Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
26 (2010), 171–207
www.emis.de/journals
ISSN 1786-0091
ON THE PROJECTIVE THEORY OF SPRAYS
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Dedicated to József Szilasi on the occasion of his 60th birthday.
Abstract. The paper aims to give a fairly self-contained survey on the
fundamentals and the basic techniques of spray geometry, using a rigorously
index-free formalism in the pull-back bundle framework, with applications
to Finslerian sprays and metrizability problems. Thus we review a number of classically well-known facts from a modern viewpoint, and prove
also known results using new ideas and tools. Among others, Laugwitz’s
metrization theorem (7.3) and the proof of the vanishing of the direction
independent Landsberg and stretch tensor (6.6, 6.7) belong to this category. We present also some results we believe are new. We mention from
this group the description of the projective factors which yield the invariance of the Berwald curvature under a projective change (5.3, 5.4) and the
sufficient conditions of the Finsler metrizability of a spray in a broad sense
deduced from the Rapcsák equations (7.7, 7.11).
1. Conventions and basic definitions
By a manifold we shall always mean a locally Euclidean, second countable,
connected Hausdorff space with a smooth structure. If M and N are manifolds,
C ∞ (M, N ) denotes the set of smooth maps from M to N ; C ∞ (M ) := C ∞ (M, R).
τ : T M → M and τT M : T T M → T M are the tangent bundles of the manifold
M and the tangent manifold T M , respectively.
X(M ) := {X ∈ C ∞ (M, T M )|τ ◦ X = 1M } is the C ∞ (M )-module of vector fields
on M , its dual X∗ (M ) is the module of 1-forms on M . If X ∈ X(M ), LX
denotes the Lie derivative with respect to X, and iX is the substitution operator
or contraction by X. d stands for the exterior derivative operator.
2000 Mathematics Subject Classification. 53C05, 53C60, 58G30.
Key words and phrases. Ehresmann connections, sprays, Berwald curvature, affine curvature, Finsler functions, projective equivalence, metrizabilities of sprays, Rapcsk equations.
171
172
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
◦
◦
◦
If o ∈ X(M ) is the zero vector field, T M := T M \ o(M ), τ := τ ↾ T M , then
◦
◦
τ : T M → M is said to be the slit tangent bundle of M . φ∗ ∈ C ∞ (T M, T N )
is the tangent linear map (or derivative) of φ ∈ C ∞ (M, N ). If I ⊂ R is an
d
open interval and c : I → M is a smooth curve then ċ := c∗ ◦ du
is the velocity
d
vector field of c. (Here du is the canonical vector field on the real line.) The
vertical lift of a function f ∈ C ∞ (M ) is f v := f ◦ τ ∈ C ∞ (T M ), the complete
lift f c ∈ C ∞ (T M ) of f is defined by f c (v) := v(f ), v ∈ T M . For any vector field
X on M there is a unique vector field X c ∈ X(T M ) such that X c f c = (Xf )c
for any function f ∈ C ∞ (M ). X c is called the complete lift of X.
◦
◦
If K is a type 11 tensor field on T M , i.e., an endomorphism of the C ∞ (T M )◦
◦
module X(T M ) and η ∈ X(T M ), then we define the Frölicher-Nijenhuis bracket
[K, η] by
◦
[K, η]ξ := [Kξ, η] − K[ξ, η] ; ξ ∈ X(T M ).
◦
Then [K, η] is also a type 11 tensor on T M ; it is just the negative of the Lie
derivative Lη K. We also associate to K two graded derivations iK and dK of
◦
the Grassmann algebra of differential forms on T M , prescribing their operation
on smooth functions and 1-forms by the following rules:
◦
(1)
iK F := 0 , iK dF := dF ◦ K ; F ∈ C ∞ (T M );
(2)
dK := iK ◦ d − d ◦ iK .
Then the degree of iK is 0, and the degree of dK is 1. On functions dK operates
by
dK F = iK dF = dF ◦ K,
◦
so for any vector field ξ on T M we have
dK F (ξ) = dF (K(ξ)) = K(ξ)F.
The main scenes of our considerations will be the pull-back bundles of the
◦
tangent bundle τ : T M → M over τ and τ , i.e., the vector bundles
◦
◦
◦
π : T M ×M T M → T M and π : T M ×M T M → T M,
respectively. (Here ×M denotes fibre product.) The sections of π are smooth
e : T M → T M ×M T M of the form
maps X
e
v ∈ T M 7→ X(v)
= (v, X(v)) ∈ T M ×M T M,
X ∈ C ∞ (T M, T M ), τ ◦ X = τ.
We have a canonical section
δ : v ∈ T M 7→ δ(v) := (v, v) ∈ T M ×M T M,
ON THE PROJECTIVE THEORY OF SPRAYS
173
and any vector field X on M induces a section
b : v ∈ T M 7→ X(v)
b
X
:= (v, X(τ (v))) ∈ T M ×M T M,
called a basic section of π or a basic vector field along τ . The C ∞ (T M )-module
Sec(π) of sections of π is generated by the basic sections. If
X(τ ) := {X ∈ C ∞ (T M, T M )|τ ◦ X = τ } ,
then X(τ ) is naturally isomorphic to Sec(π), so the two modules will be identified
◦
without any comment, whenever it is convenient. We get the C ∞ (T M )-modules
◦
◦
Sec(π) and X(τ ) in the same way. The tensor algebras of Sec(π) ∼
= X(τ ) and
◦
◦
◦
∼
Sec(π) = X(τ ) will be denoted by T (π) and T (π), respectively. The elements
◦
of these tensor algebras will frequently be mentioned as tensors along τ or τ .
◦
Obviously, T (π) may be considered as a subalgebra of T (π).
◦
◦
If A is a type 13 tensor along τ , then we define its trace trA ∈ T20 (π) by
◦
e 7→ A(Z,
e X,
e Ye ) ; X,
e Ye ∈ Sec(π).
e Ye ) := tr Z
(trA)(X,
We have a canonical injective strong bundle map i : T M ×M T M → T T M
given by
i(v, w) := ċ(0), if c(t) := v + tw,
and a canonical surjective strong bundle map
j : T T M → T M ×M T M, w ∈ Tv T M 7→ j(w) := (v, τ∗ (w))
such that the sequence
i
j
0 −→ T M ×M T M −→ T T M −→ T M ×M T M −→ 0
is an exact sequence of vector bundle maps. i and j induce C ∞ (T M )-homomorphisms at the level of sections, which will be denoted by the same letters.
So we also have the exact sequence
i
j
0 −→ X(τ ) −→ X(T M ) −→ X(τ ) −→ 0
of module homomorphisms. Xv (T M ) := iX(τ ) is the module of vertical vector
b is the vertical lift of X ∈ X(M ). If α is a 1-form on M ,
fields on T M , X v := iX
then there exists a unique 1-form αv on T M such that
αv (X v ) = 0 , αv (X c ) = (α(X))v
for all X ∈ X(M ). αv is said to be the vertical lift of α.
C := iδ is a canonical vertical vector field on T M , the Liouville vector field.
For any vector field X on M we have
(3)
[C, X v ] = −X v , [C, X c ] = 0.
174
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
J := i ◦ j is a tensor field on T M of type 11 ; it is called the vertical endomorphism. For all vector fields X on M we have
JX v = 0 , JX c = X v ;
therefore
Im(J) = Ker(J) = Xv (T M ) , J2 = 0.
The following useful relations may be verified immediately:
[J, C] = J ; [J, X v ] = [J, X c ] = 0 , X ∈ X(M ).
(4)
We define the vertical differential ∇v F ∈ T10 (π) of a function F ∈ C ∞ (T M )
by
(5)
We note that
(6)
e := (iX)F
e
e ∈ Sec(π).
∇v F (X)
,X
∇v F ◦ j = dJ F,
where dJ is the graded derivation associated to the vertical endomorphism by
(1) and (2). The vertical differential of a section Ye ∈ Sec(π) is the type 11
tensor ∇v Ye ∈ T11 (π) given by
(7)
e =: ∇v Ye := j[iX,
e η] , X
e ∈ Sec(π),
∇v Ye (X)
e
X
where η ∈ X(T M ) is such that jη = Ye . (It is easy to check that the result
does not depend on the choice of η.) Using the Leibnizian product rule as
a guiding principle, the operators ∇vXe may uniquely be extended to a tensor
derivation of the tensor algebra of Sec(π). Forming the vertical differential of a
tensor over Sec(π), we use the following convention: if, e.g., A ∈ T21 (π), then
∇v (A) ∈ T31 (π), given by
e = ∇v A(Ye , Z)
e − A(∇v Ye , Z)
e − A(Ye , ∇v Z).
e
e Ye , Z)
e := (∇v A)(Ye , Z)
∇v A(X,
e
e
e
e
X
X
X
X
◦
A type 0s or 1s tensor A along τ is said to be homogeneous of degree k,
where k is an integer, if
∇vδ A = kA.
2. Ehresmann connections and Berwald derivatives
By an Ehresmann connection over M we mean a map H : T M ×M T M →
T T M satisfying the following conditions:
(C1 ) H is fibre preserving and fibrewise linear, i.e., for every v ∈ T M ,
Hv := H ↾ {v} × Tτ (v) M is a linear map from {v} × Tτ (v) M ∼
= Tτ (v) M
into Tv T M .
(C2 ) j ◦ H = 1T M×M T M , i.e., “H splits”.
◦
(C3 ) H is smooth over T M ×M T M .
ON THE PROJECTIVE THEORY OF SPRAYS
175
(C4 ) If o : M → T M is the zero vector field, then H(o(p), v) = (o∗ )p (v), for
all p ∈ M and v ∈ Tp M .
We associate to an Ehresmann connection H
the horizontal projector h := H ◦ j, the vertical projector v := 1 ◦ − h,
◦
◦
TTM
−1
the vertical map V := i ◦ v : T T M → T M ×M T M ,
the almost complex structure F := H ◦ V − i ◦ j = H ◦ V − J.
We have the following basic relations:
h2 = h, v2 = v; J ◦ h = J, h ◦ J = 0; J ◦ v = 0, v ◦ J = J;
F2 = −1, J ◦ F = v, F ◦ J = h;
F ◦ h = −J, h ◦ F = F ◦ v = J + F, v ◦ F = −J.
The horizontal lift of a vector field X ∈ X(M ) (with respect to H) is
b =: HX
b = hX c .
X h := H ◦ X
It may be shown (see e.g. [19]) that for all vector fields X, Y on M we have
J[X h , Y h ] = [X, Y ]v , h[X h , Y h ] = [X, Y ]h .
◦
By the tension of H we mean the type 11 tensor field along τ given by
(8)
◦
e := V[HX,
e C], X
e ∈ Sec(π).
t(X)
Then
b = [X h , C], X ∈ X(M ).
it(X)
H is said to be homogeneous if its tension vanishes. We define the torsion and
the curvature of H by
e Ye ) := V[HX,
e iYe ] − V[HYe , iX]
e − j[HX,
e HYe ]
T(X,
and
◦
e Ye ) := −V[HX,
e HYe ]
R(X,
e Ye ∈ Sec(π)), respectively. Evaluating on basic vector fields, we obtain the
(X,
more expressive relations
b Yb ) := [X h , Y v ] − [Y h , X v ] − [X, Y ]v
iT(X,
and
b Yb ) = −v[X h , Y h ],
iR(X,
where X and Y are vector fields on M .
Now we recall an elementary, but crucial construction of Ehresmann connections. To do this, we need the concept of a semispray and spray. By a semispray
over a manifold M we mean a map S : T M → T T M satisfying the following
conditions:
(S1 ) τT M ◦ S = 1T M ;
176
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
◦
(S2 ) S is smooth over T M ;
(S3 ) JS = C (or, equivalently, jS = δ).
A semispray S is said to be a spray, if it satisfies the additional conditions
(S4 ) S is of class C 1 over T M ;
(S5 ) [C, S] = S, i.e., S is positive-homogeneous of degree 2.
If a spray is of class C 2 (and hence smooth) over T M , then it is called an
affine spray. Following S. Lang’s terminology [9], we say that a smooth map
S : T M → T T M is a second-order vector field over M , if it satisfies conditions
(S1 ) and (S3 ). Notice, however, that by a ’spray’ Lang means a second-order
vector field satisfying the homogeneity condition (S5 ), i.e., an ‘affine spray’ in
our sense.
Given a semispray S over M , by a celebrated result of M. Crampin [6] and J.
Grifone [7], there exists a unique Ehresmann connection H over M such that
1
(X c + [X v , S])
2
for all vector fields X on M . H is said to be the Ehresmann connection associated
to (or generated by) H. The torsion of this Ehresmann connection vanishes.
Furthermore, we have
1
H(δ) = (S + [C, S]).
2
If, in particular, S is a spray, then H(δ) = S, and H is homogeneous, i.e., its
tension also vanishes.
◦
◦
We define the h-Berwald differentials ∇h F ∈ T10 (π) (F ∈ C ∞ (T M )) and
◦
◦
∇h Ye ∈ T11 (π) (Ye ∈ Sec(π)) by the following rules:
b =
H(X)
(9)
◦
e := (HX)F,
e
e ∈ Sec(π);
∇h F (X)
X
(10)
(11)
◦
e := ∇h Ye := V[HX,
e iYe ], X
e ∈ Sec(π).
∇h Ye (X)
e
X
◦
e ∈ Sec(π)) may also uniquely be extended to the whole
The operators ∇hXe (X
◦
tensor algebra of Sec(π) as tensor derivations. Forming the h-Berwald differential
of an arbitrary tensor, we adopt the same convention as in the vertical case. We
note that the tension of H is just the h-covariant differential of the canonical
section, i.e., = ∇h δ. So the homogeneity of H means that
(12)
∇h δ = 0.
We may also consider the graded derivation dh associated to the horizontal
projector h = H ◦ j; then we have
◦
(13)
∇h F ◦ j = dh F (F ∈ C ∞ (T M )).
ON THE PROJECTIVE THEORY OF SPRAYS
177
From the operators ∇v and ∇h we build the Berwald derivative
◦
◦
◦
∇ : (ξ, Ye ) ∈ X(T M ) × Sec(π) 7→ ∇ξ Ye := ∇vVξ Ye + ∇hjξ Ye ∈ Sec(π).
Then, by (7) and (11),
∇ξ Ye = j[vξ, HYe ] + V[hξ, iYe ].
In particular,
◦
e Ye ∈ Sec(π);
∇iXe Ye = ∇vXe Ye , ∇HXe Ye = ∇hXe Ye ; X,
∇X v Yb = 0, i∇X h Yb = X h , Y v ; X, Y ∈ X(M ).
(14)
Lemma 2.1 (hh-Ricci identity for functions). Let H be a torsion-free Ehres◦
mann connection over M . If f : T M → R is a smooth function, then for any
◦
e Ye in Sec(π)
sections X,
we have
e Ye ) − ∇h ∇h f (Ye , X)
e = −iR(X,
e Ye )f.
∇h ∇h f (X,
(15)
Proof. It is enough to show that formula (15) is true for basic vector fields
◦
b Yb ∈ Sec(π).
X,
Then
b Yb ) = ∇X h (∇h f ) (Yb ) = X h Y h f − ∇h f (∇X h Yb )
(∇h ∇h f )(X,
= X h Y h f − (H∇X h Yb )f = X h Y h f − HV[X h , Y v ]f
= X h Y h f − (F + J)[X h , Y v ]f = X h Y h f − F[X h , Y v ]f,
and in the same way
So we obtain
b = Y h X h f − F[Y h , X v ]f.
(∇h ∇h f )(Yb , X)
b Yb ) − ∇h ∇h f (Yb , X)
b = [X h , Y h ]f − F([X h , Y v ] − [Y h , X v ])f
∇h ∇h f (X,
T=0
= [X h , Y h ]f − (F[X, Y ]v )f = ([X h , Y h ] − [X, Y ]h )f
b Yb )f.
= ([X h , Y h ] − h[X h , Y h ])f = v[X h , Y h ]f = −iR(X,
3. The Berwald curvature of an Ehresmann connection
In this section we specify an Ehresmann connection H over M , and consider
the Berwald derivative ∇ = (∇h , ∇v ) induced by H. We denote by R∇ the usual
curvature tensor of ∇. By the Berwald curvature of H we mean the type 13
◦
tensor field B along τ given by
(16)
e Ye )Z
e := R∇ (iX,
e HYe )Ze = ∇ e ∇ e Z
e − ∇ e ∇ eZ
e − ∇ e e Z,
e
B(X,
iX HY
HY
iX
[iX,HY ]
e Ye , Z
e are vector fields along τ◦ .
where X,
178
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Lemma 3.1. For any vector field X, Y ,Z on M we have
b Yb )Zb = j[X v , F[Y h , Z v ]] = (∇v ∇h Z)(
b X,
b Yb ),
B(X,
(17)
or, equivalently,
b Yb )Zb = [X v , [Y h , Z v ]] = [[X v , Y h ], Z v ].
iB(X,
(18)
Proof.
b Yb )Z
b := R∇ (X v , Y h )Zb = ∇X v ∇Y h Z
b − ∇Y h ∇X v Z
b − ∇[X v ,Y h ] Z
b
B(X,
= ∇X v (V[Y h , Z v ]) = j[X v , H ◦ V[Y h , Z v ]]
= j[X v , (F + J)[Y h , Z v ]] = j[X v , F[Y h , Z v ]].
On the other hand,
b X,
b Yb ) = ∇X v (∇h Z)(
b Yb ) = ∇X v ∇Y h Z
b
b = j[X v , H∇Y h Z]
(∇v ∇h Z)(
= j[X v , H ◦ V[Y h , Z v ]] = j[X v , F[Y h , Z v ]],
thus relations (17) hold. To prove the remainder, observe that
0 = [J, X v ]F[Y h , Z v ] = [v[Y h , Z v ], X v ] − J[F[Y h , Z v ], X v ]
= −[X v , [Y h , Z v ]] + J[X v , F[Y h , Z v ]],
and hence
b Yb )Z
b (17)
iB(X,
= J[X v , F[Y h , Z v ]] = [X v , [Y h , Z v ]].
Finally, using the Jacobi identity we obtain that
[X v , [Y h , Z v ]] = [[X v , Y h ], Z v ].
Lemma 3.2. The Berwald curvature of an Ehresmann connection is symmetric in its first and third variable. If the torsion of the Ehresmann connection
vanishes, then the Berwald curvature is totally symmetric.
b Yb )Z
b = [X v , [Y h , Z v ]].
Proof. Keeping the notation of the previous lemma, iB(X,
Since
0 = [X v , [Y h , Z v ]] + [Y h , [Z v , X v ]] + [Z v , [X v , Y h ]]
b Yb )Z
b − iB(Z,
b Yb )X,
b
= [X v , [Y h , Z v ]] − [Z v , [Y h , X v ]] = iB(X,
B is indeed symmetric in its first and third variable. If the Ehresmann connection has vanishing torsion, then
[X h , Z v ] − [Z h , X v ] − [X, Z]v = 0
(X, Z ∈ X(M )),
and hence
b Z
b = [Y v , [X h , Z v ]] = [Y v , [Z h , X v ]] + [Y v , [X, Z]v ]
iB(Yb , X)
b X.
b
= [Y v , [Z h , X v ]] = iB(Yb , Z)
ON THE PROJECTIVE THEORY OF SPRAYS
Thus, if the torsion vanishes,
b Yb )Z
b = iB(Z,
b Yb )X
b = iB(Z,
b X)
b Yb = iB(Yb , X)
b Z
b
iB(X,
bX
b = iB(X,
b Z)
b Yb .
= iB(Yb , Z)
179
Lemma 3.3. The tension and the Berwald curvature of an Ehresmann connection are related by
b Yb )δ = ∇v t(X,
b Yb ); X, Y ∈ X(M ).
(19)
B(X,
Proof. By an important identity, due to J. Grifone, for any vector field ξ on T M
we have
(20)
J[Jξ, S] = Jξ,
where S is an arbitrary semispray over M ([7],[21]). Since H ◦ δ is a semispray
over M , this implies that
e H ◦ δ] = X;
e X
e ∈ X(τ ).
∇ e δ = j[iX,
iX
So we obtain
b Yb )δ = R∇ (X v , Y h )δ
B(X,
= ∇X v ∇Y h δ − ∇Y h ∇X v δ − ∇[X v ,Y h ] δ
b − V[X v , Y h ]
= ∇X v (t(Yb )) − ∇Y h X
= ∇X v (t(Yb )) − V[Y h , X v ] + V[Y h , X v ] = (∇X v t)(Yb )
b Yb ).
= (∇v t)(X,
Corollary 3.4. If the torsion and the vertical differential of the tension of an
Ehresmann connection vanishes, then its Berwald curvature has the property
o
n
e Ye , Z
e ⇒ B(X,
e Ye )Ze = 0.
(21)
δ ∈ X,
Lemma 3.5. The Berwald curvature of a homogeneous Ehresmann connection
is homogeneous of degree −1, i.e.,
∇vδ B = ∇C B = −B.
Proof. Using the first relation in (4), the Jacobi identity (repeatedly) and the
homogeneity of H, for any vector fields X, Y , Z on M we get
b Yb , Z)
b = i∇C (B(X,
b Yb )Z)
b
i (∇C B) (X,
b Yb )Z]
b = [J, C]HB(X,
b Yb )Z
b − [iB(X,
b Yb )Z,
b C]
= J[C, HB(X,
b Yb )Z
b − [[X v , [Y h , Z v ]], C]
= iB(X,
b Yb )Z
b + [[[Y h , Z v ], C], X v ] + [[C, X v ], [Y h , Z v ]]
= iB(X,
= [[[Y h , Z v ], C], X v ] = −[[[Z v , C], Y h ], X v ] + [[C, Y h ], Z v ], X v ]
b Yb )Z.
b
= −[[Z v , Y h ], X v ] = −[X v , [Y h , Z v ]] = −iB(X,
180
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Lemma 3.6. (vh-Ricci formulae for functions and sections). If F is a smooth
◦
e is a section along τ◦ , then for any sections X,
e Ye in
function on T M and Z
◦
Sec(π) we have
e Ye ) = ∇h ∇v F (Ye , X);
e
∇v ∇h F (X,
(22)
(23)
e X,
e Ye ) − ∇h ∇v Z(
e Ye , X)
e = B(X,
e Ye )Z.
e
∇v ∇h Z(
Proof. The expression on the left-hand side of (22) is
e Ye ) = (∇ e ∇h F )(Ye ) = (iX)(H
e
∇v ∇h F (X,
Ye )F − ∇h F (∇iXe Ye )
iX
e
e
e HYe ]F
= (iX)(H
Ye )F − (H∇iXe Ye )F = (iX)(H
Ye )F − h[iX,
The right-hand side of (22) can be written in the form
e = (∇ e ∇v F )(X)
e = (HYe )(iX)F
e − ∇v F (∇ e X)
e
∇h ∇v F (Ye , X)
HY
HY
e − (i∇ e X)F
e
= (HYe )(iX)F
HY
so their difference is
e − v[HYe , iX]F,
e
= (HYe )(iX)F
e HYe ]F + v[HYe , iX]F
e − h[iX,
e HYe ]F = 0.
[iX,
This proves relation (22). Relation (23) may be checked by a similar calculation.
We have, on the one hand,
e X,
e Ye ) = ∇ e ∇ e Ze − ∇
e
∇v ∇h Z(
e Z.
iX HY
H∇ fY
iX
On the other hand,
e Ye , X)
e = ∇ e∇ eZ
e−∇
∇h ∇v Z(
HY iX
i∇
e
HY
e
e Z.
X
e = h[iX,
e HYe ] − v[HYe , iX]
e = [iX,
e HYe ], it follows that
Since H∇iXe Ye − i∇HYe X
e Ye )Z.
e
the difference of the left-hand sides is indeed B(X,
◦
Lemma 3.7 (vh-Ricci formula for covariant tensors). Let A ∈ Ts0 (π), s ≥ 1.
e Ye , Z
f1 , . . . Z
fs along τ◦ we have
For any sections X,
(24)
e Ye , Z
f1 , . . . Z
fs ) − ∇h ∇v A(Ye , X,
e Z
f1 , . . . Z
fs )
∇v ∇h A(X,
=−
s
X
i=1
f1 , . . . , B(X,
e Ye )Z
fi , . . . , Z
fs ).
A(Z
Proof. For brevity, we sketch the argument only for a type 02 tensor A. It
may easily be shown that the left-hand side of (24) is tensorial in its first two
ON THE PROJECTIVE THEORY OF SPRAYS
181
e and Ye
variables (actually, in all variables), so we may chose in the role of X
b Yb . Then
basic vector fields X,
b Yb , Z
f1 , Z
f2 ) = X v (Y h A(Z
f1 , Z
f2 )) − X v A(∇Y h Z
f1 , Z
f2 )
∇v ∇h A(X,
f1 , ∇Y h Z
f2 ) − Y h A(∇X v Z
f1 , Z
f2 ) + A(∇Y h ∇X v Z
f1 , Z
f2 )
− X v A(Z
f1 , ∇Y h Z
f2 ) − Y h A(Z
f1 , ∇X v Z
f2 ) + A(∇Y h Z
f1 , ∇X v Z
f2 )
+ A(∇X v Z
f1 , ∇Y h ∇X v Z
f2 );
+ A(Z
b Z
f1 , Z
f2 ) = Y h (X v A(Z
f1 , Z
f2 )) − Y h A(∇X v Z
f1 , Z
f2 )
∇h ∇v A(Yb , X,
f1 , ∇X v Z
f2 ) − [Y h , X v ]A(Z
f1 , Z
f2 ) + A(∇[Y h ,X v ] Z
f1 , Z
f2 )
− Y h A(Z
f1 , ∇[Y h ,X v ] Z
f2 ) − X v A(∇Y h Z
f1 , Z
f2 ) + A(∇X v ∇Y h Z
f1 , Z
f2 )
+ A(Z
f1 , ∇X v Z
f2 ) − X v A(Z
f1 , ∇Y h Z
f2 ) + A(∇X v Z
f1 , ∇Y h Z
f2 )
+ A(∇Y h Z
f1 , ∇X v ∇Y h Z
f2 ),
+ A(Z
and after subtraction we get
b Yb , Z
f1 , Z
f2 ) − ∇h ∇v A(Yb , X,
b Z
f1 , Z
f2 )
∇v ∇h A(X,
f1 − ∇X v ∇Y h Z
f1 − ∇[Y h ,X v ] Z
f1 , Z
f2 )
= A(∇Y h ∇X v Z
f1 , ∇Y h ∇X v Z
f2 − ∇X v ∇Y h Z
f2 − ∇[Y h ,X v ] Z
f2 ) =
+ A(Z
b Yb )Z
f1 , Z
f2 ) − A(Z
f1 , B(X,
b Yb )Z
f2 ).
− A(B(X,
Proposition 3.8. An Ehresmann connection H over M has vanishing Berwald
curvature, if and only if, there exists a (necessarily unique) covariant derivative
operator D on the base manifold M such that for any vector fields X, Y on M
we have
(25)
[X h , Y v ] = (DX Y )v .
Proof. The sufficiency of the condition is immediate: if there exists a covariant
derivative operator D on M satisfying (25), then for all vector fields X, Y , Z
on M we have
(25)
v
b Yb , Z)
b (18)
iB(X,
= [X v , [Y h , Z v ]] = [X v , (DY Z) ] = 0,
since the Lie bracket of vertically lifted vector fields vanishes.
Conversely, if H has vanishing Berwald curvature, then for all vector fields
X, Y, Z in X(M ),
[X v , [Y h , Z v ]] = 0.
This implies that [Y h , Z v ] is a vertical lift, so we may define a map
D : X(M ) × X(M ) → X(M ), (Y, Z) 7→ DY Z
by
v
(DY Z) := [Y h , Z v ].
182
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
It is easy to check, that D is a covariant derivative operator on M . For example,
if f ∈ C ∞ (M ), then
(DY f Z)v := [Y h , (f Z)v ] = [Y h , f v Z v ] = (Y h f v )Z v + f v [Y h , Z v ]
v
v
= (Y f )v Z v + (f DY Z) = ((Y f )Z + f DY Z) ,
hence
DY f Z = (Y f )Z + f DY Z.
Similarly,
(Df Y Z)v := (f Y )h , Z v = [f v Y h , Z v ] = −(Z v f v )Y h + f v [Y h , Z v ]
= f v [Y h , Z v ] = (f DY Z)v ,
which implies that
Df Y Z = f DY Z.
The other rules are immediate consequences of the definition of D.
Relation (25) can also be written in the form
\
∇hXb Yb = D
XY ,
so it is reasonable to call an Ehresmann connection h-basic or briefly basic,
if it has vanishing Berwald curvature: then the Christoffel symbols of the hcovariant derivative do not depend on the direction. More generally, we say that
an Ehresmann connection is weakly Berwald if the trace of its Berwald curvature
vanishes.
We shall use similar terminology for sprays. A spray will be called Berwald, if
its associated Ehresmann connection has vanishing Berwald curvature, and will
be called weakly Berwald if the Berwald curvature of its associated Ehresmann
connection is traceless.
4. The affine curvature of an Ehresmann connection
We continue to assume that an Ehresmann connection H is specified over M ,
and consider the Berwald derivative ∇ = (∇h , ∇v ) determined by H. By the
◦
affine curvature of H we mean the type 13 tensor H along τ given by
◦
e Ye )Z
e := R∇ (HX,
e HYe )Z;
e X,
e Ye , Z
e ∈ Sec(π).
H(X,
This tensor was essentially introduced by L. Berwald ([5]) in terms of the classical tensor calculus and in the more specific context of an Ehresmann connection
associated to a spray. So we think that it is appropriate to preserve his terminology. To indicate the meaning of the affine curvature, we remark, that if an
Ehresmann connection is basic with base covariant derivative D on M , then its
affine curvature may be identified with the curvature of D. More precisely, we
have
b Yb )Zb = (RD (X, Y )Z)v ; X, Y, Z ∈ X(M ).
iH(X,
ON THE PROJECTIVE THEORY OF SPRAYS
183
Now we formulate and prove some basic relations found by Berwald in our
setting. The first observation, roughly speaking, is that the affine curvature is
just the vertical differential of the curvature of H. The exact relation between
H and R is formulated in
◦
e Ye , Z
e ∈ Sec(π),
Lemma 4.1. For all X,
e Ye )Z
e = ∇v R(Z,
e X,
e Ye ).
H(X,
(26)
b Yb , Z
b along
Proof. It is enough to check that (26) is true for basic vector fields X,
◦
τ . Then, on the one hand,
b Yb )Z
b := ∇X h ∇Y h Z
b − ∇Y h ∇X h Z
b − ∇[X h ,Y h ] Z
b
H(X,
(14)
= ∇X h V[Y h , Z v ] − ∇Y h V[X h , Z v ] − ∇h[X h ,Y h ] Zb
(11),(8)
=
V[X h , [Y h , Z v ]] − V[Y h , [X h , Z v ]] − V[[X, Y ]h , Z v ]
= V([X h , [Y h , Z v ]] + [Y h , [Z v , X h ]] + [Z v , [X, Y ]h ])
= V([−Z v , [X h , Y h ]] + [Z v , [X, Y ]h ])
b Yb )].
= V[Z v , [X, Y ]h − [X h , Y h ]] = V[Z v , iR(X,
On the other hand,
b X,
b Yb ) = (∇Z v R) (X,
b Yb )
∇v R(Z,
(14)
b Yb )) (7)
b Yb )] = −j[Z v , HV[X h , Y h ]]
= ∇Z v (R(X,
= j[Z v , HR(X,
= −j[Z v , F[X h , Y h ] + J[X h , Y h ]] = −j[Z v , F[X h , Y h ]].
Now, taking into account the second relation in (4),
0 = [J, Z v ]F[X h , Y h ] = [JF[X h , Y h ], Z v ] − J[F[X h , Y h ], Z v ]
= [v[X h , Y h ], Z v ] − J[F[X h , Y h ], Z v ]
b Yb )] + J[Z v , F[X h , Y h ]]
= [Z v , iR(X,
hence
b Yb )] + J[Z v , F[X h , Y h ]],
= iV[Z v , iR(X,
b X,
b Yb ) = −j[Z v , F[X h , Y h ]] = V[Z v , iR(X,
b Yb )] = H(X,
b Yb )Z.
b
∇v R(Z,
Lemma 4.2. If H is a homogeneous Ehresmann connection, then the curvature
of H may be reproduced from the affine curvature, namely, we have
(27)
◦
e Ye ) = H(X,
e Ye )δ ; X,
e Ye ∈ Sec(π).
R(X,
184
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Proof.
e Ye )δ := ∇ e ∇ e δ − ∇ e ∇ e δ − ∇ e e δ
H(X,
HX HY
HY
HX
[HX,HY ]
(12)
e
e
= −∇v[HX,H
e Y
e ] δ = −j[v[HX, HY ], H ◦ δ].
Since H ◦ δ = S is a spray, we obtain that
e Ye )δ = −i−1 J[JF[HX,
e HYe ], S]
H(X,
(20)
e HYe ] = −V[HX,
e HYe ] = R(X,
e Ye ),
= −i−1 JF[HX,
as we claimed.
Corollary 4.3. If an Ehresmann connection is homogeneous, then its curvature
R is homogeneous of degree 1, i.e., ∇C R = R.
Proof. For any vector fields X, Y on M ,
b Yb ) = ∇v R(δ, X,
b Yb ) (26)
b Yb )δ (27)
b Yb ).
(∇C R)(X,
= H(X,
= R(X,
Lemma 4.4. The affine curvature of a homogeneous Ehresmann connection is
homogeneous of degree zero, i.e., ∇C H = 0.
Proof. By a similar technique as above, we have for any vector fields X, Y, Z on
M:
b Yb , Z)
b = i∇C (H(X,
b Yb )Z)
b
i(∇C H)(X,
(26)
b Yb )) = i∇C j[Z v , HR(X,
b Yb )]
= i∇C ∇Z v (R(X,
b Yb )]] = J[C, [Z v , HR(X,
b Yb )]]
= J[C, h[Z v , HR(X,
since
b Yb ), C]] + [HR(X,
b Yb ), [C, Z v ]])
= −J([Z v , [HR(X,
b Yb )], Z v ] − [HR(X,
b Yb ), Z v ]) = 0,
= −J([h[C, HR(X,
b Yb ) (4.3)
b Yb )) = Hj[C, HR(X,
b Yb )] = h[C, HR(X,
b Yb )]. HR(X,
= H∇C (R(X,
Lemma 4.5. If H is a torsion-free Ehresmann connection, then its affine curvature satisfies the Bianchi identity
(28)
e Ye )Z
e + H(Ye , Z)
eX
e + H(Z,
e X)
e Ye = 0
H(X,
◦
e Ye , Z
e ∈ Sec(π)).
(X,
b Yb , Z
b again. Then,
Proof. In our calculations we may use basic vector fields X,
taking into account the first partial result in the proof of 4.1, we get
b Yb )Zb = [Z v , iR(X,
b Yb )]
iH(X,
= [Z v , −v[X h , Y h ]] = [Z v , [X, Y ]h ] − [Z v , [X h , Y h ]]
= [Z v , [X, Y ]h ] + [X h , [Y h , Z v ]] + [Y h , [Z v , X h ]].
ON THE PROJECTIVE THEORY OF SPRAYS
185
In the same way,
bX
b = [X v , [Y, Z]h ] + [Y h , [Z h , X v ]] + [Z h , [X v , Y h ]],
iH(Yb , Z)
b X)
b Yb = [Y v , [Z, X]h ] + [Z h , [X h , Y v ]] + [X h , [Y v , Z h ]].
iH(Z,
Now we add these three relations, and apply the vanishing of the torsion repeatedly:
b Yb )Zb + H(Yb , Z)
b X
b + H(Z,
b X)
b Yb ) = [X h , [Y h , Z v ] − [Z h , Y v ]]
i(H(X,
+ [Y h , [Z v , X h ] − [X v , Z h ]] + [Z h , [X h , Y v ] − [Y h , X v ]] + [X v , [Y, Z]h ]
+ [Y v , [Z, X]h ] + [Z v , [X, Y ]h ]
= [X h , [Y, Z]v ] − [[Y, Z]h , X v ] + [Y h , [Z, X]v ] − [[Z, X]h , Y v ] + [Z h , [X, Y ]v ]
v
− [[X, Y ]h , Z v ] = ([X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]]) = 0.
Now we suppose that the Ehresmann connection H is associated to a spray S.
Then, as we have already mentioned, H is homogeneous and torsion-free. The
◦
type 11 tensor field K along τ defined by
◦
e := V[S, HX],
e X
e ∈ Sec(π)
K(X)
(29)
is said to be the affine deviation tensor (L. Berwald [5]) or the Jacobi endomorphism (W. Sarlet et al. [12]) of the spray S, or of the Ehresmann connection
associated to S. The homogeneity of S implies that S = Hδ, so it follows that
e = V[Hδ, HX]
e = −R(δ, X)
e = R(X,
e δ).
K(X)
(30)
This means that the affine deviation can immediately be obtained from the
curvature of the Ehresmann connection. The converse is also true:
Proposition 4.6. Let S be a spray over M , and let H be the Ehresmann connection associated to S. Then the curvature and the affine deviation of H are
related by
e Ye ) =
R(X,
1 v e e
◦
e Ye )) ; X,
e Ye ∈ Sec(π).
(∇ K(Y , X) − ∇v K(X,
3
Proof. Let X, Y be vector fields on M . Then
b = (∇Y v K) (X)
b = ∇Y v (K(X))
b
∇v K(Yb , X)
b ∇Y v δ
b δ)) = (∇Y v R) (X,
b δ) + R ∇Y v X,
b δ + R X,
= ∇Y v (R(X,
(30)
Similarly,
b δ) + R(X,
b Yb ).
= ∇v R(Yb , X,
b Yb ) = ∇v R(X,
b Yb , δ) + R(Yb , X),
b
∇v K(X,
186
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
therefore
b − ∇v K(X,
b Yb ) = ∇v R(Yb , X,
b δ) − ∇v R(X,
b Yb , δ) + 2R(X,
b Yb )
∇v K(Yb , X)
(26)
b δ)Yb − H(Yb , δ)X
b + 2R(X,
b Yb ) = H(X,
b δ)Yb + H(δ, Yb )X
b + 2R(X,
b Yb )
= H(X,
4.5
b + 2R(X,
b Yb ) = H(X,
b Yb )δ + 2R(X,
b Yb ) 4.2
b Yb ).
= −H(Yb , X)δ
= 3R(X,
Remark. Historically, Berwald’s starting point in his above mentioned, famous
posthumous paper was a SODE of form
(xi )′′ + 2Gi (x, x′ ) = 0, i ∈ {1, . . . , n} ,
(31)
where
◦ −1
Gi ∈ C 1 (τ −1 (U)) ∩ C ∞ (τ
(U)), CGi = 2Gi
(U ⊂ M is a coordinate neighborhood).
As a first step, Berwald deduces the following ‘equation of affine deviation’:
D2 ξ i
dx
i
+ Kj x,
ξ j = 0, i ∈ {1, . . . , n} ,
ds
ds
where
dξ i
∂Gi
Dξ i
:=
+ Gir ξ r , Gir :=
,
ds
ds
∂y r
and the functions
∂Gi ∂Gij r
−
y + 2Gijr Gr − Gir Grj , (i, j ∈ {1, . . . , n})
∂xj
∂xr
are the components of a type 11 tensor, called affinen Abweichungstensor by
Berwald. It may easily be checked that our tensor K is indeed an intrinsic form
of the tensor obtained by him.
In the second step, Berwald introduces the ‘Grundtensor der affinen Krmmung’, giving its components by
!
∂Kji
1 ∂Kki
i
−
.
Kjk :=
3 ∂y j
∂y k
Kji := 2
Proposition 4.6 shows that this is just the curvature of the Ehresmann connection
which may be associated to the SODE (31).
Finally, in the third step, Berwald defines the ‘affine Krmmungstensor’ by its
components
i
∂Kjk
i
.
:=
Khjk
∂y h
In view of Lemma 4.1, this is just our tensor H.
ON THE PROJECTIVE THEORY OF SPRAYS
187
Lemma 4.7 (hh-Ricci formulae for sections and 1-forms). If H is a torsion-free
◦
◦
e ∈ Sec(π)
Ehresmann connection, then for any section Z
and 1-form α
e ∈ T10 (π)
we have
e X,
b Yb ) − ∇h ∇h Z(
e Yb , X)
b = H(X,
b Yb )Ze − ∇v Z(R(
e
b Yb )),
(32)
∇h ∇h Z(
X,
(33)
Proof.
b Yb , Z)
b − ∇h ∇h α
b Z)
b
∇h ∇h α
e(X,
e(Yb , X,
b Yb )Z)
b − ∇v α
b Yb ), Z),
b (X, Y, Z ∈ X(M )).
= −e
α(H(X,
e(R(X,
e X,
b Yb ) = ∇X h (∇h Z)(
e Yb ) = ∇X h ∇Y h Ze − ∇h Z
e ∇X h Yb
∇h ∇h Z(
e−∇
= ∇X h ∇Y h Z
H∇
Xh
and, similarly,
Hence
e = ∇X h ∇Y h Z
e − ∇HV[X h ,Y v ] Z,
e
bZ
Y
e Yb , X)
b = ∇Y h ∇X h Z
e − ∇HV[Y h ,X v ] Z.
e
∇h ∇h Z(
e X,
b Yb ) − ∇h ∇h Z(
e Yb , X)
b
∇h ∇h Z(
e − ∇Y h ∇X h Ze + ∇HV([Y h ,X v ]−[X h ,Y v ]) Ze
= ∇X h ∇Y h Z
b Yb )Z
e + ∇[X h ,Y h ]−(F+J)[X,Y ]v Z
e
= H(X,
b Yb )Z
e + ∇[X h ,Y h ]−[X,Y ]h Z
e = H(X,
b Yb )Z
e−∇ b b Z
e
= H(X,
iR(X,Y )
b Yb )Z
e − ∇v Z(R(
e
b Yb )),
= H(X,
X,
which proves relation (32). Relation (33) can be checked in the same way.
5. Projectively related sprays
We recall (for details, see [16], [23], [24], [21]) that two sprays S and S over M
are said to be (pointwise) projectively related, if there exists a function P : T M →
◦
R, C 1 on T M , smooth on T M , such that
(34)
S = S − 2P C.
The projective factor P in (34) is necessarily positive-homogeneous of degree
1, i.e., CP = P . If A is a geometric object associated to S, then we denote by A
the corresponding geometric object determined by S. The following relations are
well-known ([21]), and may easily be checked. If H is the Ehresmann connection
associated to S, then
(35)
H = H − P i − ∇v P ⊗ C,
(36)
h = h − P J − (∇v P ◦ j) ⊗ C = h − P J − dJ P ⊗ C,
188
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
(37)
X h = X h − P X v − (X v P )C (X ∈ X(M )),
(38)
V = V + P j + (∇v P ◦ j) ⊗ δ = V + P j − dJ P ⊗ δ.
We also have the less immediate
Lemma 5.1.
h
∇ = ∇h − P ∇v − ∇v P ⊗ ∇C + ∇v P ⊙ 1 + ∇v ∇v P ⊗ δ,
(39)
where the symbol ⊙ denotes symmetric product (without any numerical factor),
and 1 ∈ T11 (π) is the unit tensor.
e and Ye be vector fields along τ◦ . Then
Proof. Let X
e iYe ] (35)
e − P iX
e − ∇v P (X)C,
e
∇HXe Ye = V[HX,
= V[HX
iYe ]
e iYe ] + V iYe (P )iX
e − P [iX,
e iYe ]
= V[HX,
e )C − iX(P
e )[C, iYe ]
+ V iYe (iXP
(38)
e iYe ] − P j[iYe , HX]
e − (J[iYe , HX]P
e )δ + iYe (P )X
e
= V[HX,
e iYe ] + iYe (iXP
e )δ − iX(P
e )i−1 [iδ, iYe ].
− P i−1 [iX,
An easy calculation shows that
e iYe ] = ∇
i−1 [iX,
e − ∇ e X,
e
iY
eY
iX
so we obtain
h
i−1 [iδ, iYe ] = ∇C Ye − ∇iYe δ = ∇C Ye − Ye ,
e Ye ) = ∇ e Ye = ∇ e Ye − P ∇ e X
e − (i∇ e X)P
e δ + iYe (P )X
e
∇ (X,
HX
iY
iY
HX
e + (iYe (iX)P
e )δ − iX(P
e )∇C Ye + iX(P
e )Ye
− P ∇iXe Ye + P ∇iYe X
v
v
v
v e
e
e
e e
e e
= ∇HXe Ye − P ∇X
e Y − ∇ P (X)∇C Y + ∇ P (X)Y + ∇ P (Y )X
e P δ + i∇ e X
e − i∇ e Ye P δ + (iX(i
e Ye )P )δ
− i∇iYe X
iY
iX
e Ye ). = (∇h − P ∇v − ∇v P ⊗ ∇C + ∇v P ⊙ 1 + ∇v ∇v P ⊗ δ)(X,
Corollary 5.2. For all vector fields X, Y on M ,
b + X v (Y v P )δ,
∇X h Yb = ∇X h Yb + (X v P )Yb + (Y v P )X
(40)
(41)
Proof. Since
∇S Yb = ∇S Yb + P Yb + (Y v P )δ.
b Yb ) = ∇X v Yb = 0,
∇v (X,
∇C Yb = j[C, Y v ] = −jY v = 0,
ON THE PROJECTIVE THEORY OF SPRAYS
189
b Yb relation (39) leads to (40). As to the second relation,
for basic vector fields X,
observe that
hS = (h − P J − (∇v P ◦ j)C)(S − 2P C) = S − P C − (CP )C = S − 2P C = S.
Hence
h
(39)
∇S Yb = ∇h S Yb = ∇δ Yb = ∇hδ Yb − P ∇vδ Yb − ∇v P (δ)∇C Yb
since
+ ∇v P (δ)Yb + ∇v P (Yb )δ + ∇v ∇v P (δ, Yb )Yb = ∇S Yb + P Yb + (Y v P )δ,
∇v ∇v P (δ, Yb ) = ∇C (∇v P )(Yb ) = C(Y v P ) = [C, Y v ]P + Y v (CP )
= −Y v P + Y v P = 0.
It was shown in [21], that the Berwald curvatures of S and S and their traces
are related by
(42)
B = B − ∇v ∇v P ⊙ 1 − ∇v ∇v ∇v P ⊗ δ
and
(43)
trB = trB − (n + 1)∇v ∇v P,
respectively.
From relation (43) it follows at once that the trace of the Berwald curvature
is a projective invariant, if and only if, the projective factor satisfies the PDE
(44)
∇v ∇v P = 0.
However, this relation gives also the criterion of the projective invariance of the
Berwald curvature.
Indeed, if P satisfies (44), then (42) yields B = B. Conversely, if B = B,
then trB = trB, and (43) implies that ∇v ∇v P = 0.
The solutions of this PDE may be described easily, so we obtain
Proposition 5.3. The Berwald curvature of a spray S is invariant under a
projective change S = S − 2P C, if and only if, the projective factor is of form
(45)
P = iξ αv =: α, α ∈ X∗ (M ),
where ξ is an arbitrary second-order vector field over M .
Proof. First we check that the functions given by (45) solve (44). To do this,
observe that for any vector field Y on M ,
(46)
Y v α = (α(Y ))v .
Indeed,
Y v α = Y v iξ αv = LY v iξ αv = iξ LY v αv + i[Y v ,ξ] αv .
The first term on the right-hand side vanishes, since for any vector field X on
M we have
(LY v αv )(X v ) = Y v (αv (X v )) − αv ([Y v , X v ]) = 0,
190
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
v
(LY v αv )(X c ) = Y v (αv (X c )) − αv ([Y v , X c ]) = Y v (α(X))v − αv ([Y, X] ) = 0.
As for the second term, it is known (see e.g. [19], 3.2, Corollary) that
[Y v , ξ] = Y c + η, η ∈ Xv (T M ),
therefore
Y v α = αv ([Y v , ξ]) = αv (Y c ) + αv (η) = (α(Y ))v ,
as we claimed.
Now, for any vector fields X, Y on M ,
(46)
b Yb ) = (∇X v ∇v α) (Yb ) = X v (∇v α(Yb )) = X v (Y v α) = X v (α(Y ))v = 0,
∇v ∇v α(X,
therefore the functions α, α ∈ X∗ (M ) solve our PDE (44). We show that these
solutions satisfy the homogeneity condition Cα = α.
We may suppose that ξ is homogeneous of degree two, i.e., [C, ξ] = ξ, since
the definition of α does not depend on the choice of ξ. Then we obtain
Cα = LC iξ αv = iξ LC αv − i[ξ,C] αv = i[C,ξ] αv = iξ αv = α,
since LC αv = 0.
Conversely, if ∇v ∇v P = 0, then for all X, Y ∈ X(M ),
b Yb ) = (∇X v (∇v P ))(Yb ) = X v (Y v P ).
0 = ∇v ∇v P (X,
Then Y v P is the vertical lift of a smooth function on M , therefore it is of form
(46)
Y v P = (α(Y ))v = Y v α, α ∈ X∗ (M ).
Since Y is arbitrary, this implies that
P = α + f v , f ∈ C ∞ (M ).
However, the homogeneity condition CP = P forces that f = 0, since Cα = α
and Cf v = 0.
Corollary 5.4. A Berwald or a weakly Berwald spray remains of that type under
a projective change, if and only if, the projective factor is given by (45).
6. Basic objects associated to a Finsler function
By a Finsler function over M we mean a function F : T M → R satisfying the
following conditions:
◦
(F1 ) F is smooth on T M .
(F2 ) F is positive-homogeneous of degree 1 in the sense that for each nonnegative real number λ and each vector v ∈ T M , we have
F (λv) = λF (v).
ON THE PROJECTIVE THEORY OF SPRAYS
191
(F3 ) The metric tensor
1 v v 2
◦
∇ ∇ F ∈ T20 (π)
2
is (fibrewise) non-degenerate.
A Finsler manifold is a pair (M, F ) consisting of a manifold M and a Finsler
function on its tangent manifold.
By (F1 ) and (F2 ), F is continuous on T M and identically zero on σ(M ). The
function E := 12 F 2 is called the energy function of the Finsler manifold (M, F ).
g :=
◦
It is continuous on T M , smooth on T M , and also identically zero on σ(M ). By
(F2 ), E satisfies
E(λv) = λ2 E(v)
for all v ∈ T M and non-negative λ ∈ R, i.e., E is positive-homogeneous of degree
◦
2. Over T M this holds, if and only if, CE = 2E. It may be shown (see e.g.
[25]) that, actually, E is of class C 1 on T M .
For any vector fields X, Y on M we have
b Yb ) = X v (Y v E),
(47)
g(X,
e and
from which it follows immediately that g is symmetric. More generally, if X
◦
Ye are in Sec(π), then
(48)
e Ye ) = (iX)(i
e Ye )E − (i∇ e Ye )E = (iX)(i
e Ye )E − J[iX,
e HYe ]E,
g(X,
iX
where H is an arbitrary Ehresmann connection over M . In particular, we get
(49)
g(δ, δ) = 2E.
A further elementary property of the metric tensor is that it is homogeneous of
degree 0, i.e.,
(50)
∇vδ g = ∇C g = 0.
Indeed, for any vector fields X, Y on M ,
(47)
b Yb ) = Cg(X,
b Yb ) = C(X v (Y v E)) = [C, X v ](Y v E) + X v (C(Y v E))
(∇C g) (X,
= −X v Y v E + X v ([C, Y v ]E + Y v (CE))
= −2X v (Y v E) + 2X v (Y v E) = 0.
A Finsler manifold (M, F ) is said to be positive definite if the condition
◦
(F4 ) F (v) > 0 whenever v ∈ T M
is also satisfied. It may be shown ([11]) that in this case the metric tensor is
(fibrewise) positive
definite.
0
The type 3 tensor
(51)
C♭ :=
1 v
1
∇ g = ∇v ∇v ∇v E
2
2
192
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
is said to be the Cartan tensor of the Finsler manifold (M, F ). It may easily
◦
e1 , X
e2 , X
e3 in Sec(π)
be seen that C♭ is totally symmetric: for any sections X
and
any permutation σ : {1, 2, 3} → {1, 2, 3} we have
eσ(1) , X
eσ(2) , X
eσ(3) ) = C♭ (X
e1 , X
e2 , X
e3 ).
C♭ (X
Since g is homogeneous of degree 0, it follows that C♭ is homogeneous of degree
−1, i.e.,
(52)
∇vδ C♭ = ∇C C♭ = −C♭ .
As a consequence of the total symmetry of C♭ and the 0-homogeneity of g we get
o
n
e Ye , Z
e ⇒ C♭ (X,
e Ye , Z)
e = 0.
(53)
δ ∈ X,
Indeed,
e = ∇v g(δ, Ye , Z)
e = (∇C g) (Ye , Z)
e = 0.
2C♭ (δ, Ye , Z)
It is also known (see e.g. [25] again) that the following assertions are equivalent
for a positive definite Finsler manifold (M, F ):
(i) The energy function E of (M, F ) is of class C 2 (and hence smooth) on
T M.
(ii) E is the norm associated to a Riemannian metric on M .
(iii) There exists a Riemannian metric γ on M , such that
b Yb ) = γ(X, Y ) ◦ τ,
g(X,
for all vector fields X, Y ∈ X(M ).
(iv) The Cartan tensor of (M, F ) vanishes.
The 1-form
◦
◦
e ∈ Sec(π)
e := g(X,
e δ) ∈ C ∞ (T M )
θ: X
7→ θ(X)
is said to be the canonical 1-form or Hilbert 1-form of (M, F ). It may be seen
immediately that
(54)
θ = F ∇v F = ∇v E.
The section
1
δ
F
is traditionally called the normalized support element field of (M, F ). Its dual
form is
1
(55)
ℓ♭ := θ = ∇v F
F
since
1
1
(49) 1
ℓ♭ (ℓ) = 2 θ(δ) = 2 g(δ, δ) =
· 2E = 1.
F
F
2E
By the angular metric tensor of (M, F ) we mean the type 02 tensor
ℓ :=
(56)
η := g − ℓ♭ ⊗ ℓ♭ = g − ∇v F ⊗ ∇v F
ON THE PROJECTIVE THEORY OF SPRAYS
193
◦
along τ . We obtain:
1
η = ∇v ∇v F.
F
Indeed, for any vector fields X, Y , Z on M we have
1 b b
1 b b
b v F (Yb )
η(X, Y ) =
g(X, Y ) − ∇v F (X)∇
F
F 1 v v 2
(47) 1
v
v
X (Y F ) − (X F )(Y F )
=
F 2
1
b Yb ).
= (X v ((Y v F )F ) − (X v F )(Y v F )) = X v (Y v F ) = (∇v ∇v F )(X,
F
We note that (55) and (57) imply
1
(58)
∇v ℓ♭ = η.
F
Lemma 6.1. If (M, F ) is a Finsler manifold, then for any vector fields X, Y ,
Z on M we have
b Yb , Z)
b = 2 C♭ (X,
b Yb , Z)
b − 1
b Yb , Z)
b
(59)
∇v ∇v ∇v F (X,
S ℓ♭ ⊗ η(X,
b Y
b ,Z)
b
F
F 2 (X,
(57)
where the symbol
S
b Y
b ,Z)
b
(X,
Proof.
(57)
b Yb , Z.
b
means cyclic sum over X,
1
b
η (Yb , Z)
F
b Yb , Z)
b = ∇X v
∇ ∇ ∇ F (X,
1
v
b
b + 1 ∇X v (g − ℓ♭ ⊗ ℓ♭ )(Yb , Z)
η(Yb , Z)
=X
F
F
1
b + 1 (∇X v g) (Yb , Z)
b − 1 (∇X v ℓ♭ ) (Yb )ℓ♭ (Z)
b
= − 2 (X v F )η(Yb , Z)
F
F
F
1
b Yb , Z)
b − 1 ∇v F (X)η(
b Yb , Z)
b
b = 1 ∇v g(X,
− ℓ♭ (Yb ) (∇X v ℓ♭ ) (Z)
F
F
F2
1
2
b Yb )ℓ♭ (Z)
b − 1 ℓ♭ (Yb )∇v ℓ♭ (X,
b Z)
b (51),(55),(58)
b Yb , Z)
b
− (∇v ℓ♭ ) (X,
=
C♭ (X,
F
F
F
1 b Yb , Z)
b + ℓ♭ ⊗ η(Z,
b X,
b Yb ) + ℓ♭ ⊗ η(Yb , X,
b Z)
b
− 2 ℓ♭ ⊗ η(X,
F
2
b Yb , Z)
b − 1
b Yb , Z),
b
S ℓ♭ ⊗ η(X,
= C♭ (X,
b Y
b ,Z)
b
F
F 2 (X,
v
v
v
taking into account in the last step that
b Z)
b = 1 η(X,
b Z)
b = 1 (g(X,
b Z)
b − (X v F )(Z v F ))
∇v ℓ♭ (X,
F
F
1
b X)
b − (Z v F )(X v F )) = ∇v ℓ♭ (Z,
b X).
b
= (g(Z,
F
194
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Remark. Let Sym denote the symmetrizer defined by
◦
e X,
e Ye , Z)
e :=
(SymA)(
if A ∈ T30 (π). Let λ :=
∇v F
F
S
e Y
e ,Z)
e
(X,
e Ye , Z),
e
A(X,
, µ := ∇v ∇v F . Then
∇v F
1
ℓ
⊗
η
=
⊗ µ,
♭
F2
F
and (59) may be written in the more concise form
2
∇v µ = C♭ − Sym(λ ⊗ µ).
(60)
F
If (M, F ) is a Finsler manifold, then the 2-form
1
d(∇F 2 ◦ j) = ddJ E
2
◦
is (fibrewise) non-degenerate on T M by (F3 ). So there exists a unique map
S: TM → TTM
◦
defined to be zero on o(M ), and defined on T M to be the unique vector field
such that
(61)
iS ddJ E = −dE.
Then, actually, S is a spray over M , i.e., has the properties (C1 )-(C6 ). A very
instructive, but quite forgotten proof of this fundamental fact may be found in
F. Warner’s quoted paper [25]. The spray S will be called the canonical spray
of the Finsler manifold (M, F ). If H is the Ehresmann connection associated to
S according to (9), then
(i) H is homogeneous and torsion-free;
(ii) H is conservative in the sense that
(62)
dF ◦ H = 0 ⇔ X h F = X h E = 0, X ∈ X(M ).
Property (i), as we have already learnt, holds for any Ehresmann connection
associated to a spray. Here the new and surprising phenomenon is property
(ii), which expresses that the Finsler function (and hence the energy function)
is a first integral of the horizontally lifted vector fields. We call this Ehresmann
connection the canonical connection of the Finsler manifold. We note that other
terms - Barthel connection, Cartan’s nonlinear connection, Berwald connection are also frequently used in the literature. A recent index-free proof of (ii) can be
found in [22]. In the next section we shall show that the canonical connection of a
Finsler manifold (M, F ) is unique in the sense that there is only one Ehresmann
connection over M which satisfies conditions (i), (ii).
Warning. The Berwald connection H : T M ×M T M → T T M associated
to the canonical spray of a Finsler manifold and the Berwald derivative ∇ =
ON THE PROJECTIVE THEORY OF SPRAYS
195
(∇h , ∇v ) induced by H are essentially different objects: the latter is a covariant
derivative operator (’linear connection’) in a (special) vector bundle. It will
sometimes be mentioned as the Finslerian Berwald derivative.
Using the Finslerian h-Berwald covariant derivative, we define the Landsberg
tensor P of a Finsler manifold (M, F ) by the following formula:
1
P := − ∇h g.
2
(63)
Lemma 6.2. For all vector fields X, Y, Z ∈ X(M ) we have
b Yb , Z)
b = (iB(X,
b Yb )Z)E,
b
∇h g(X,
(64)
therefore the Berwald curvature and the Landsberg tensor of a Finsler manifold
are related by
∇v E ◦ B = −2P.
(65)
Proof.
b
b Yb , Z)
b = (∇X h g)(Yb , Z)
∇h g(X,
b − g(∇X h Yb , Z)
b − g(Yb , ∇X h Z)
b
= X h g(Yb , Z)
(47),(48)
=
X h (Y v (Z v E))) − (i∇X h Yb )Z v E + (i∇i∇
Xh
b
b Z)E
Y
b + J[Y v , H∇X h Z]E
b
− Y v (i∇X h Z)E
= X h (Y v (Z v E)) − [X h , Y v ]Z v E − Y v ([X h , Z v ]E)
+ J[Y v , HV[X h, Z v ]].
Here, as we have already seen in the proof of 3.1,
b Z
b = iB(X,
b Yb )Zb
J[Y v , HV[X h , Z v ]] = [Y v , [X h , Z v ]] = iB(Yb , X)
therefore
b Yb , Z)
b = (iB(X,
b Yb )Z)E
b + Y v (X h (Z v E))
∇h g(X,
b Yb )Z,
b
− Y v (X h (Z v E) − Z v (X h E)) = iB(X,
taking into account that H is conservative. Thus we have proved relation (64).
Relation (65) is merely a reformulation of (64).
Corollary 6.3. The Landsberg tensor of a Finsler manifold has the following
properties:
(i) it isntotally symmetric;
o
e Ye , Z
e ⇒ P(X,
e Ye , Z)
e = 0;
(ii) δ ∈ X,
(iii) ∇C P = 0, i.e., P is homogeneous of degree zero.
196
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Indeed, (i) and (ii) are immediate consequences of (64) and the corresponding
property of the Berwald tensor. Taking into account our calculations in the proof
of 3.5, we get for any vector fields X, Y , Z on M
b Yb )Z)E
b
b Yb , Z)
b = C(P(X,
b Yb , Z))
b = − 1 C(iB(X,
(∇C P) (X,
2
1
b Yb )Z]E
b + (iB(X,
b Yb )Z)CE)
b
= 0,
= − ([C, iB(X,
2
which proves (iii).
Corollary 6.4. If g is the metric tensor, S is the canonical spray of a Finsler
manifold, then ∇S g = 0.
◦
e Ye in Sec(π),
Proof. For any sections X,
e Ye ) = ∇hδ g (X,
e Ye ) = ∇h g(δ, X,
e Ye ) = −2P(δ, X,
e Ye ) 6.3=(ii) 0.
(∇S g) (X,
Now we can easily deduce an important relation between the Cartan tensor
and the Landsberg tensor of a Finsler manifold.
Proposition 6.5. ∇S C♭ = P.
Proof. Let X, Y , Z be vector fields on M . Applying the hv-Ricci formula (24),
property (53) and Corollary 6.3 (ii), we obtain:
b Yb , Z)
b = ∇hδ ∇v g(X,
b Yb , Z)
b = ∇h ∇v g(δ, X,
b Yb , Z)
b
2 (∇S C♭ ) (X,
b δ, Yb , Z)
b = ∇X v ∇h g(δ, Yb , Z)
b = −2 (∇X v P) (δ, Yb , Z)
b
= ∇v ∇h g(X,
b + 2P(∇X v δ, Yb , Z)
b = 2P(X,
b Yb , Z).
b
= −2X v P(δ, Yb , Z)
Now we are able to prove in an index-free manner Proposition 3.1 in [2].
Proposition 6.6. If the Landsberg tensor of a Finsler manifold depends only
on the position, then it vanishes identically, i.e., ∇v P = 0 implies that P = 0.
Proof. Applying the preceding Proposition, the Ricci formula (24), and taking
into account Corollary 3.4, for any vector fields X, Y , Z on M we have
b Yb , Z)
b = (∇S C♭ )(X,
b Yb , Z)
b = (∇h C♭ )(δ, X,
b Yb , Z)
b = 1 (∇h ∇v g)(δ, X,
b Yb , Z)
b
P(X,
2
1
b δ, Yb , Z)
b = −(∇v P)(X,
b δ, Yb , Z)
b = 0.
= (∇v ∇h g)(X,
2
By the stretch tensor of a Finsler manifold we mean the type 04 tensor Σ
◦
along τ given by
1
e Ye , Z,
e U
e ) := ∇h P(X,
e Ye , Z,
e U
e ) − ∇h P(Ye , X,
e Z,
e U)
e
Σ(X,
2
([2],[3]). Next we verify by an index-free argument the following result of [2]:
ON THE PROJECTIVE THEORY OF SPRAYS
197
Proposition 6.7. If the stretch tensor of a Finsler manifold depends only on
the position, then it vanishes identically, i.e., ∇v Σ = 0 implies that Σ = 0.
Proof. Let X, Y , Z, U , V be vector fields on M .
Step 1 By our assumption
b Yb , Z,
b U
b , Vb ) = X v (Σ(Yb , Z,
b U
b , Vb ))
0 = (∇v Σ)(X,
hence
b U
b , Vb ) − ∇h P(Z,
b Yb , U
b , Vb )))
= 2(X v (∇h P(Yb , Z,
b Yb , Z,
b U
b , Vb ) − ∇v ∇h P(X,
b Z,
b Yb , U
b , Vb ))),
= 2(∇v ∇h P(X,
b Yb , Z,
b U
b , Vb ) = ∇v ∇h P(X,
b Z,
b Yb , U
b , Vb ).
∇v ∇h P(X,
Since this is a tensorian relation, we also have
b Yb , Z,
b U
b , δ) = ∇v ∇h P(X,
b Z,
b Yb , U
b , δ).
(66)
∇v ∇h P(X,
Now, by the Ricci formula (24) and Corollary 6.3 (ii),
b Yb , Z,
b U
b , δ) = ∇v ∇h P(Yb , X,
b Z,
b U
b , δ)
∇h ∇v P(X,
(66)
b X,
b U
b , δ) (24)
b Yb , X,
b U
b , δ),
= ∇v ∇h P(Yb , Z,
= ∇h ∇v P(Z,
so ∇h ∇v P(., ., ., ., δ) is symmetric in its first and third variables:
(67)
b Yb , Z,
b U
b , δ) = ∇h ∇v P(Z,
b Yb , X,
b U
b , δ).
∇h ∇v P(X,
Step 2 We show that
b Yb , Z,
b U)
b + ∇h ∇v P(X,
b Yb , Z,
b U
b , δ) = 0.
(68)
∇h P(X,
b U
b , δ) = 0. Operating on both sides first by Y v ,
We start out the identity P(Z,
h
and next by X , we obtain
b U
b , δ)) = ∇v P(Yb , Z,
b U
b , δ) + P(Z,
b U
b , Yb )
0 = Y v (P(Z,
b U
b ) + ∇v P(Yb , Z,
b U
b , δ);
= P(Yb , Z,
b U
b )) + X h (∇v P(Yb , Z,
b U
b , δ))
0 = X h (P(Yb , Z,
b Yb , Z,
b U)
b + (∇h ∇v P)(X,
b Yb , Z,
b U
b , δ)
= (∇h P)(X,
b U)
b + P(Yb , ∇X h Z,
b U
b ) + P(Yb , Z,
b ∇X h U
b)
+ P(∇X h Yb , Z,
b U
b , δ) + ∇v P(Yb , ∇X h Z,
b U
b , δ) + ∇v P(Yb , Z,
b ∇X h U
b , δ).
+ ∇v P(∇X h Yb , Z,
Since, for example,
b U
b , δ) = [X h , Y v ]P(Z,
b U
b , δ) − P(Z,
b U
b , ∇[X h ,Y v ] δ)
∇v P(∇X h Yb , Z,
b U
b , V[X h , Y v ]) = −P(Z,
b U
b , ∇X h Yb ) = −P(∇X h Yb , Z,
b U
b ),
= −P(Z,
the last six terms cancel in pairs on the right-hand side of the above relation.
So we get (68).
198
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
b and Yb in (68), we find
Step 3 Interchanging X
b Z,
b U
b ) + ∇h ∇v P(Yb , X,
b Z,
b U
b , δ)
0 = ∇h P(Yb , X,
b Z,
b U
b ) + ∇h ∇v P(X,
b Yb , Z,
b U
b , δ),
= ∇h P(Yb , X,
since, by Step 1, the second term does not change under the permutations
b Z)
b → (Yb , Z,
b X)
b → (X,
b Z,
b Yb ) → (X,
b Yb , Z).
b
(Yb , X,
The last relation and (68) imply that
whence Σ = 0.
b Yb , Z,
b U)
b = ∇h P(Yb , X,
b Z,
b U
b)
∇h P(X,
Proposition 6.8. Let (M, F ) and (M, F ) be Finsler manifolds, and let the
geometric data arising from F be distinguished by bar. Suppose that the canonical sprays S and S of (M, F ) and (M, F ) are projectively related, namely
S = S − 2P C. Then
2P =
(69)
SF
,
F
and
∇S C ♭ = P C ♭ + P.
(70)
Proof. Since S is horizontal with respect to the canonical connection of (M, F ),
we obtain
0 = S F = (S − 2P C)F = SF − 2P F ,
so (69) is valid. To prove the second relation, let X, Y , Z be arbitrary vector
fields on M . Then, applying (41) and (53),
b Yb , Z)
b
∇S C ♭ (X,
b Yb , Z))
b − C ♭ (∇S X,
b Yb , Z)
b − C ♭ (X,
b ∇S Yb , Z)
b − C ♭ (X,
b Yb , ∇S Z)
b
= S(C ♭ (X,
b Yb , Z)
b
b Yb , Z))
b + 2P C(C ♭ (X,
b Yb , Z))
b − C ♭ (∇ X,
= S(C ♭ (X,
S
b ∇ Yb , Z)
b Yb , ∇ Z)
b Yb , Z).
b
b − C ♭ (X,
b + 3P C ♭ (X,
− C ♭ (X,
S
S
Since C ♭ is homogeneous of degree −1,
b Yb , Z)
b = −C ♭ (X,
b Yb , Z)
b = ∇C C ♭ (X,
b Yb , Z),
b
CC ♭ (X,
so we get
b Yb , Z)
b = ∇ C ♭ (X,
b Yb , Z)
b + P C ♭ (X,
b Yb , Z)
b 6.5
b Yb , Z),
b
∇S C ♭ (X,
= (P + P C ♭ )(X,
S
thus proving Proposition 6.8.
ON THE PROJECTIVE THEORY OF SPRAYS
199
7. Rapcsák’s equations: some consequences and applications
Following the terminology of [24] we say that a spray is Finsler-metrizable
in a broad sense or, after Z. Shen [16], projectively Finslerian, if there exists a
Finsler function whose canonical spray is projectively related to the given spray.
If, in particular, the projective relation is trivial in the sense that the projective
factor vanishes, then the spray will be called Finsler metrizable in a natural
sense or Finsler variational. In this section we deal with different conditions
concerning both types of metrizability of a spray.
Lemma 7.1. Let a spray S over M be given. Let H be the Ehresmann connection associated to S, and let h := H ◦ j be the horizontal projector associated to
◦
H. Then for any smooth function F on T M we have
(71)
2dh F = d(F − CF ) − iS ddJ F + dJ iS dF.
This important relation was found by J. Klein, see [8], section 3.2. Its validity
may be checked by brute force, evaluating both sides of (71) on vertical lifts X v
and complete lifts X c , X ∈ X(M ). It is possible, however, to verify (71) also by
a more elegant, completely ‘argumentum-free’ reasoning, see again [8], and [19].
Proposition 7.2. Let (M, F ) be a Finsler-manifold with energy function E :=
◦
1 2
2F
and canonical spray S, given by iS ddJ E = −dE on T M . Suppose S is a
further spray over M , and let H be the Ehresmann connection associated to S.
S is projectively related to S, if and only if, for each vector field X ∈ X(M )
we have
b
(R1 ) 2X hF = X v (SF ) (X h := HX).
Proof. Let H be the canonical connection of (M, F ). Suppose first that S and
◦
S are projectively related, namely S = S − 2P C, P ∈ C ∞ (T M ) ∩ C 1 (T M ). If
b X h = H(X),
b then
X ∈ X(M ), X h = H(X),
X h = X h − P X v − (X v P )C
. Since H is conservative, then
by (37). By Proposition 6.8 we have 2P = SF
F
we obtain:
SF v
SF
0 = 2X h F = (2X h −
X − X v(
)C)F
F
F
SF
1
1
= 2X h F −
(X v F ) − X v (SF )F + 2 (SF )(X v F )F = 2X h F − X v (SF ).
F
F
F
This proves the validity of (R1 ) if S and S are projectively related.
Conversely, suppose that (R1 ) is satisfied. Then, for all X ∈ X(M ),
(72)
(R )
2X hE = 2F (X h F ) =1 F (X v (SF )) = X v (SE) − (X v F )(SF ).
200
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Since E − CE = −E, we obtain by Lemma 7.1
(73)
dJ iS dE − 2dh E = iS ddJ E + dE.
Next we prove that the left-hand side of (73) equals to SF
i ddJ E.
F C
For any X ∈ X(M ) an easy calculation shows that, on the one hand,
(dJ iS dE − 2dh E)(X v ) = 0 =
SF
iC ddJ E(X v ).
F
On the other hand,
(dJ iS dE − 2dh E)(X c ) = JX c (iS dE) − 2(hX c )E
(72)
= X v (SE) − 2X h E = (X v F )(SF ),
while
SF
SF
iC ddJ E(X c ) =
ddJ E(C, X c )
F
F
SF
=
CdJ E(X c ) − X c dJ E(C) − dJ E[C, X c ]
F
SF
SF
C(X v E) =
([C, X v ]E + X v (CE))
=
F
F
SF
(X v E) = (SF )(X v F ),
=
F
hence
(74)
dJ iS dE − 2dh E =
SF
iC ddJ E,
F
as we claimed. (73) and (74) imply that
iS ddJ E + dE = i SF C ddJ E
F
whence
iS− SF C ddJ E = −dE.
F
◦
Since S is uniquely determined on T M by the ‘Euler-Lagrange equation’ iS ddJ E =
C, which proves the Proposition.
−dE, we conclude that S = S − SF
F
(R1 ) provides a necessary and sufficient condition for the Finsler-metrizability
of a spray in a broad sense. In terms of classical tensor calculus, it was first
formulated by A. Rapcsák [15], so it will be quoted as Rapcsák’s equation for F
with respect to S.
Now we derive a ‘more intrinsic’ expression of (R1 ), showing that it can also
be written in the form
(R2 ) ∇S ∇v F = ∇h F .
ON THE PROJECTIVE THEORY OF SPRAYS
201
Indeed, for any vector field X ∈ X(M ) we have
b = S(X v F ) − ∇v F (∇S X)
b
∇S ∇v F (X)
b
= S(X v F ) − (i∇S X)F
= S(X v F ) − v[S, X v ]F
(9)
= S(X v F ) − (vX c )F = S(X v F ) − X c F + X h F
= [S, X v ]F + X v (SF ) − X c F + X h F
(9)
= (X c − 2X h )F + X v (SF ) − X c F + X h F = X v (SF ) − X h F ,
hence
∇S ∇v F = ∇h F ⇔ X v (SF ) − X h F = X h F for all X ∈ X(M ).
This proves the equivalence of (R1 ) and (R2 ). Rapcsák equations (R1 ), (R2 )
have several further equivalents, we collect here some of them:
(R3 ) iS ddJ F = 0;
(R4 ) iδ ∇h ∇v F = ∇h F ;
(R5 ) dh dJ F = 0;
e Ye ) = ∇h ∇v F (Ye , X)
e ;
(R6 ) ∇h ∇v F (X,
v h
v h
e
e
e
e
(R7 ) ∇ ∇ F (X, Y ) = ∇ ∇ F (Y , X)
e and Ye are arbitrary sections along τ◦ ).
(in (R6 ) and (R7 ) X
Details on a proof of the equivalence of conditions (R1 )-(R7 ) can be found in
[24], [19], [20], [18]. We note only that the equivalence of (R6 ) and (R7 ) is an
immediate consequence of the Ricci identity (22), while the equivalence of (R2 )
and (R4 ) follows from the identity iδ ∇h ∇v F = ∇S ∇v F .
Proposition 7.3 (Criterion for Finsler variationality). Let S be a spray over
M , and let H be the Ehresmann connection associated to S. S is the canonical
spray of a Finsler manifold (M, F ), if and only if, dF ◦ H = 0.
Proof. The necessity is obvious since the canonical connection of (M, F ) is conservative. To prove the sufficiency, suppose that dF ◦ H = 0. Then for all
X ∈ X(M ) we have
(75)
b = dF (X h ) = X h F = 0.
dF ◦ H(X)
◦
Since the horizontal lifts X h , X ∈ X(M ) generate the C ∞ (T M )-module of Hhorizontal vector fields, this implies that for any H-horizontal vector field ξ on
◦
T M we have ξF = 0. In particular, S is also H-horizontal, so SF = 0 holds
too. Then Rapcsák’s equation (R1 ) is valid trivially: both sides of the relation
vanish identically. By Proposition 7.2, from this it follows that S and S are
projectively related:
◦
S = S − 2P C, P ∈ C 1 (T M ) ∩ C ∞ (T M ).
202
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
(69) 1 SF
2 F
However, P =
= 0, so we obtain the required equality S = S.
Remark. In his excellent textbook [10], written in a ‘semi-classical style’, D. Laugwitz formulates and proves the following theorem: The paths of a system (xi )′′ +
2H i (x, x′ ) = 0 (i ∈ {1, . . . n}) are the geodesics of a Finsler function F , if and
only if, F is invariant under the parallel displacement
dξ i
dxr
∂H i
+ Hri (x, ξ)
= 0, Hri :=
dt
dt
∂y r
associated with the sytem of paths. ([10], Theorem 15.8.1.) Here we slightly
modified Laugwitz’s formulation and notation. The ’system’, actually a SODE,
is given in a chart (τ −1 (U), (xi , y i )) on T M , induced by a chart (U, (ui )) on M :
xi := ui ◦ τ = (ui )v , y i := (ui )c ; i ∈ {1, . . . , n} .
It may be easily seen that our Proposition 7.3 is just an intrinsic reformulation
of Laugwitz’s metrization theorem. Laugwitz’s proof takes more than one page
and applies a totally different argument.
Corollary 7.4 (The uniqueness of the canonical connection.). Let (M, F ) be
a Finsler manifold. If H is a torsion-free, homogeneous Ehresmann connection
over M such that dF ◦ H = 0, then H is the canonical connection of (M, F ).
Proof. Since H is torsion-free and homogeneous, Corollary 6 in section 3 of [19]
assures that H is associated to a spray. Then the condition dF ◦ H = 0 implies
by the preceding Proposition that this spray is the canonical spray, and hence
H is the canonical connection of (M, F ).
Remark. The uniqueness proof presented here is based, actually, on the Rapcsák
equations. The idea that they may be applied also in this context is due to
Z. I. Szabó [17].
Our next results may be considered as necessary conditions for the Finsler
metrizability of a spray in a broad sense.
Proposition 7.5. Let S be a spray over M , and let ∇ = (∇h , ∇v ) be the Berwald
derivative induced by the Ehresmann connection associated to S. If a Finsler
function F : T M → R satisfies one (and hence all) of the Rapcsák equations with
respect to S, then
(76)
∇S ∇v ∇v F = 0.
ON THE PROJECTIVE THEORY OF SPRAYS
203
Proof. For any vector fields X, Y on M we have
b Yb ) = ∇h ∇v (∇v F )(δ, X,
b Yb )
∇S ∇v ∇v F (X,
(24)
b δ, Yb ) + ∇v F (B(X,
b δ)Yb )
= ∇v ∇h (∇v F )(X,
(21)
b δ, Yb ) = X v ∇h ∇v F (δ, Yb ) − ∇h ∇v F (X,
b Yb )
= ∇v ∇h (∇v F )(X,
(22)
b
= X v ∇v ∇h F (Yb , δ) − ∇v ∇h F (Yb , X)
= X v (Y v (SF ) − ∇h F (Yb )) − Y v (X h F )
(R1 )
= X v (2Y h F − Y h F ) − Y v (X h F )
b Yb ) − ∇v ∇h F (Yb , X)
b (R=7 ) 0.
= X v (Y h F ) − Y v (X h F ) = ∇v ∇h F (X,
Corollary 7.6. Under the assumptions of the Proposition above, let
(57)
µ := ∇v ∇v F =
1
η,
F
where η is the angular metric tensor of the Finsler manifold (M, F ). Then
∇S ∇v µ + ∇h µ = 0.
(77)
Proof. Let Y, Z ∈ X(M ). Then by Proposition 7.5,
b = ∇S ∇v ∇v F (Yb , Z)
b = 0.
∇h µ(δ, Yb , Z)
Operating on both sides by X v , where X ∈ X(M ), we obtain
b = ∇X v ∇h µ (δ, Yb , Z)
b + ∇h µ(X,
b Yb , Z)
b
0 = X v (∇h µ(δ, Yb , Z))
b δ, Yb , Z)
b + ∇h µ(X,
b Yb , Z)
b
= ∇v ∇h µ(X,
(24),(21)
=
b Yb , Z)
b + ∇h µ(X,
b Yb , Z)
b
∇h ∇v µ(δ, X,
b Yb , Z).
b
= (∇S ∇v µ + ∇h µ)(X,
Theorem 7.7. Let (M, F ) be a Finsler manifold with canonical spray S; let
v
λ := ∇FF , µ := ∇v ∇v F ; and let C ♭ and P be the Cartan and the Landsberg
tensor of (M, F ), respectively. If F satisfies one (and hence all) of the Rapcsák
equations with respect to a spray S, and P is the projective factor between S and
S, then
(78)
∇h µ = Sym(λ ⊗ µ) +
2
(P C ♭ − P).
F
204
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Proof.
h
(77)
v
(60)
∇ µ = −∇S ∇ µ = −2∇S
2SF
1
C♭
F
+ ∇S Sym(λ ⊗ µ)
2
∇S C ♭ + Sym(∇S λ ⊗ µ + λ ⊗ ∇S µ)
F
F
2P
2
(69),(70),(76) 4P
C♭ −
C♭ − P + Sym(∇S λ ⊗ µ)
=
F
F
F
2
= Sym(∇S λ ⊗ µ) + (P C ♭ − P).
F
=
2
C♭ −
Corollary 7.8. If a Finsler function F satisfies a Rapcsák equation with respect
to a spray S, and (∇h , ∇v ) is the Berwald derivative induced by S, then the tensor
∇h µ = ∇h ∇v ∇v F is totally symmetric.
Proof. The total symmetry of ∇h µ can be read from (78).
Remark. Relation (78) is an intrinsic, index and argumentum free version of
formula (2.4) in [1]. The total symmetry of ∇h µ (modulo a Rapcsák equation)
may also be verified immediately, independently of (78).
Proposition 7.9. Let S be a spray over M , and suppose that a Finsler function
F : T M → R satisfies a Rapcsák equation with respect to S. If R is the curvature
of the Ehresmann connection associated to H, then
(79)
S
b Y
b ,Z)
b
(X,
b Yb ), Z)
b = 0.
µ(R(X,
Proof. Let (∇h , ∇v ) be the Berwald derivative determined by H. First we show
that
b Yb , Z)
b
∇h ∇h ℓ♭ (X,
(80)
(ℓ♭ := ∇v F ; X, Y, Z ∈ X(M )) is symmetric in its last two arguments. Indeed,
b Yb , Z)
b = ∇X h (∇h ℓ♭ ) (Yb , Z)
b
∇h ∇h ℓ♭ (X,
b − ∇h ℓ♭ (∇X h Yb , Z)
b − ∇h ℓ♭ (Yb , ∇X h Z)
b
= X h (∇h ℓ♭ (Yb , Z))
(R6 )
b Yb )) − ∇h ℓ♭ (∇X h Z,
b Yb ) − ∇h ℓ♭ (Z,
b ∇X h Yb )
= X h (∇h ℓ♭ (Z,
b Yb ) = ∇h ∇h ℓ♭ (X,
b Z,
b Yb ),
= ∇X h (∇h ℓ♭ ) (Z,
which proves our claim.
Next we apply the Ricci identity (33) to (80):
b Yb , Z)
b = ∇h ∇h ℓ♭ (Yb , X,
b Z)
b − ℓ♭ (H(X,
b Yb )Z)
b − µ(R(X,
b Yb ), Z).
b
∇h ∇h ℓ♭ (X,
ON THE PROJECTIVE THEORY OF SPRAYS
205
b Yb and Z
b cyclically:
Interchanging X,
b X)
b = ∇h ∇h ℓ♭ (Z,
b Yb , X)
b − ℓ♭ (H(Yb , Z)
b X)
b − µ(R(Yb , Z),
b X),
b
∇h ∇h ℓ♭ (Yb , Z,
b X,
b Yb ) = ∇h ∇h ℓ♭ (X,
b Z,
b Yb ) − ℓ♭ (H(Z,
b X)
b Yb ) − µ(R(Z,
b X),
b Yb ).
∇h ∇h ℓ♭ (Z,
We add these three relations. Then, using the Bianchi identity (28) and the
symmetry of ∇h ∇h ℓ♭ in its last two variables, relation (79) drops.
Remark. In the language of classical tensor calculus, relation (79) was first formulated by A. Rapcsák [15]. For another index-free treatment, using Grifone’s
formalism, we refer to [24].
Lemma 7.10. Let a spray S : T M → T T M and a Finsler function
F : T M → R be given. If µ := ∇v ∇v F ; R is the curvature, and K is the Jacobi
endomorphism of the Ehresmann connection associated to S, then relation (79)
is equivalent to the condition
b Yb ) = µ(X,
b K(Yb )) ; X, Y ∈ X(M ).
(81)
µ(K(X),
Proof. We recall that by (30), K and R are related by
◦
e = R(X,
e δ) , X
e ∈ Sec(π).
K(X)
(79)⇒(81) By assumption, for any vector fields X, Y on M we have
b δ), Yb ) + µ(R(δ, Yb ), X)
b + µ(R(Yb , X),
b δ) = 0,
µ(R(X,
or, equivalently,
b Yb ) − µ(X,
b K(Yb )) = µ(R(X,
b Yb ), δ).
µ(K(X),
We show that the right-hand side vanishes.
b Yb ), δ) = ∇v ∇v F (δ, R(X,
b Yb ))
µ(R(X,
b Yb )) = C(iR(X,
b Yb )F ) − ∇v F (∇C (R(X,
b Yb )))
= (∇C ∇v F )(R(X,
b Yb ))
b Yb )F ) − ∇v F (∇C R(X,
= C(iR(X,
(4.3)
b Yb )F ) − iR(X,
b Yb )F
= C(iR(X,
b Yb )]F = [C, [X, Y ]h − [X h , Y h ]]
= [C, iR(X,
= −[C, [X h , Y h ]] = [X h , [Y h , C]] + [Y h , [C, X h ]] = 0,
taking into account the homogeneity of the associated Ehresmann connection.
b =
(81)⇒(79) We operate by X v on both sides of the relation µ(K(Yb ), Z)
b and permute the variables cyclically. Then we obtain:
µ(Yb , K(Z)),
b = X v (µ(Yb , K(Z))),
b
X v (µ(K(Yb ), Z))
b X))
b = Y v (µ(Z,
b K(X))),
b
Y v (µ(K(Z),
b Yb )) = Z v (µ(X,
b K(Yb ))).
Z v (µ(K(X),
206
SÁNDOR BÁCSÓ AND ZOLTÁN SZILASI
Applying the product rule,
b K(Yb ), Z)
b − ∇v µ(X,
b Yb , K(Z))
b = µ(Yb , ∇v K(X,
b Z))
b − µ(∇v K(X,
b Yb ), Z),
b
∇v µ(X,
b X)
b − ∇v µ(Yb , Z,
b K(X))
b = µ(Z,
b ∇v K(Yb , X))
b − µ(∇v K(Yb , Z),
b X),
b
∇v µ(Yb , K(Z),
b K(X),
b Yb ) − ∇v µ(Z,
b X,
b K(Yb )) = µ(X,
b ∇v K(Z,
b Yb )) − µ(∇v K(Z,
b X),
b Yb ).
∇v µ(Z,
Now we add these three relations. Since ∇v µ = ∇v ∇v ∇v F is totally symmetric,
we obtain
b − ∇v K(X,
b Yb ), Z)
b + µ(∇v K(Z,
b Yb )
0 = µ(∇v K(Yb , X)
b X)
b + µ(∇v K(X,
b Z)
b − ∇v K(Z,
b X),
b Yb )
− ∇v K(Yb , Z),
(4.6)
b Yb ), Z)
b + µ(R(Yb , Z),
b X)
b + µ(R(Z,
b X),
b Yb ))
= 3(µ(R(X,
=3
S
b Y
b ,Z)
b
(X,
b Yb ), Z).
b
µ(R(X,
Corollary 7.11 (The self-adjointness condition). If a Finsler function F : T M →
R satisfies a Rapcsák equation with respect to a spray over M , then the Jacobi
endomorphism K determined by the spray is self-adjoint with respect to the symmetric type 02 tensor µ = ∇v ∇v F , i.e.,
◦
e Ye ) = µ(X,
e K(Ye )) ; X,
e Ye ∈ Sec(π).
µ(K(X),
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Faculty of Informatics,
University of Debrecen,
H-4010 Debrecen, Hungary
E-mail address: bacsos@inf.unideb.hu
Institute of Mathematics,
University of Debrecen,
H-4010 Debrecen, Hungary
E-mail address: szilasi.zoltan@inf.unideb.hu