Finsler manifolds with non-Riemannian holonomy

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by  Zoltan Muzsnay

Finsler manifolds with non-Riemannian holonomy
Zolt´an Muzsnay and P´eter T. Nagy
Institute of Mathematics, University of Debrecen
H-4010 Debrecen, Hungary, P.O.B. 12
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1 E-mail: [email protected], [email protected]
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2

n
a
J
Abstract

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1 The aimofthis paperis to showthatholonomypropertiesofFinslermanifolds
can be very different from those of Riemannian manifolds. We prove that the

]
holonomygroupofapositivedefinitenon-RiemannianFinslermanifoldofnon-zero
G
constant curvature with dimension > 2 cannot be a compact Lie group. Hence
D
this holonomy group does not occur as the holonomy group of any Riemannian
.
h manifold. In addition, we provide an example of left invariant Finsler metric on
t the Heisenberg group, so that its holonomy groupis not a (finite dimensional) Lie
a
m group. These results give a positive answer to the following problem formulated
by S. S. Chern and Z. Shen: Is there a Finsler manifold whose holonomy group is
[
not the holonomy group of any Riemannian manifold?

4
v
0 1 Introduction
7
4
0 The notion of the holonomy group of a Riemannian manifold can be generalized very
. naturally for a Finsler manifold (cf. e.g. S. S. Chern and Z. Shen, [2], Chapter 4):
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0 it is the group at a point x generated by the canonical homogeneous (nonlinear) par-
9
allel translations along all loops emanated from x. Until now the holonomy groups
0
: of non-Riemannian Finsler manifolds have been described only in special cases: for
v
Berwald manifolds there exist Riemannian metrics with the same holonomy group (cf.
i
X
Z. I. Szabo´, [11]), for positive definite Landsberg manifolds the holonomy groups are
r compact Lie groupsconsisting of isometries of theindicatrix with respecttoan induced
a
Riemannian metric(cf. L.Kozma, [4], [5]). Athoroughstudyof theholonomy groupof
homogeneous (nonlinear) connections was initiated by W. Barthel in his basic work [1]
in 1963; he gave a construction for a holonomy algebra of vector fields on the tangent
space. A general setting for the study of infinite dimensional holonomy groups and
holonomy algebras of nonlinear connections was initiated by P. Michor in [7]. However
the introduced holonomy algebras could not be used to estimate the dimension of the
2000 Mathematics Subject Classification: 53B40, 53C29
Key words and phrases: Finsler geometry, holonomy.
This research was supported bythe Hungarian Scientific Research Fund(OTKA) Grant K 67617.
1
holonomy group since their tangential properties to the holonomy group were not clar-
ified.
The aim of this paper is to show that holonomy properties of Finsler manifolds can
be very different from those of Riemannian manifolds. We prove that if the holonomy
group of a non-Riemannian Finsler manifold of non-zero constant curvature with di-
mension n> 2 is a (finite dimensional) Lie group then its dimension is strictly greater
than the dimension of the orthogonal group acting on the tangent space and hence
it can not be a compact Lie group. An estimate for the dimension of the holonomy
group will be obtained by investigation of a Lie algebra of tangent vector fields on the
indicatrix, algebraically generated by curvature vector fields of the Finsler manifold.
We call this Lie algebra the curvature algebra and prove that its elements are tangent
to one-parameter families of diffeomorphisms contained in the holonomy group. For
non-Riemannian Finsler manifolds of constant curvature = 0 with dimension n > 2 we
n(n 1) 6
construct more than − linearly independent curvature vector fields.
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Inaddition, weprovideanexampleofaleftinvariantsingular(nony-global) Finsler
metric of Berwald-Mo´or-type on the Heisenberg group which has infinite dimensional
curvaturealgebraandhenceitsholonomy isnota(finitedimensional)Liegroup. These
results give a positive answer to the following problem formulated by S. S. Chern and
Z. Shen in [2] (p. 85): Is there a Finsler manifold whose holonomy group is not the
holonomy group of any Riemannian manifold? This question is contained also in the
list of open problems in Finsler geometry by Z. Shen [10], (March 8, 2009, Problem
34).
2 Preliminaries
Finsler manifold and its canonical connection
A Minkowski functional on a vector space V is a continuous function , positively
F
homogeneous of degree two, i.e. (λy) = λ2 (y), smooth on Vˆ := V 0 , and for any
F F \{ }
y Vˆ the symmetric bilinear form g : V V R defined by
y
∈ × →
1∂2 (y+su+tv)
g : (u,v) g (y)uivj = F
y ij
7→ 2 ∂s∂t t=s=0
(cid:12)
(cid:12)
is non-degenerate. If gy is positive definite for an(cid:12)y y Vˆ then is said positive
∈ F
definite and (V, ) is called positive definite Minkowski space. A Minkowski functional
F
is called semi-Euclidean if there exists a symmetric bilinear form , on V such
F h i
that g (u,v) = u,v for any y Vˆ and u,v V. A semi-Euclidean positive definite
y
h i ∈ ∈
Minkowski functional is called Euclidean.
A Finsler manifold is a pair (M, ) where M is an n-dimensional manifold and
F
: TM R is a function (called Finsler metric, cf. [9]) defined on the tangent bundle
oFf M, sm→ooth on TˆM := TM 0 and its restriction = is a Minkowski
\{ } Fx F|TxM
functional on T M for all x M. If the restriction = of the Finsler metric
x ∈ Fx F|TxM
: TM R is positive definite on T M for all x M then (M, ) is called positive
x
F → ∈ F
definite Finsler manifold. Apointx M iscalled (semi-)Riemannian iftheMinkowski
∈
2
functional is (semi-)Euclidean.
x
F
We remark that in many applications the metric is smooth only on an open cone
F
M TM 0 , where M= M is a fiber bundle over M such that each M
x M x x
C ⊂ \{ } C ∪ ∈ C C
is an open cone in T M 0 . In such case (M, ) is called singular (or non y-global)
x
\{ } F
Finsler space (cf. [9]).
Geodesics of Finsler manifolds are determined by a system of 2nd order ordinary
differential equation:
x¨i +2Gi(x,x˙) = 0, i= 1,...,n
where Gi(x,x˙) are locally given by
1 ∂g ∂g
Gi(x,y) := gil(x,y) 2 jl(x,y) jk(x,y) yjyk.
4 ∂xk − ∂xl
(cid:16) (cid:17)
The associated homogeneous (nonlinear) parallel translation can be defined as follows:
a vector field X(t) = Xi(t) ∂ along a curve c(t) is said to be parallel if it satisfies
∂xi
dXi(t) ∂
X(t) := +Γi(c(t),X(t))c˙j(t) , (1)
∇c˙ dt j ∂xi
(cid:16) (cid:17)
where Γi = ∂Gi.
j ∂yj
Horizontal distribution, curvature
The geometric structure associated to can be given on TM in terms of the hor-
∇
izontal distribution. Let TM TTM denote the vertical distribution on TM,
V ⊂
TM := Kerπ . The horizontal distribution TM TTM associated to (1) is
y ,y
V ∗ H ⊂
locally generated by the vector fields
∂ ∂ ∂
l := +Γk(x,y) , i = 1,...,n. (2)
(x,y) ∂xi ∂xi i ∂yk
(cid:16) (cid:17)
For any y TM we have the decomposition T TM = TM TM. The projectors
y y y
∈ H ⊕V
corresponding to this decomposition will be denoted by h and v . The isomorphism
y y
l : T M TM definedby theformula(2)is called horizontal lift. Thena vector
(x,y) x y
→ H
field X(t) along a curve c(t) is parallel if and only if it is a solution of the differential
equation
d
X(t) = l (c˙(t)). (3)
dt X(t)
The curvature tensor field characterizes the integrability of the horizontal distribution:
R (ξ,η) := v[hξ,hη], ξ,η T TM. (4)
(x,y) (x,y)
∈
Using local coordinate system we have
∂Γk ∂Γk ∂Γk ∂Γk ∂
R = i j +Γm j Γm i dxi dxj .
(x,y) ∂xj − ∂xi i ∂ym − j ∂ym ⊗ ⊗ ∂yk
!
3
Themanifoldiscalledofconstantcurvaturec R,ifforanyx M thelocalexpression
∈ ∈
of the curvature is
∂
R = c δkg (y)ym δkg (y)ym dxi dxj . (5)
(x,y) i jm − j im ⊗ ⊗ ∂yk
(cid:16) (cid:17)
In this case the flag curvature of the Finsler manifold (cf. [2], Section 2.1 pp. 43-46)
does not depend either on the point or on the 2-flag.
Indicatrix bundle
Let (M, ) be an n-dimensional Finsler manifold. The indicatrix I M at x M is a
x
F ∈
hypersurface of T M defined by
x
I M := y T M; (y) = 1 .
x x
{ ∈ F ± }
If the Finsler manifold (M, ) is positive definite then the indicatrix I M is a compact
x
F
hypersurface in the tangent space T M, diffeomorphic to the standard (n 1)-sphere.
x
−
In this case the group Diff(I M) of all smooth diffeomorphisms of I M is a regular
x x
infinite dimensional Lie group modeled on the vector space X(I M) of smooth vector
x
fields on I M. The Lie algebra of the infinite dimensional Lie group Diff(I M) is the
x x
vector space X(I M), equipped with the negative of the usual Lie bracket, (c.f. A.
x
Kriegl and P. W. Michor [6], Section 43).
Let (IM,π,M) denote the indicatrix bundle of (M, ) and i :IM ֒ TM the natural
F →
embedding of the indicatrix bundle into the tangent bundle (TM,π,M).
Parallel translation and holonomy
Let (M, ) be a Finsler manifold. The parallel translation τ :T M T M along
c c(0) c(1)
F →
a curve c : [0,1] R is defined by vector fields X(t) along c(t) which are solutions
→
of the differential equation (1). Since τ : T M T M is a differentiable map
c c(0) c(1)
→
between Tˆ M and Tˆ M preserving the value of the Finsler metric, it induces a
c(0) c(1)
map
τI: I M I M (6)
c c(0) c(1)
−→
between the indicatrices.
Definition 1 The holonomy group Hol(x) of a Finsler space (M, ) at x M is the
F ∈
subgroup of the group of diffeomorphisms Diff(I M) of the indicatrix I M determined
x x
by parallel translation of I M along piece-wise differentiable closed curves initiated at
x
the point x M.
∈
We note that the holonomy group Hol(x) is a topological subgroup of the regular
infinite dimensional Lie group Diff(I M), but its differentiable structure is not known
x
in general.
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3 Tangent Lie algebras to subgroups of Diff (M)
∞
LetH beasubgroupofthediffeomorphismgroupDiff (M)ofadifferentiablemanifold
∞
M and let X (M) be the Lie algebra of smooth vector fields on M.
∞
Definition 2 A vector field X X (M) is called strongly tangent to H, if there exists
∞
∈
a -differentiable k-parameter family φ H of diffeomorphisms such
C∞ { (t1,...,tk) ∈ }ti∈(−ε,ε)
that
(i) φ =Id, if t = 0 for some 1 j k;
(t1,...,tk) j ≤ ≤
∂kφ
(ii) (t1,...,tk) = X.
∂t1···∂tk (t1,...,tk)=(0,...,0)
A vector field X(cid:12) X (M) is called tangent to H, if there exists a 1-differentiable
(cid:12) ∞
∈ C
1-parameter family φ H of diffeomorphisms of M such that φ = Id and
t t ( ε,ε) 0
∂φt = X. { ∈ }∈ −
∂t t=0
A Lie subalgebra g of X (M) is called tangent to H, if all elements of g are tangent
(cid:12) ∞
vect(cid:12)or fields to H.
Theorem 3 Let be a set of vector fields strongly tangent to the subgroup H of
V
Diff (M). The Lie subalgebra v of X (M) generated by is tangent to H.
∞ ∞
V
Proof. First, we investigate some properties of vector fields strongly tangent to the
group H.
Lemma 4 Let ψ Diff (U) be a -differentiable h-parameter fam-
ily of (local) diff{eo(mt1o,.r..p,thh)is∈ms on∞a ne}igtih∈b(o−uε,rεh)ood UC∞Rn, satisfying ψ = Id, if
⊂ (t1,...,th)
t = 0 for some 1 j h. Then
j
≤ ≤
∂i1+...+ihψ
(i) (t1,...,th) (x) = 0, if i = 0 for some 1 p h;
∂ti1 ... ∂tih (cid:12) p ≤ ≤
1 h (cid:12)(0,...,0)
(cid:12)
(ii) ∂h(ψ(t1,...,th))−1 (cid:12)(cid:12) (x) = ∂hψ(t1,...,th) (x);
∂t1 ... ∂th (0,...,0) − ∂t1 ... ∂th (0,...,0)
(cid:12) (cid:12)
(iii) ∂hψ(t1,...,th) (cid:12)(cid:12) (x) = ∂ψ√ht,...,√ht) (x(cid:12)(cid:12))
∂t1 ... ∂th (0,...,0) ∂t t=0
(cid:12)
(cid:12)
at any point x U(cid:12) . (cid:12)
∈ (cid:12)
Proof. Assertions (i) and (ii) can be obtained by direct computation. It follows from
∂hψ
(i) that (t1,...,th) (x) is the first non-necessarily vanishing derivative of the dif-
∂t1...∂th (0,...,0)
feomorphism family(cid:12) ψ at any point x M. Using
(cid:12) { (t1,...,th)} ∈
(cid:12)
ψ (x) = x+t t (X(x)+ω(x,t ,...,t )),
(t1,...,tk) 1··· k 1 k
where lim ω(x,t ,...,t ) = 0 we obtain, that
1 k
ti 0
→
∂ ∂
ψ (x) = x+t X(x)+ω(x,√k t,...,√k t) = X(x),
∂t t=0 (√kt,...,√kt) ∂t t=0
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12) 5
which proves (iii).
We remark that the assertion (iii) means that any vector field strongly tangent to H
is tangent to H.
Now, we generalize a well-known relation between the commutator of vector fields and
the commutator of their induced flows.
Lemma 5 Let φ and ψ be -differentiable k-parameter, respec-
{ (s1,...,sk)} { (t1,...,tl)} C∞
tively l-parameter families of (local) diffeomorphisms defined on a neighbourhood U
⊂
Rn. Assume that φ = Id, respectively ψ = Id, if some of their variables
(s1,...,sk) (t1,...,tl)
equals 0. Then the family of (local) diffeomorphisms [φ ,ψ ] defined by
(s1,...,sk) (t1,...,tl)
the commutator of the group Diff (U) fulfills [φ ,ψ ] = Id, if some of its
∞ (s1,...,sk) (t1,...,tl)
variables equals 0. Moreover
∂k+l[φ ,ψ ] ∂kφ ∂lψ
(s1...sk) (t1...tl) (x) = (s1...sk) , (t1...tl) (x)
∂s1 ... ∂sk ∂t1 ... ∂tl (0...0;0...0) −"∂s1 ... ∂sk (0...0) ∂t1 ... ∂tl (0...0)#
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
at any point x U. (cid:12) (cid:12) (cid:12)
∈
Proof. The group theoretical commutator φ ,ψ of the families of dif-
(s1,...,sk) (t1,...,tl)
feomorphisms satisfies [φ ,ψ ] = Id, if some of its variables equals 0. Hence
(s1,...,sk) (t1,...,tl) (cid:2) (cid:3)
∂i1+...+ik+j1+...+jl[φ ,ψ ]
(s1,...,sk) (t1,...,tl) = 0,
∂si1 ... ∂sik∂tj1 ... ∂til (0,...,0;0,...,0)
1 k 1 l (cid:12)
(cid:12)
if i =0 or j =0 for some index 1 p k o(cid:12)r 1 q l. The families of diffeomorphisms
p q
≤ ≤ ≤ ≤
φ , ψ , φ 1 and ψ 1 are the constant family Id, if some
{ (s1,...,sl)} { (t1,...,tl)} { −(s1,...,sl)} { (−t1,...,tl)}
of their variables equals 0. Hence one has
∂k+l[φ ,ψ ]
(s1...sk) (t1...tl) (x) = (7)
∂s1 ... ∂sk ∂t1 ... ∂tl (0,...,0;0,...,0)
(cid:12)
= ∂k ∂l(cid:12)(cid:12)φ−(s11...sk)◦ψ(−t11...tl)◦φ(s1...sk)◦ψ(t1...tl)(x)
∂s1...∂sk (0...0) (cid:16) ∂t1...∂tl (cid:17) (0...0)!
(cid:12) (cid:12)
(cid:12) (cid:12)
= ∂k (cid:12) d(φ 1 ) ∂lψ(−t11...tl) (φ ((cid:12)x)) ,
∂s1...∂sk (0...0) −(s1...sk) φ(s1...sk)(x) ∂t1...∂tl (cid:12) (s1...sk) 
(cid:12) (cid:12)(0,...,0)
(cid:12) (cid:12)
(cid:12)  (cid:12) 
where d φ−(s11,...,sk) φ(s1,...,sk)(x) denotes the Jacobi operato(cid:12)r of the map φ−(s11,...,sk) at the
point φ (x). Using the fact, that φ is the constant family Id, if some
((cid:0)s1,...,sk) (cid:1) { (s1,...,sk)}
of its variables equals 0, and the relation d(φ 1 ) = Id, we obtain that (7)
−(0,...,0) φ(s1,...,sk)(x)
can be written as
∂kφ−1 ∂lψ−1 (x) ∂lψ−1 ∂kφ (x)
d (s1...sk) (t1...tl) +d (t1...tl) (s1...sk) .
∂s1...∂sk (0...0) x ∂t1...∂tl (0...0) ∂t1...∂tl (0...0) x ∂s1...∂sk (0,...,0)
(cid:16) (cid:12) (cid:17) (cid:12) (cid:16) (cid:12) (cid:17) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
According to asser(cid:12)tion (ii) of Lemma(cid:12)4 the last formula(cid:12)gives (cid:12)
∂kφ ∂lψ (x) ∂lψ ∂kφ (x)
d (s1...sk) (t1...tl) d (t1...tl) (s1...sk) ,
∂s1 ... ∂sk (0...0) x ∂t1 ...∂tl (0...0)− ∂t1 ...∂tl (0...0) x ∂s1 ...∂sk (0...0)
(cid:16) (cid:12) (cid:17) (cid:12) (cid:16) (cid:12) (cid:17) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
6
which is the Lie bracket of vector fields
∂lψ ∂kφ
(t1,...,tl) , (s1,...,sk) : U Rn.
" ∂t1 ... ∂tl (0,...,0) ∂s1 ... ∂sk (0,...,0)# →
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Lemma 6 The Lie algebra v has a basis consisting of vector fields strongly tangent to
the group H.
Proof. The iterated Lie brackets of vector fields belonging to linearly generate the
V
vector space v. It follows from Lemma 5 that iterated Lie brackets of vector fields
belonging to are strongly tangent to the group H. Hence v is linearly generated by
V
vector fields strongly tangent to H.
Lemma 7 Linear combinations of vector fields tangent to H are tangent to H.
Proof. If X and Y are vector fields tangent to H then there exist 1-differentiable
C
1-parameter families of diffeomorphisms φ H and ψ H such that
t t
{ ∈ } { ∈ }
∂ ∂
φ =ψ =Id, φ = X, ψ = Y.
0 0 t t
∂t t=0 ∂t t=0
(cid:12) (cid:12)
(cid:12) (cid:12)
Considering the 1-differe(cid:12)ntiable 1-parameter(cid:12) families of diffeomorphisms φt ψt and
C { ◦ }
φ one has
ct
{ }
∂ ∂
X +Y = (φ ψ ), cX = φ , for all c Rn,
∂t t=0 t ◦ t ∂t t=0 (ct) ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
which proves the assertion.
(cid:12) (cid:12)
Lemmas 4 – 7 prove Theorem 3.
4 Curvature algebra
Definition 8 A vector field ξ X(I M) on the indicatrix I M is called a curvature
x x
∈
vector field of the Finsler manifold (M, ) at x M, if there exists X,Y T M such
x
F ∈ ∈
that ξ = r (X,Y), where
x
r (X,Y)(y) := R (l X,l Y) (8)
x (x,y) y y
The Lie subalgebra R := r (X,Y); X,Y T M of X(I M) generated by the curva-
x x x x
∈
ture vector fields is called the curvature algebra of the Finsler manifold (M, ) at the
(cid:10) (cid:11) F
point x M.
∈
7
SincetheFinslermetricispreservedbyparalleltranslations,itsderivatives withrespect
to horizontal vector fields are identically zero. Using (4) we obtain, that the derivative
of the Finsler metric with respect to (8) vanishes, and hence
g y,R (l X,l Y) = 0, for any y,X,Y T M
(x,y) (x,y) y y x
∈
(cid:0) (cid:1)
(c.f.[9],eq.(10.9)). Thismeansthatthecurvaturevectorfieldsξ=r (X,Y)aretangent
x
to theindicatrix. Inthesequel weinvestigate thetangential properties of thecurvature
algebra to the holonomy group of the canonical connection of a Finsler manifold.
∇
Proposition 9 Anycurvature vector field at x M is strongly tangent to the holonomy
∈
group Hol(x).
Proof. Indeed, let us consider the curvature vector field r (X,Y) X(I M), X,Y
x x
∈ ∈
T M and let Xˆ,Yˆ X(M) be commuting vector fields i.e. [Xˆ,Yˆ]=0 such that Xˆ =X,
x x
Yˆ =Y. By the geo∈metric construction, the flows φ and ψ of the horizontal lifts
x t s
{ } { }
l(Xˆ) and l(Yˆ) are fiber preserving diffeomorphisms of the bundle IM for any t R,
∈
corresponding to parallel translations along integral curves of Xˆ and Yˆ respectively.
Then the commutator
θ = [φ ,ψ ] = φ 1 ψ 1 φ ψ : IM IM
t,s t s −t s− t s
◦ ◦ ◦ →
is also a fiber preserving diffeomorphism of the bundle IM for any t,s R. Therefore
∈
for any x M the restriction
∈
θ (x) = θ :I M I M
t,s t,s IxM x → x
(cid:12)
to the fiber IxM (cid:12)is a 2-parameter C∞-differentiable family of diffeomorphisms con-
tained in the holonomy group Hol(x) such that
∂2
θ (x) =Id, θ (x) = Id, and θ (x) = r (X,Y),
0,s t,0 t,s x
∂t∂s t=0,s=0
(cid:12)
(cid:12)
which proves that the curvature vector field rx(X,Y)(cid:12)is strongly tangent to the holon-
omy group Hol(x) and hence we obtain the assertion.
Theorem 10 The curvature algebra R of a Finsler manifold (M, ) is tangent to the
x
F
holonomy group Hol(x) for any x M.
∈
Proof. Since by Proposition 9 the curvature vector fields are strongly tangent to
Hol(x) and the curvature algebra R is algebraically generated by the curvature vector
x
fields, the assertion follows from Theorem 3.
Proposition 11 The curvature algebra R of a Riemannian manifold (M,g) at any
x
point x M is isomorphic to the linear Lie algebra over the vector space T M generated
x
∈
by the curvature operators of (M,g) at x M.
∈
8
Proof. The curvature tensor field of a Riemannian manifold given by the equation (4)
is linear with respect to y T M and hence
x
∈
∂
R (ξ,η) = (R (ξ,η))kyl ,
(x,y) x l ∂yk
whereR (ξ,η))k isthematrixofthecurvatureoperatorR (ξ,η): T M T M withre-
x l x x → x
specttothenaturalbasis ∂ ,..., ∂ .Henceanycurvaturevector fieldr (ξ,η)(y)
∂x1|x ∂xn|x x
with ξ,η T M has the shape r (ξ,η)(y) = (R (ξ,η))kyl ∂ . It follows that the flow
∈ x (cid:8) x (cid:9) x l ∂yk
of r (ξ,η)(y) on the indicatrix I M generated by the vector field r (ξ,η)(y) is induced
x x x
by the action of the linear 1-parameter group exptR (ξ,η)) on T M, which implies the
x x
assertion.
Remark 12 The curvature algebra of Finsler surfaces is one-dimensional.
Proof. For Finsler surfaces the curvature vector fields form a one-dimensional vector
space and hence the generated Lie algebra is also one-dimensional.
5 Constant curvature
Now, we consider a Finsler manifold (M, ) of non-zero constant curvature. In this
F
case for any x M the curvature vector field r (X,Y)(y) has the shape (cf. (5))
x
∈
∂
r(X,Y)(y) = c δig (y)ym δig (y)ym XjYk , 0 = c R.
j km − k jm ∂yi 6 ∈
(cid:0) (cid:1)
Puttingy = g (y)ym wecan writer(X,Y)(y) = c δiy δiy XjYk ∂ .Anylinear
j jm j k − k j ∂yi
combination of curvature vector fields has the form(cid:16) (cid:17)
∂
r(A)(y) = Ajk δiy δiy ,
j k − k j ∂yi
(cid:0) (cid:1)
where A= Ajk ∂ ∂ T M T M is arbitrary bivector at x M.
∂xj ∧ ∂xk ∈ x ∧ x ∈
Lemma 13 Let (M, ) be a Finsler manifold of non-zero constant curvature. The
F
curvature algebra R at any point x M satisfies
x
∈
n(n 1)
dimR − , (9)
x
≥ 2
where n =dimM.
Proof. Let us consider the curvaturevector fields r = r ( ∂ , ∂ )(y) at a fixed point
jk x ∂yj ∂yk
x M. If a linear combination
∈
∂ ∂ ∂
Ajkr = Ajk(δiy δiy ) = (Aiky Ajiy ) = 2Aiky
jk j k − k j ∂yi k − j ∂yi k∂yi
9
of curvature vector fields r with constant coefficients Ajk = Akj satisfies Ajkr = 0
jk jk
−
for any y T M then one has the linear equation Aiky = 0 for any fixed index i.
x k
∈
Since the covector fields y ,...,y are linearly independent we obtain Ajk = 0 for all
1 n
j,k 1,...,n . It follows that the curvaturevector fields r are linearly independent
jk
∈ { }
for any j < k and hence dimRx ≥ n(n2−1).
Corollary 14 Let (M,g) be a Riemannian manifold of non-zero constant curvature
with n = dimM. The curvature algebra R at any point x M is isomorphic to the
x
∈
orthogonal Lie algebra o(n).
Proof. Theholonomy group of aRiemannian manifold is a subgroupof the orthogonal
group O(n) of the tangent space T M and hence the curvature algebra R is a sub-
x x
algebra of the orthogonal Lie algebra o(n). Hence the previous assertion implies the
corollary.
Theorem 15 Let (M, ) be a Finsler manifold of non-zero constant curvature with
F
n = dimM > 2. If the point x M is not (semi-)Riemannian then the curvature
∈
algebra R at x M satisfies
x
∈
n(n 1)
dimR > − . (10)
x
2
Proof. We assume dimRx = n(n2−1). For any constant skew-symmetric matrices
Ajk and Bjk the Lie bracket of vector fields Aiky ∂ and Biky ∂ has the
{ } { } k∂yi k∂yi
shape Ciky ∂ , where Cik is a constant skew-symmetric matrix, too. Using the
k∂yi { }
homogeneity of g we obtain
hl
∂y ∂g
h = hl yl+g = g (11)
∂ym ∂ym hm hm
and hence
∂ ∂ ∂y ∂y ∂
Amky ,Bihy = AmkBih h BmkAih h y
k ∂ym h ∂yi ∂ym − ∂ym k ∂yi
(cid:20) (cid:21) (cid:18) (cid:19)
∂ ∂
= Bihg Amk Aihg Bmk y = Ciky .
hm − hm k ∂yi k ∂yi
(cid:16) (cid:17)
Particularly, for the skew-symmetric matrices Eij = δiδj δiδj, a,b 1,...,n , we
ab a b− b a ∈ { }
have
∂ ∂ ∂ ∂
Eij y , Ekly = Eihg Emk Eihg Emk y = Λim y ,
ab j ∂yi cd l ∂yk cd hm ab − ab hm cd k ∂yi ab,cd m ∂yi
(cid:20) (cid:21)
(cid:16) (cid:17)
ij ij ji ij ij ij
wheretheconstantsΛ satisfyΛ = Λ = Λ = Λ = Λ .Putting
ab,cd ab,cd − ab,cd − ba,cd − ab,dc − cd,ab
i = a and computing the trace for these indices we obtain
(n 2)(g y g y ) = Λl y , (12)
bd c bc d b,cd l
− −
10