On geometric vector fields of Minkowski spaces and their applications
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ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI
SPACES AND THEIR APPLICATIONS
CS.VINCZE
Abs ac . As i is well-known, a Minkowski space is a ini e dimen-
sional eal ec o space equipped wi h a Minkowski unc ional F. By
he help o i s second o de pa ial de i a i es we can in oduce a Rie-
mannian me ic on he ec o space and he indica ix hype su ace
S:= F−1(1) can be in es iga ed as a Riemannian submani old in he
usual sense.
Ou aim is o s udy a ine ec o ields on he ec o space which a e,
a he same ime, a ine wi h espec o he Funk me ic associa ed wi h
he indica ix hype su ace. We gi e an uppe bound o he dimen-
sion o hei ( eal) Lie algeb a and i is p o ed ha equali y holds i and
only i he Minkowski space is Euclidean. C i e ia o he exis ence is also
gi en in lowe dimensional cases. No e ha in case o a Euclidean ec-
o space he Funk me ic educes o he s anda d Cayley-Klein me ic
pe u bed wi h a nonze o 1- o m.
As an applica ion o ou esul s we p esen he gene al solu ion o
Ma sumo o’s p oblem on con o mal equi alen Be wald and locally
Minkowski mani olds. The easoning is based on he heo y o ha monic
ec o ields on he angen spaces as Riemannian mani olds o , in an
equi alen way, as Minkowski spaces. Ou main esul s a es ha he
con o mal equi alence be ween wo Be wald mani olds mus be i ial
unless he mani olds a e Riemannian.
1. P elimina ies
1.1. Minkowski unc ionals. [1], [15]. Le Vbe an n-dimensional (n≥2)
eal ec o space. The elemen s o Vwill be in e p e ed bo h as poin s p, q, ...
and ec o s , w, ... as usual. A Minkowski unc ional on Vis a unc ion
F:V→Rwi h he ollowing p ope ies:
(F0) ∀p∈V {0}:F(p)>0 and F(0) = 0.
(F1) Fis posi i e homogeneous o deg ee 1, i.e. ∀ ∈R+:F( p) = F(p).
(F2) Fis con inuous on Vand smoo h o e he se V {0}.
(F3) ∀p∈V {0}:
gp:= E00(p): V×V→R
is an inne p oduc on V, whe e E:= 1
2F2is he ene gy unc ion.
The condi ion (F1) implies he ene gy unc ion E o be homogeneous o
deg ee 2 and we ha e
(1) gp(p, ) = E0(p)( ), gp(p, p) = 2E(p).
1991 Ma hema ics Subjec Classi ica ion. 53C60, 58B20.
Key wo ds and ph ases. Minkowski spaces, A ine ec o ields, Finsle mani olds, Con-
o mal equi alence, Be wald mani olds.
Suppo ed by FKFP (0184/2001), Hunga y.
1
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 2
1.2. Ca an enso s. Le (V, F) be a Minkowski space and conside he
mappings
(2) C[(p):=E000(p): V×V×V→R,Cp:V×V→V
de ined by he o mula
(3) gp(Cp( , w), z) = C[(p)( , w, z);
Cis called he i s Ca an enso . The i s Ca an enso , as well as i s
lowe ed enso C[is o ally symme ic and, o cou se, mul ilinea . This
means ha he mapping
Cp( , ·): V→V, Cp( , ·)(w):=Cp( , w)
is a sel -adjoin linea ope a o wi h espec o he inne p oduc gp. Since
he ene gy unc ion is homogeneous o deg ee 2 i ollows ha
(4) Cp(p, ·) = 0.
I is well-known ha he anishing o he i s Ca an enso implies he
Minkowski space o be Euclidean. The con ac ed Ca an enso is de ined
by he o mula
(5) ˜
Cp( ) := Cp( , ·);
Deicke’s classical heo em s a es ha he con ac ed Ca an enso anishes
i and only i he space is Euclidean; see e.g. [2], [3] and [1].
1.3. The associa ed Funk me ic. [12], [15]. Le (V, F) be a Minkowski
space and conside he se
(6) B◦:= {p∈V|F(p)<1};
he associa ed Funk me ic
(7) L:TB◦→R
is de ined by he p ope y
Fp+
L( p)= 1,
whe e p∈TpB◦is an a bi a y nonze o angen ec o a he poin p∈B◦.
Then, o cou se, he pai (B◦, L) is a Finsle mani old in he usual sense,
i.e. o any poin p∈B◦ he es ic ion
(8) Lp:= L|TpB◦
is a Minkowski unc ional. Le e1, . . . , enbe an a bi a y basis o he ec o
space V wi h he dual basis u1, . . . , unand conside he s anda d induced
coo dina e sys em (xi, yi)n
i=1 on he angen mani old T V . Okada’s heo em
s a es ha o any indeces i∈ {1, . . . , n}:
∂
∂xiL=L∂
∂yiL;
o a p oo see e.g. [15], Lemma 2.3.1. In e ms o di e en ial geome ic
s uc u es we can w i e he p e ious o mula in he o m
(9) dhL=LdJL,
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 3
whe e his he ho izon al dis ibu ion de e mined by he i s ncoo dina e
ec o ields
∂
∂x1, . . . , ∂
∂xn
and Jis he canonical almos angen s uc u e on he angen mani old
TB◦. We ha e ha
(10) dJdhL=dJL∧dJL+Ld2
JL= 0,
which is jus he coo dina e- ee exp ession o he classical Rapcs´ak equa-
ion o p ojec i e equi alence; see [16]. This means ha Vas a ine ec o
space and he associa ed Funk mani old (B◦, L) a e p ojec i ely equi alen ;
simply pu he Funk mani old is p ojec i ely la . No e ha he geodesics
o Vas a ine ec o space a e he usual pa ame ized lines
(11) c:R→V, −→ c( ):=p+ .
Using he undamen al ela ion we can w i e he o mula o p ojec i e equi -
alence be ween he canonical sp ay ξo he Funk mani old and ξVin he
o m
(12) ξ=ξV−LC,
whe e Cis he so-called Liou ille ec o ield; [16], 3.8. P oposi ion, o he
de ails o sp ay geome y see also [4], [5] and [15].
De ini ion. Le (M, ξ)be an a bi a y sp ay mani old; he ec o ield
X∈X(M)is called an a ine ec o ield i i s local 1-pa ame e g oup
consis s o geodesic-p ese ing maps. The ec o ield is p ojec i e i he
local 1-pa ame e g oup consis s o maps p ese ing he geodesics up o a
s ic ly inc easing epa ame iza ion.
Fo lo s o equi alen cha ac e iza ions see e.g. [9] and [13].
1.4. Example. Suppose ha Xis an a ine ec o ield on he ec o space
Vand conside a poin p oge he wi h i s open neighbou hood U⊂Vsuch
ha any in eg al cu e s a ing om a poin q∈Uis de ined on he open
in e all (−, ); he mapping
ϕ: ∈(−, )→ϕ
deno es he local 1-pa ame e g oup o he ec o ield X. We se
c(s):=q+s ,
whe e he pa ame e sis small enough sa is ying he condi ion Im c ⊂U.
Since Xis a ine, he cu e ˜c:= ϕ ◦cis a geodesic, i.e. i s second o de
de i a i e anishes; especially
(13) ˜c00(0) = 0 ⇒(ϕ )00(q)( , ) = 0
and, consequen ly, ϕ00
= 0. In o he wo ds, ϕ0
is independen o he poin q
which implies he ec o ial pa
(14) q∈U→lim
→0
(ϕ− )0(q)(ei)−ei
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 4
o he Lie b acke [X, ∂
∂ui] o be cons an . Taking he Lie b acke again i
ollows ha
(15) [X, ∂
∂ui],∂
∂uj= 0
and we ha e he ollowing simple di e en ial equa ion
∂2
∂ui∂ujXk= 0
o he coe icien s o he ec o ield X. The e o e
(16) X= (αi
juj+βi)∂
∂ui,
whe e A:= (αj
i)1≤i,j≤nis a ma ix o eal numbe s and β1, . . . , βn∈R. As
a ou ine calcula ion shows,
(17) ϕ (q) = e Aq+w ,
whe e he pa o ansla ion is independen o q.
1.5. Riemannian quan i ies. Le (V, F ) be a Minkowski space; acco ding
o he egula i y p ope y (F3), he ec o space can be conside ed as a
Riemannian mani old in he usual sense. A e iden i ying he angen spaces
wi h V, conside he ollowing special ec o ields:
X:V→V, p −→ Xp:= x,
Y:V→V, p −→ Yp:= y,
Z:V→V, p −→ Zp:= z,
whe e x, y and z∈Va e a bi a ily ixed ec o s. I can be easily seen ha
he L´e i-Ci i a connec ion ∇associa ed wi h gac s as ollows:
(18) ∇XpY=Cp(x, y)
and, consequen ly, he cu a u e enso has he ollowing simple o m:
(19) Qp(x, y)z=Cp(Cp(x, z), y)− Cp(x, Cp(y, z)).
We se
(20) Rp(x, y) :=
n
X
i=1
gp(Qp(ei, x)y, ei),
whe e e1, . . . , en∈Vis a gp-o hono mal sys em; as usual Ris called he
Ricci enso o he Riemannian mani old V {0}.
2. A ine ec o ields o he associa ed Funk me ic
In wha ollows Vdeno es a Minkowski ec o space equipped wi h he
Minkowski unc ional F. As we ha e seen abo e Vas a ine ec o space and
he Funk mani old (B◦, L) a e p ojec i ely equi alen and, consequen ly, he
es ic ion o p ojec i e ec o ields on he ec o space a e p ojec i e wi h
espec o he Funk me ic and ice e sa. In wha ollows we a e going o
s udy a ine ec o ields on he ec o space which a e, a he same ime,
a ine wi h espec o he Funk me ic. Suppose ha Xis one o hem; i
c: ∈R→c( ):=q+ ,
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 5
hen, by he p ojec i e equi alence, he e is a s ic ly inc easing epa ame i-
za ion θsuch ha he cu e ˜c:= c◦θis a geodesic o he Funk mani old.
Acco ding o he o mula (12), he epa ame iza ion is jus he solu ion o
he di e en ial equa ion
(21) θ00 =−(θ0)2L( q);
see e.g. [7]. Unde he ini ial condi ions θ(0) = 0 and θ0(0) = 1 we ha e
ha
(22) θ(s) = 1
L( q)ln(1 + sL( q)),
i.e.
˜c(s) = q+1
L( q)ln(1 + sL( q))
is a geodesic o he Funk mani old. Le ϕ: ∈R→ϕ be he 1-pa ame e
g oup o he ec o ield X; using he o mula o epa ame iza ion i ollows
ha Xis a ine wi h espec o he Funk me ic i and only i
(23) ϕ (q+θ(s) ) = ϕ (q) + 1
L◦Tϕ ( q)ln(1 + sL ◦Tϕ ( q)) ( ),
whe e := (ϕ )0(q) which is ac ually independen o he poin q∈B◦as we
ha e seen abo e. On he o he hand
(24) ϕ (q+θ(s) )(17)
=ϕ (q) + θ(s) ( ),
i.e.
θ(s) = 1
L◦Tϕ ( q)ln(1 + sL ◦Tϕ ( q)).
Di e en ia ing by s, i can be easily seen ha
(25) L◦Tϕ ( q) = L( q)
p o ided, o cou se, ha he pa ame e is small enough sa is ying he
condi ion ϕ (q)∈B◦. Since he mapping
(26) Tϕ: ∈R→Tϕ
is jus he 1-pa ame e g oup o he comple e li Xc, he ela ion
(27) Xc
qL= lim
→0
L◦Tϕ ( q)−L( q)
= 0
ollows immedia ely.
P oposi ion 1. Suppose ha Xis an a ine ec o ield on he ec o space
V; hen he ollowing condi ions a e equi alen 1:
(i) Xis an a ine ec o ield wi h espec o he Funk me ic.
(ii) XcL= 0.
1Fo he implica ion (i)⇒(ii) o P oposi ion 1 and 2 we should e e o he lec u e
A ine and p ojec i e ec o ields on sp ay mani olds p esen ed by L. R. Lo as; Wo kshop
on Finsle Geome y and i s Applica ions, Augus 11-15, 2003, Deb ecen, Hunga y. See
also [9].
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 6
In e ms o coo dina es we ha e he exp ession
(28) Xc= (αi
jxj+βi)∂
∂xi+αi
jyj∂
∂yi.
Conside now he p ojec ion
(29) ρ:TB◦→S, q→ρ( q):=q+
L( q);
i is clea ha F◦ρ= 1 and, consequen ly, T F ◦T ρ = 0. On he o he
hand, as a s aigh o wa d calcula ion shows
(30) TF ◦Tρ(Xc)( q) = −1
L2( q)F0(ρ( q))( )Xc
qL+Xρ( q)F.
The s ic ly con exi y o he indica ix hype su ace implies ha
(31) F0(ρ( q))( )6= 0;
in a geome ical in e p e a ion his means ha couldn’ be angen ial o
he indica ix hype su ace a he poin ρ( q)∈S. Using he p e ious esul
we ha e he ollowing p oposi ion immedia ely.
P oposi ion 2. Suppose ha Xis an a ine ec o ield on he ec o space
V; hen he ollowing condi ions a e equi alen :
(i) Xis an a ine ec o ield wi h espec o he Funk me ic.
(ii) XF ◦ρ= 0.
Since ρis su jec i e, (ii) means ha he es ic ion X|Smus be angen ial
o he indica ix hype su ace. In o he wo ds, i cis an in eg al cu e o
he ec o ield Xs a ing om a poin p∈S, hen Im c ⊂S.
P oposi ion 3. Suppose ha Xis an a ine ec o ield on he ec o space
Vwhich is, a he same ime, a ine wi h espec o he Funk me ic. Then
(32) A:=
n
X
i=1
αi
i= 0.
I he Minkowski unc ional is e e sibile, hen Xis a linea ec o ield,
i.e. i s 1-pa ame e g oup consis s o linea ans o ma ions and Xcan be
w i en in he o m
(33) X=αi
juj∂
∂ui.
i.e. he 1-pa ame e g oup consis s o special linea ans o ma ions.
P oo . Since o any ∈R he indica ix hype su ace is in a ian unde
he ans o ma ion ϕ p ese ing he a ine (especially con ex) combina ion
i ollows ha B◦is also in a ian . The e o e
ZB◦
du1. . . dun=Zϕ− (B◦)
de ϕ0
du1. . . dun=ZB◦
e Adu1. . . dun;
di e en ia ing by , we ha e
(34) 0 = AZB◦
e Adu1. . . dun⇒ A= 0
as was o be s a ed.
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 7
Suppose ha Fis e e sible, i.e.
(35) F( ) = F(− );
hen
Theo em 1. Suppose ha Xis an a ine ec o ield on he ec o space V
which is, a he same ime, a ine wi h espec o he Funk me ic. Then X
is a Killing ec o ield on he ec o space Vas Riemannian mani old and,
a he same ime, i is a Killing ec o ield on he indica ix hype su ace
wi h espec o he induced Riemannian s uc u e.
P oo . As an easy calcula ion shows, o any indeces i, j ∈ {1, . . . , n}
LXg(∂
∂ui,∂
∂uj) = ∂2
∂ui∂uj(XE)−[X, ∂
∂ui],∂
∂uj(E).
By he ela ion (ii) o P oposi ion 2, XF = 0 on he indica ix hype su ace;
since X is ac ually a linea o , in an equi alen way, i is a homogeneous
ec o ield, he ela ion XF = 0 holds on he whole ec o space V. This
means ha XE = 0 and he i s e m anishes. The anishing o he Lie
b acke ollows immedia ely om he o mula (15).
P oposi ion 4. Suppose ha Xis an a ine ec o ield on he ec o space
Vwhich is, a he same ime, a ine wi h espec o he Funk me ic; hen
˜
C(X) = 0.
P oo . Since LXg= 0, i ollows ha he di e gence o he ec o ield X
anishes; indeed, o any ec o ields Yand Z
0 = Xg(Y, Z)−g([X, Y ], Z)−g(Y, [X, Z]) = g(∇XY−[X, Y ], Z)+
+g(∇XZ−[X, Z], Y ) = g(∇YX, Z) + g(∇ZX, Y ),
i.e. he Hesse o m (∇X)[is an isymme ic and, o cou se, di X = 0. Le
now e1, . . . , enbe a basis o he ec o space; he ela ion (18) shows ha
he pa ame e s o he L´e i-Ci i a connec ion wi h espec o he dual basis
u1, . . . , una e jus he componen s o he Ca an enso . Thus we ha e
di X =
n
X
i=1
∂
∂uiXi+˜
C(X) =
n
X
i=1
αi
i+˜
C(X)(35)
=˜
C(X)
and he anishing o ˜
C(X) ollows immedia ely.
Theo em 2. Suppose ha Vis o dimension n≥3and le A◦(V)be he
( eal) Lie algeb a o a ine ec o ields on he ec o space which a e, a he
same ime, a ine wi h espec o he Funk me ic; hen
(36) dim A◦(V)≤n(n−1)
2
and equali y holds i and only i he Minkowski space is Euclidean.
P oo . Since he elemen s o A◦(V) a e angen ial o he indica ix hy-
pe su ace and dim S=n−1, he es ima ion is a di ec consequence o
Theo em 1; see [13], sec ion 3.53. Suppose ha dim A◦(V) = n(n−1)
2and
conside a basis
X1, . . . , Xk
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 8
whe e, o he sake o simplici y, k=n(n−1)
2. Le p∈Sbe an a bi a ily
ixed poin ; since n≥3 he ec o ields X1, . . . , Xkis linea ly dependen a
he poin p, i.e. he e exis eal numbe s 1, . . . , ksuch ha 16= 0 and
(37) 1X1(p) + . . . + kXk(p) = 0.
The ec o ield
(38) Y1:= 1X1+. . . + kXk
is, o cou se, non i ial. On he o he hand, Y1 anishes a he poin pwhich
means ha i s 1-pa ame e g oup consis ing o isome ies wi h espec o
he Riemannian me ic ghas a ixpoin and, consequen ly,
(39) gp( , w) = (ϕ∗
g)p( , w) = gϕ (p)(ϕ0
( ), ϕ0
(w)) = gp(ϕ0
( ), ϕ0
(w)) =
=gp(ϕ ( ), ϕ (w)),
i.e. he g oup consis s o o hogonal ans o ma ions wi h espec o he
inne p oduc gp. I also ollows ha Y1can be in e p e ed as a non i ial
elemen o A◦(H), whe e he subspace His o hogonal o he poin pwi h
espec o gp. Indeed, he in a iance o Hunde he ans o ma ions o
he 1-pa ame e g oup implies he ec o ield Y1 o be angen ial o he
subspace H. On he o he hand, i he es ic ion Y1|H anishes hen he
ans o ma ions o he 1-pa ame e g oup ha e u he ixpoin s; by se ing
a basis o hem we can see ha he g oup is i ial and, consequen ly, Y1= 0
which is a con adic ion.
Conside now he basis Y1, X2, . . . , Xk; i he ec o ields X2, . . . , Xka e
linea ly dependen a he poin p, hen he e exis eal numbe s 2, . . . , k
such ha 26= 0 and
(40) 2X2(p) + . . . + kXk(p) = 0.
In a simila way as abo e we de ine he non i ial ec o ield
(41) Y2:= 2X2+. . . + kXk.
Since Y2 anishes a he poin p, i s 1-pa ame e g oup consis s o o hogonal
ans o ma ions wi h espec o he inne p oduc gp. I also ollows ha
Y2can be in e p e ed as a non i ial elemen o A◦(H). Using his p occess
as a as possible we can cons uc he ec o ields Y1, Y2, . . . , Yl; in wha
ollows i is p o ed ha hei es ic ions o he subspace Ha e linea ly
independen . Suppose, in con a y, ha
(42) s1Y1+. . . +slYl|H= 0
is a non i ial combina ion; i s16= 0, hen Y1|H∈ L(Y2, . . . , Yl) and, by he
cons uc ing p occess, he ela ion
(43) X1|H∈ L(X2, . . . , Xk)
ollows immedia ely. Le us in oduce he ec o ield
(44) X:= X1−η2X2+. . . −ηkXk
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 9
whe e (η2)2+. . . + (ηk)26= 0 and X|H= 0. Then Xis a linea ec o ield
and i s 1-pa ame e g oup can be ep esen ed in he o m
(45)
1 0 ...0 0
0 1 ...0 0
. . .
. . .
. . .
0 0 ...1 0
0 0 ...0α( )
n×n
whe e he condi ion de ϕ = 1 should be also sa is ied. This means ha
α≡1 and, consequen ly, he 1-pa ame e g oup o he ec o ield Xis
i ial, i.e. X= 0 which is a con adic ion.
In case o s1= 0, he easoning is simila o he i s non i ial coe -
icien ; he con adic ion shows ha Y1, . . . , Yla e linea ly independen as
he elemen s o A◦(H). Since he p occess ends a he s ep
(46) l= dim A◦(H) = (n−1)(n−2)
2,
we ha e ha he X’s block o he basis Y1, . . . , Yl, Xl+1, . . . , Xkmus be
linea ly independen a he poin p. He e
(47) k−l=n(n−1)
2−(n−1)(n−2)
2=n−1
and, by P oposi ion 4,
˜
C(Xl+1) = . . . =˜
C(Xk) = 0.
This means ha ˜
Cp anishes on a basis o he angen space TpSand, con-
sequen ly, ˜
Cp= 0; Deicke’s heo em implies he space o be Euclidean.
We ha e a mo e anspa en pic u e in case o lowe dimensional spaces
as he ollowing heo em shows.
Theo em 3. Suppose ha dim V= 2; hen he Lie algeb a A◦(V)is i ial
unless he space is Euclidean.
I Vis o dimension 3, hen we ha e he ollowing cases:
(i) dim A◦(V) = 0.
(ii) dim A◦(V)=1and he indica ix is a o a ion su ace wi h espec
o he inne p oduc gp, whe e p∈Sis a ze o o any ec o ield
X∈ A◦(V).
(iii) dim A◦(V) = 3 and he space is Euclidean.
P oo . Le X∈ A◦(V) be a non i ial ec o ield and dim V= 2; i X
has no ze o excep he o igin hen, by P oposi ion 4, ˜
C= 0 and Deicke’s
heo em implies he space o be Euclidean.
I X(p) = 0, hen i s 1-pa ame e g oup consis s o o hogonal ans o -
ma ions wi h espec o he inne p oduc gp:
(48) gp( , w) = (ϕ∗
g)p( , w) = gϕ (p)(ϕ0
( ), ϕ0
(w)) = gp(ϕ0
( ), ϕ0
(w)) =
=gp(ϕ ( ), ϕ (w)).
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 16
Riemannian, i.e. he e is a neighbou hood Uo he poin psuch ha he
es ic ed ene gy unc ion E|T U is quad a ic.
P oo . Since he h -cu a u e enso o he Be wald connec ion is in a i-
an , o any ec o ield Y, Z and W∈X(M) i ollows ha
0 =◦
Pα(Yc, Zc)Wc−◦
P(Yc, Zc)Wc=
= [[Yhα, Z ], W ]−[[Yh, Z ], W ] = [[Yhα, Z ]−[Yh, Z ], W ].
This means ha he ec o ield
[Yhα, Z ]−[Yh, Z ] = [Yhα−Yh, Z ]
is a e ical li and, consequen ly, he di e ence ec o ield Yhα−Yhis
linea on any angen space TpM. As an easy calcula ion shows
Yhα−YhE=−(Y α) E.
Conside he ec o ield
X:= Yhα−Yh+1
2(Y α) C;
since i is angen ial o he indica ix hype su ace, he es ic ion X|TpM
is an elemen o he Lie algeb a A◦(TpM). This ollows immedia ely om
P oposi ion 2. The e o e, by P oposi ion 4, ˜
C(FX) = 0, whe e ˜
Cis he
semibasic ace o he i s Ca an enso . O cou se, we ha e a well-known
ans o ma ion o mula o changing o he Ba hel endomo phism unde a
con o mal change, namely,
Yhα=Yh−1
2αcY −1
2(Y α) C−EC(F g ad α , Y c) + 1
2Y Eg ad α ;
[6], see also [17], [19] and [20] o he coo dina e- ee exp ession. I ollows
ha
X=1
2Y Eg ad α −1
2αcY −EC(F g ad α , Y c)
and, consequen ly,
0 = 1
2Y E˜
C(F g ad α )−1
2αc˜
C(Yc)−E˜
C(FC(F g ad α , Y c)).
Since i is a enso ial ela ion, he subs i u ion o he canonical sp ay S
ins ead o Ycshows ha
(60) ˜
C(F g ad α ) = 0 ⇒ −1
2αc˜
C(Yc)−E˜
C(FC(F g ad α , Y c)) = 0.
By subs i u ing he ec o ield F g ad α ins ead o Ycwe ha e ha
(61) ˜
C(FC(F g ad α ,F g ad α )) = 0.
Le now ∈TpMbe a nonze o angen ec o ; since he lowe ed i s Ca an
enso is o ally symme ic, he mapping
C(F g ad α ,·)( ): T TM →T T M
is ”sel -adjoin ” wi h espec o he me ic g in he ollowing sense: we can
conside a g -o hono mal sys em Y
1, . . . , Y
na he poin such ha
C(F g ad α , Y c
i)( ) = λiY
i( ).
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 17
Then, by he o mula (62), i ollows ha
(62)
0 =
n
X
i=1
g(C(FC(F g ad α ,F g ad α ), Y c
i), Y
i)( ) =
=
n
X
i=1
g(Q(F g ad α , Y c
i)F g ad α , Y
i)( )+
+
n
X
i=1
g(C(F g ad α ,FC(Yc
i,F g ad α )), Y
i)( ) =
=−R(F g ad α ,F g ad α ) + λ2
1+. . . +λ2
n,
whe e Ris he e ical Ricci enso o he Ca an connec ion. The e o e
R(F g ad α ,F g ad α )≥0.
Since he e ical co a ian di e en ia ion wi h espec o he Ca an con-
nec ion is jus he same as ha wi h espec o he L´e i-Ci i a connec ion
∇on he ”mani old” TpM, he o mula 3.4 (iii) shows ha di g ad α = 0
whe e he di e gence ope a o , o cou se, is aken wi h espec o he con-
nec ion ∇. On he o he hand, he Hesse o m ∇g ad α is au oma ically
sel -adjoin . This means, by a heo em due o G. de Rham (see [13], sec ion
5.4) ha g ad α is a ha monic ec o ield. Mo eo e , de Rham’s heo em
s a es ha
g( ∇2g ad α ,·) = R(F g ad α ,·)
and, by a heo em due o S. Bochne (see [13], sec ion 4.18) we ha e ha
(63) 2g( ∇2g ad α ,g ad α )+2k∇ g ad α k2+ ∆kg ad α k2= 0,
whe e he no m, o cou se, is aken wi h espec o he me ic g. The e o e
(64) ∆kg ad α k2≤0.
Since he unc ion kg ad α k2is homogeneous o deg ee 0 i a ains bo h i s
maximum and minimum on he ec o space TpM. In his case a subha -
monic unc ion mus be cons an as he Hop ’s maximum p inciple s a es;
see [21], Theo em 2.1. This means ha we can w i e he unc ion kg ad α k2
in he o m
(65) kg ad α k2=β◦π
and he p oo can be inished as ollows. The hypho esis on he h -cu a u e
enso o he Be wald connec ion implies ha he di e ence o he canonical
sp ays is a quad a ic ec o ields. O cou se, we ha e a well-known ans-
o ma ion o mula o changing o he canonical sp ay unde a con o mal
change, namely,
Sα=S−αcC+Eg ad α ;
[6]; see also [17], [19] and [20] o he coo dina e- ee exp ession. I ollows
ha he unc ion
Ekg ad α k2=Sα−Sαc+ (αc)2
is quad a ic. Since dpα6= 0, he le hand side is non i ial on a neighbou -
hood Uo he poin p. The e o e, by he o mula (66), he es ic ion E|T U
mus be he ene gy unc ion o a Riemannian mani old.
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 18
Theo em 5. The con o mal equi alence be ween wo Be wald mani olds
mus be i ial unless he mani olds a e Riemannian.
P oo . I emains only o show ha i a Be wald mani old is locally
Riemannian, hen i is a Riemannian mani old; bu his is i ial. The local
p ope y can be easily ex ended by he help o he (linea ) pa allel anspo
p o ided, o cou se, ha Mis a connec ed mani old.
Exe cise. Using he same echnic on he ”mani old” TpMas in he p oo
o heo em 4 p o e Deicke’s classical heo em o Finsle mani olds. (Hin :
Subs i u e an a bi a y e ical li ed ec o ield in o he o mulas (61)-(66)
ins ead o g ad α .)
Exe cise. Find a sho p oo o heo em 4 in case o dimension 2. (Hin :
Suppose ha A◦(TpM) is i ial; hen X= 0, i.e. g ad α and he Liou ille
ec o ield Ca e linea ly dependen on he ”mani old” TpM. By he help
o Lemma 1 we ge a con adic ion immedia ely.)
Re e ences
[1] D. Bao, S. S. Che n and Z. Shen, An in oduc ion o Riemann-Finsle Geome y,
Sp inge -Ve lag, 2000.
[2] F. B ickell, A new p oo o Deicke’s heo em on homogeneous unc ions, P oc o AMS
16, 1965, 190-191.
[3] A. Deicke, ¨
Ube he Finsle -R¨aume mi Ai= 0, A ch. Ma h. 4, 1953, 45-51.
[4] J. G i one, S uc u e p esque- angen e e connexions I, Ann. Ins . Fou ie , G enoble
22 no.1 (1972), 287-334.
[5] J. G i one, S uc u e p esque- angen e e connexions II, Ann. Ins . Fou ie , G enoble
22 no.3 (1972), 291-338.
[6] M. Hashiguchi, On con o mal ans o ma ions o Finsle me ics, J. Ma h. Kyo o
Uni . 16 (1976), 25-50.
[7] J. Klein and A. Vou ie , Fo mes ex ´e ieu es g´en´e a ices de sp ays, Ann. Ins . Fou ie
(G enoble), 18 (1968) 1, 241-260.
[8] M. de Le`on and P. R. Rod igues, Me hods o Di e en ial Geome y in Anali ical
Mechanics, No h-Holland, Ams e dam, 1989.
[9] R. L. Lo as, A ine and p ojec i e ec o ield on spay mani olds, o appea in he
P oceedings o he Wo kshop on Finsle Geome y and i s Applica ions, Augus 11-
15, 2003, Deb ecen, Hunga y.
[10] M. Ma sumo o, Founda ions o Finsle Geome y and Special Finsle Spaces,
Kaisheisha P ess, Japan, 1986.
[11] M. Ma sumo o, Con o mally Be wald and con o mally la Finsle spaces, Publ.
Ma h. Deb ecen, 58 (1-2) (2001), 275-285.
[12] T. Okada, On models o p ojec i ely la Finsle spaces wi h cons an nega i e cu a-
u e, Tenso (N.S.) 40 (1983), 117-123.
[13] W. A. Poo , Di e en ial Geome ic S uc u es, McG aw-Hill, New Yo k, 1981.
[14] H. Rund, The Di e en ial Geome y o Finsle Spaces, Sp inge -Ve lag, Be lin, 1958.
[15] Z. Shen, Di e en ial Geome y o Sp ay and Finsle spaces, Kluwe Academic Pub-
lishe s, Do d ech , 2001.
[16] J. Szilasi and Sz. Va am´any, On he Finsle -me izabili ies o sp ay mani olds, Pe i-
odica Ma hema ica Hunga ica, Vol 44(1), (2002), 81-100.
[17] J. Szilasi and Cs. Vincze, On con o mal equi alence o Riemann-Finsle me ics, Publ.
Ma h. Deb ecen 52 (1-2) (1998), 167-185.
[18] J. Szilasi and Cs. Vincze, A new look a Finsle connec ions and special Finsle man-
i olds, www.emis.de/jou nals AMAPN 16 (2000), 33-63.
[19] Cs. Vincze, On con o mal equi alence o Riemann-Finsle me ics and special Finsle
mani olds, Ph.D. disse a ion, Uni e si y o Deb ecen, Deb ecen, Hunga y, 2000.
ON GEOMETRIC VECTOR FIELDS OF MINKOWSKI SPACES ... 19
[20] Cs. Vincze, On con o mal equi alence o Be wald mani olds all o whose indica ices
ha e posi i e cu a u e, SUT Jou nal o ma hema ics, Vol 39, No. 1 (2003), 15-40.
[21] K. Yano and S. Bochne , Cu a u e and Be y numbe s, Annals o Ma hema ics S ud-
ies No. 32, P ince on Uni . P ess, 1953.
[22] K. Yano and S. Ishiha a, Tangen and Co angen Bundles: Di e en ial Geome y,
Ma cel Decke Inc. New Yo k, 1973.
Ins i u e o Ma hema ics and In o ma ics, Uni e si y o Deb ecen, H-4010
Deb ecen, P.O.Box 12, Hunga y
E-mail add ess:[email p o ec ed]