Mo e exac ly, we will use i o he implica ions ii) ⇒iii), iii) ⇒i ), and ) ⇒i). Also, he
assump ion dim M⩾3 will be needed only o he implica ion iii) ⇒i ).
In o de o p o e ha condi ion i) implies condi ion ii) we assume ha he sp ay Sis Finsle me izable. We will p o e
ha he co esponding Finsle space has scala flag cu a u e. This esul coincides wi h Lemma 8.2.2 in [21], whe e he
p oo uses di e en (local) echniques. Ou p oo is based on he di e en ial calculus on T0Massocia ed o a sp ay wi hin
he F öliche –Nijenhuis o malism.
We conside Fa Finsle unc ion ha me icizes he sp ay S. I ollows ha Sis he unique solu ion o he equa ion
LSdJF2=dF2. The e o e, he co esponding Helmhol z condi ions a e sa isfied. One o hese Helmhol z condi ions in ol es
he Jacobi endomo phism Φ, and can be exp essed using he semi-basic 1- o m θ=dJF2as ollows, see [6, Theo em 4.1.],
0=dΦθ=dρJ−α⊗Cθ=ρdJθ−dα⊗Cθ=−α∧LCθ=−α∧θ. (4.3)
In he abo e o mula we used dJθ=d2
JF2=0, since d2
J=d[J,J]=0, and LCθ=θ,sinceθis a 1-homogeneous 1- o m.
Helmhol z condi ion (4.3) implies α∧dJF2=0 and hence he e exis s a unc ion κ∈C∞(T0M)such ha α=κdJF2/2=
κFdJF. I ollows ha ρ=iSα=κF2and he Jacobi endomo phism (2.5) can be w i en now as in o mula (3.3), which
shows ha he Finsle space (M,F)has scala flag cu a u e κ.
In o de o p o e ha condi ion ii) implies condi ion iii) we make use o he di e en ial assump ion dJα=0. Acco ding
o Lemma 2.2 i ollows dJα=0isequi alen o2α=dJρ. Using he ac ha ρ=κF2and 2α=κdJF2i ollows
2α=dJρ=F2dJκ+κdJF2=F2dJκ+2α.
Abo e o mulae imply ha dJκ=0 and hence he scala flag cu a u e κdoes no depend on he flagpole y. I ollows ha
ρ(x,y)=λ(x)F2(x,y), whe e λ(x)=κ(x). Hence, he Finsle Func ion Fis an Eins ein me ic ha me icizes he sp ay S.
In o de o p o e ha condi ion iii) implies condi ion i ) we will also make use o Schu ’s Lemma. We know ha he
sp ay Sis Finsle me izable by an Eins ein me ic F. Then, he e exis s a non- anishing unc ion λ∈C∞(M)such ha
ρ(x,y)=λ(x)F2(x,y).SinceSis iso opic, i ollows ha i s Jacobi endomo phism is gi en by o mula
Φ=λF2J−α⊗C.
Now we use he assump ion ha dJα=0, which by Lemma 2.2 implies ha 2α=dJρ=λdJF2. This implies ha he Jacobi
endomo phism is gi en by o mula (3.3), whe e κ(x)=λ(x)is he scala flag cu a u e. Since dim M⩾3, we use Schu ’s
Lemma and ob ain ha κis a non-ze o cons an . This implies ha he sp ay Sis me izable by he Finsle unc ion Fo
cons an flag cu a u e.
The implica ion i ) ⇒ ) is s aigh o wa d. Conside S he geodesic sp ay o Finsle me ic Fo non-ze o cons an flag
cu a u e κ. I ollows ha he Ricci scala is gi en by ρ=κF2, whe e κis a cons an . The e o e, dhρ=κdhF2=0since
dhF2=0.
Fo he las implica ion ) ⇒i), we use he fi s implica ion o Theo em 4.1.SinceSis iso opic and sa isfies dJα=0
i ollows ha he Jacobi endomo phism Φsa isfies equa ion D1)in(4.1). The ac ha Sis Ricci cons an means ha Φ
sa isfies also equa ion D2)in(4.1).ByTheo em 4.1 we ob ain ha he sp ay Sis Finsle me izable.
To conclude he p oo , we ha e o show ha i a sp ay Ssa isfies one o he fi e equi alen condi ions o he heo em,
hen necessa ily Sis iso opic, sa isfies he algeb aic condi ion A), as well as he condi ion dJα=0. We assume ha
condi ion i ) is sa isfied and hence he sp ay Sis Finsle me izable by a Finsle me ic Fo non-ze o cons an cu a u e κ.
I ollows ha i s Jacobi endomo phism is gi en by o mula (3.3) and hence α=κFdJF=κdJF2.SinceFis a Finsle
unc ion we ob ain ha ankdd JF2=2nand hence he algeb aic condi ion A) is sa isfied. The condi ion dJα=0isalso
sa isfied. 2
I a Finsle unc ion is educible o a Riemannian me ic, and Sis i s iso opic geodesic sp ay, i ollows ha α=
κdJF2, whe e dJκ=0, and hence we always ha e dJα=0. This shows ha he equi alence o condi ions i), ii) and i ) o
Theo em 4.2 gene alize o he Finsle ian con ex he equi alence o he h ee condi ions o Rema k 3.4.
In Theo em 4.2, he condi ion dim M⩾3 was e y impo an , since i allowed us o use he Finsle ian e sion o Schu ’s
Lemma.
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4.3. dim M =2
In his subsec ion we pay a en ion o he Finsle me izabili y p oblem on 2-dimensional mani olds. I is known ha
any sp ay on a 2-dimensional mani old is iso opic. This esul allows us o simpli y he wo condi ions (4.1) o Theo em 4.1.
Theo em 4.3. Conside S a sp ay o non- anishing Ricci cu a u e ha sa isfies he algeb aic condi ion A), o n =2.
i) The sp ay S is iso opic, which means ha i s Jacobi endomo phism is gi en by o mula (2.5), whe e he semi-basic 1- o m α=
αidxihas he componen s
α1=R2
2
y1=−R2
1
y2,α2=−R1
2
y1=R1
1
y2.(4.4)
ii) The sp ay S is me izable by a Finsle unc ion o non-ze o cons an flag cu a u e i and only i sa isfies he ollowing wo condi-
ions
dJα=0,dhρ=0.(4.5)
P oo . i) Due o he homogenei y condi ions o he sp ay we ha e ha Φ(S)=0 and by o mula (2.3) we ob ain Rijyj=0.
I ollows ha Φcan be w i en as in o mula (2.5), whe e ρ=R1
1+R2
2and he semi-basic 1- o m α=αidxihas he
componen s (4.4).
Since any sp ay Sis iso opic we ha e ha he Jacobi endomo phism Φsa isfies Eq. D1)in(4.1) i and only i i sa isfies
fi s equa ion in (4.5). This pa is hen a consequence o Theo em 4.1.2
The fi s pa o Theo em 4.3 coincides wi h Lemma 8.1.10 in [21] and o mulae (4.4) coincide wi h o mulae (8.37) and
(8.38) in [21].
The wo condi ions (4.5) can be w i en as ollows
dJα=0⇔2α=dJρ⇔∂α1
∂y2=∂α2
∂y1⇔2α1=∂ρ
∂y1and 2α2=∂ρ
∂y2,
dhρ=0⇔δρ
δx1=δρ
δx2=0,whe e δ
δxi=h∂
∂xi=∂
∂xi−∂Gj
∂yi
∂
∂yj.(4.6)
We will use he abo e condi ions in he nex sec ion o es whe he o no some sp ays on a wo-dimensional mani old
a e me izable by Finsle unc ions o cons an cu a u e.
5. Examples
In his sec ion we p o ide examples o show he consis ency o he condi ions we discussed so a .
5.1. The case d Jα=0
I is well known ha he class o iso opic sp ays is in a ian unde p ojec i e de o ma ions o sp ays. We s a wi h a
Finsle me izable iso opic sp ay, S0, which sa isfies he condi ion dJα=0. We will s udy p ojec i e de o ma ions o S0,
which also sa is y he condi ion dJα=0. Wi hin his p ojec i e de o ma ions, we will seek o hose which do no p ese e
he condi ion o being Ricci cons an . Using Theo em 4.2, hiswillleadus oexampleso non-Finsle me izableiso opic
sp ays.
Le S0be he geodesic sp ay o a Finsle unc ion F0, which has cons an flag cu a u e κ0.Deno ebyh0 he ho izon al
p ojec o induced by he sp ay S0and ∇0 he co esponding dynamical co a ian de i a i e. The sp ay S0is iso opic, i s
Jacobi endomo phism is gi en by
Φ0=κ0F2
0J−F0dJF0⊗C.
Conside he p ojec i ely equi alen sp ay S=S0−2PC, whe e P∈C∞(T0M)is a 1-homogeneous unc ion. Acco ding o
o mulae (4.8) in [8, P oposi ion 4.4], he ho izon al p ojec o and he Jacobi endomo phism o he sp ay Sa e gi en by
h=h0−2(PJ+dJP⊗C),
Φ=Φ0+P2−S0(P)J+(2dh0P−PdJP−∇
0dJP)⊗C.
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The sp ay Sis also iso opic. We will s udy now when a p ojec i e de o ma ion p ese es he condi ion dJα=0, whe e
he 1- o m αis gi en by α=κ0F0dJF0−2dh0P+PdJP+∇
0dJP.The2- o mdJαhas been compu ed in he p oo o
P oposi ion 4.4 in [8] and i is gi en by
dJα=3dJdh0P=−3dh0dJP.(5.1)
The e o e dJα=0 i and only i dh0P=dJg, o some unc ion g∈C∞(T0M).Fo g=P2/2 he unc ion Pis called a Funk
unc ion,see[21, Defini ion 12.1.4]. I has been shown in [21, P oposi ion 12.1.3] ha p ojec i e de o ma ions by a Funk
unc ion p ese e he Jacobi endomo phism.
In o de o simpli y some o he calcula ions we assume ha he unc ion Psa isfies dh0P=0 and hence dJα=0. I
ollows ha S0(P)=0 and using he commu a ion o mula (4.11) in [8] we ha e ∇0dJP=dJ∇0P−dh0P=0. The e o e, he
Jacobi endomo phism Φo he sp ay Sis gi en by
Φ=κ0F2
0+P2J−(κ0F0dJF0+PdJP)⊗C.(5.2)
I ollows ha he Ricci scala is gi en by
ρ=κ0F2
0+P2.
We check now he las condi ion o Theo em 4.2. Using he assump ion ha dh0P=0 i ollows ha dh0ρ=0 and hence
we ha e
dhρ=−2dPJ+dJP⊗Cρ=−2(PdJρ+2ρdJP)=−2
PdJρP2.(5.3)
F om he abo e o mula, we see ha we can choose a unc ion Psuch ha dhρ= 0, which will imply ha he sp ay Sis
no Ricci cons an . Acco ding o Theo em 4.2 we conclude ha he sp ay Sis no Finsle me izable. Indeed, we can ake
P=λF0, whe e λis a cons an . I we eplace his in o mula (5.3), we ob ain
dhρ=−4λκ0+λ2dJF2
0.(5.4)
In his case we ha e ha he sp ay Sis Ricci cons an , and hence i is Finsle me izable, i and only i ei he λ=0o
λ2+κ0=0. See also Theo em 5.1 in [8]. When he sp ay S0is p ojec i ely fla , we can iew his as an al e na i e p oo o
Theo em 1.2 in [24].
5.2. The case d Jα= 0.
We p esen now an example o a Finsle me izable, iso opic sp ay, which does no sa is y he condi ion dJα=0. This
also shows ha he assump ion dJα=0, which we made in Theo em 4.2, is he bes assump ion we could conside in o de
o ha e he equi alence o he fi e condi ions.
We will use he ollowing Rande s me ic s udied by Shen, see Example 11.2 in [20]. Conside a domain M⊂Rn, whe e
(x)=1−|a|2|x|4>0. Deno e by β(x,y)=2a,xx,y−|x|2a,y. The Finsle unc ion F:M×Rn→R,gi enby
F(x,y)=β2(x,y)+(x)|y|2+β(x,y)
(x)
has scala flag cu a u e gi en by
κ(x,y)=3a,y
F+3a,x2−2|a|2|x|2.
The geodesic sp ay So he Finsle unc ion Fis iso opic and he 1- o m α,in o mula(2.5),isgi enbyα=κFdJFand
hence
dJα=FdJκ∧dJF.
The scala flag cu a u e κis 0-homogeneous and he e o e 0 =C(κ)=iSdJκ. Mo eo e he flag cu a u e κdepends on
he flagpole y, which means ha dJκ= 0. The e o e,
iSdJα=FiSdJκdJF−FiSdJFdJκ=−F2dJκ= 0,
and his implies ha dJα= 0. The e o e, he sp ay Sis Finsle me izable and iso opic. Howe e , he fi e condi ions in
Theo em 4.2 a e no equi alen and his is due o he ac ha dJα= 0.
5.3. Two-dimensional examples
We conside now some examples o sp ays on a wo-dimensional mani old and use he condi ions (4.5) o es i hey
a e me izable by a Finsle unc ion o cons an cu a u e.