<< . .

. 42
( : 71)



. . >>

Since this construction has geometrical character, V is an operator J 1 J 1V
natural on the category FMm,n .
Proposition. The vertical prolongation V is the only natural operator J 1
J 1V .
We start the proof with ¬nding the equations of V“. If

dy p = Fip (x, y)dxi
(2)
is the coordinate form of “ and Y p = dy p are the additional coordinates on V Y ,
then (1) implies that the equations of V“ are (2) and
‚Fip q i
p
(3) dY = Y dx .
‚y q
31.2. The standard ¬ber of V on the category FMm,n is Rn . Let S1 =
J0 (J 1 (Rm — Rn ’ Rm ) ’ Rm — Rn ) and Z = J0 (V (Rm — Rn ) ’ Rm ),
1 1

0 ∈ Rm — Rn . By 18.19, the ¬rst order natural operators are in bijection with
G2 -maps S1 — Rn ’ Z over the identity of Rn . The canonical coordinates on
m,n
p p p
S1 are yi , yiq , yij and the action of G2 can be found in 27.3. The action of
m,n
2 n
Gm,n on R is
¯
Y p = ap Y q .
(1) q

The coordinates on Z are Y p , zi = ‚y p /‚xi , Yip = ‚Y p /‚xi . By standard
p

evaluation we ¬nd the following action of G2
m,n

zi = ap zj aj + ap aj
¯p q
¯
Y p = ap Y q , ˜i j ˜i
q q
(2)
Yip = ap Yjq aj + ap zj Y q aj + ap Y q aj .
¯ r
˜i ˜i ˜i
q qr qj


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
256 Chapter VII. Further applications


The coordinate form of any map S1 —Rn ’ Z over the identity of Rn is Y p = Y p
and

zi = fip (Y q , yj , ykt , ylm )
p rs u

Yip = gi (Y q , yj , ykt , ylm ).
p rs u


First we discuss fip . The equivariance with respect to base homotheties yields
kfip = fip (Y q , kyj , kykt , k 2 ylm ).
r s u
(3)
By the homogeneous function theorem, if we ¬x Y q , then fip is linear in yj , ykt
r s
u
and independent of ylm . The ¬ber homotheties then give
kfip = fip (kY q , kyj , ykt ).
rs
(4)
By (3) and (4), fip is a sum of an expression linear in yi and bilinear in Y p and
p

yiq . Since fip is GL(m) — GL(n)-equivariant, the generalized invariant tensor
p

theorem implies it has the following form
p q p
ayi + bY p yqi + cY q yqi .
(5)
The equivariance on the kernel K of the projection G2 ’ G1 — G1 yields
m,n m n

ap = aap + bY p (aq + aq yi ) + cY q (ap + ap yi ).
r r
(6) qr qr
i i qi qi

This implies a = 1, b = c = 0, which corresponds to 31.1.(2).
p
For gi , the above procedure leads to the same form (5). Then the equivariance
with respect to K yields a = b = 0, c = 1. This corresponds to 31.1.(3). Thus
we have proved that V is the only ¬rst order natural operator J 1 J 1V .
31.3. By 23.7, every natural operator A : J 1 J 1 V has a ¬nite order r. Let f =
(fip , gi ) : Sr — Rn ’ Z be the associated map of A, where S r = J0 (J 1 (Rm+n ’
p r

Rm ) ’ Rm+n ). Consider ¬rst fip (Y q , yj±β ) with the same notation as in the
r

second step of the proof of proposition 27.3. The base homotheties yield
kfip = fip (Y q , k |±|+1 yj±β ).
r
(1)
By the homogeneous function theorem, if we ¬x Y p , then fip are independent
p p
of yi±β with |±| ≥ 1 and linear in yiβ . Hence the only s-th order term is
•s = •pjβ (Y r )yjβ , |β| = s, s ≥ 2. Using ¬ber homotheties we ¬nd that •s is of
q
iq
degree s in Y p . Then the generalized invariant tensor theorem implies that •s
is of the form
p q1
as yiq1 ...qs Y q1 . . . Y qs + bs Y p yiq1 q2 ...qs Y q2 . . . Y qs .
Consider the equivariance with respect to the kernel of the jet projection Gr+1 ’
m,n
Gr . Using induction we deduce the transformation law
m,n

yiq1 ...qr = yiq1 ...qr + ap 1 ...qr yi + ap 1 ...qr ,
¯p p t
tq iq

while the lower order terms remain unchanged. By direct evaluation we ¬nd
p
ar = br = 0. The same procedure takes place for gi . Hence A is an operator
of order r ’ 1. By recurrence we conclude A is a ¬rst order operator. This
completes the proof of proposition 31.1.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
31. Two problems on general connections 257


31.4. Remark. In [Kol´ˇ, 81a] it was clari¬ed geometrically that the vertical
ar
prolongation V“ plays an important role in the theory of the original connec-
tion “. The uniqueness of V“ proved in proposition 31.1 gives a theoretical
justi¬cation of this phenomenon.
It is remarkable that there is another construction of V“ using ¬‚ow pro-
longations of vector ¬elds, see 45.4. The equivalence of both de¬nitions is an
interesting consequence of the uniqueness of operator V.
31.5. Natural operators transforming second order di¬erential equa-
tions into general connections. We recall that a second order di¬erential
equation on a manifold M is usually de¬ned as a vector ¬eld ξ : T M ’ T T M on
T M satisfying T pM —¦ ξ = idT M , where pM : T M ’ M is the bundle projection.
Let LM be the Liouville vector ¬eld on T M , i.e. the vector ¬eld generated by
the homotheties. If [ξ, LM ] = ξ, then ξ is said to be a spray. There is a classi-
cal bijection between sprays and linear symmetric connections, which is used in
several branches of di¬erential geometry. (We shall obtain it as a special case of
a more general construction.)
A. Dekr´t, [Dekr´t, 88], studied the problem whether an arbitrary second
e e
order di¬erential equation on M determines a general connection on T M by
means of the naturality approach. He deduced rather quickly a simple analytical
expression for all ¬rst order natural operators. Only then he looked for the
geometrical interpretation. Keeping the style of this book, we ¬rst present the
geometrical construction and then we discuss the naturality problem.
According to 9.3, the horizontal projection of a connection “ on an arbitrary
¬bered manifold Y is a vector valued 1-form on Y , which will be called the
horizontal form of “.
On the tangent bundle T M , we have the following natural tensor ¬eld VM of
type 1 . Since T M is a vector bundle, its vertical tangent bundle is identi¬ed
1
with T M • T M . For every B ∈ T T M we de¬ne

(1) VM (B) = (pT M B, T pM B).

(A general approach to natural tensor ¬elds of type 1 on an arbitrary Weil
1
bundle is explained in [Kol´ˇ, Modugno, 92].)
ar
Given a second order di¬erential equation ξ on M , the Lie derivative Lξ VM
is a vector valued 1-form on T M . Let 1T T M be the identity on T T M . The
following result gives a construction of a general connection on T M determined
by ξ.
Lemma. For every second order di¬erential equation ξ on M , 1 (1T T M ’Lξ VM )
2
is the horizontal form of a connection on T M .
Proof. Let xi be local coordinates on M and y i = dxi be the induced coordinates
on T M . The coordinate expression of the horizontal form of a connection on
T M is
‚ ‚
+ Fij (x, y)dxi — j .
dxi —
(2)
‚xi ‚y

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
258 Chapter VII. Further applications


By (1), the coordinate expression of VM is


dxi —
(3) .
‚y i

Having a second order di¬erential equation ξ of the form

‚ ‚
yi + ξ i (x, y) i
(4)
‚xi ‚y

we evaluate directly for Lξ VM

‚ξ i j
‚ ‚ ‚
’dx — i ’ j dx — i + dy i — i .
i
(5)
‚x ‚y ‚y ‚y

Hence 1 (1T M ’ Lξ VM ) has the required form
2

1 ‚ξ i j
‚ ‚
i
dx — i + dx — i .
(6)
2 ‚y j
‚x ‚y

31.6. Denote by A the operator from lemma 31.5. By the general theory, the
di¬erence of two general connections on T M ’ M is a section T M ’ V T M —
T — M = (T M • T M ) — T — M . The identity tensor of T M — T — M determines a
natural section IM : T M ’ V T M — T — M . Hence A + kI is a natural operator
for every k ∈ R.
Proposition. All ¬rst order natural operators transforming second order dif-
ferential equations on a manifold into connections on the tangent bundle form
the one-parameter family
k ∈ R.
A + kI,

Proof. We have the case of a morphism operator from 18.17 with C = Mfm ,
F1 = G1 = T , q = id, F2 = T1 , G2 = J 1 T and the additional conditions
2

that we consider the sections of T1 ’ T and J 1 T ’ T . Let S be the ¬ber of
2

J 1 (T1 Rm ’ T Rm ) over 0 ∈ Rm and Z be the ¬ber of J 1 T Rm over 0 ∈ Rm . By
2

18.19 we have to determine all G3 -equivariant maps f : S ’ Z over the identity
m
of T0 Rm .
i 2i
Denote by X i = dx , Y i = ddtx the induced coordinates on T1 Rm and by
2
2
dt
Xj = ‚Y i /‚xj , Yji = ‚Y i /‚X j the induced coordinates on S. By direct evalu-
i

ation, one ¬nds the following action of G3m


¯ ¯
X i = ai X j , Y i = ai X j X k + ai Y j
(1) j jk j
¯i
Xj = ai am X k X l + ai al Y k + 2ai ak am X l X n + ai Xlk al +
(2) klm ˜j kl ˜j kl ˜mj n ˜j
k

ai al am X n Ylk
k ˜mj n
¯
Yji = 2ai al X k + ai Ylk al
(3) kl ˜j ˜j
k


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
32. Jet functors 259


The standard coordinates Z i , Zj on Z have the transformation law
i


¯ ¯i
Z i = ai Z j , Zj = ai ak Z l + ai Zlk al .
(4) kl ˜j ˜j
j k


Let Z i = X i and Zj = fj (X p , Y q , Xn , Ylk ) be the coordinate expression of
i i m

f . The equivariance with respect to the homotheties in GL(m) ‚ G3 yields m

fj (X p , Y q , Xn , Ylk ) = fj (kX p , kY q , Xn , Ylk ).
i m i m
(5)

Hence fj do not depend on X p and Y q . Let ai = δj , ai = 0. Then the
i i
j jk
equivariance condition reads

fj (Xn , Ylk ) = fj (Xn + am X p X q , Ylk ).
i m i m
(6) npq


This implies fj are independent of Xn . Putting ai = δj , we obtain
i m i
j


fj (Ylk ) + ai X m = fj (Ylk + 2ak X m )
i i
(7) mj lm


with arbitrary ai . Di¬erentiating with respect to Ylk , we ¬nd ‚fj /‚Ylk = const.
i
jk
i
Hence fj are a¬ne functions. By the Invariant tensor theorem, we deduce

fj = k1 Yji + k2 δj Yk + k3 δj .
i ik i
(8)

1
Using (7) once again, we obtain k1 = 2 , k2 = 0. This gives the coordinate form
of our assertion.
31.7. Remark. If X is a spray, then the operator A from lemma 31.5 deter-
mines the classical linear symmetric connection induced by X. Indeed, 31.5.(4)
satis¬es the spray condition if and only if

‚ξ i j
y = 2ξ i .
‚y j

This kind of homogeneity implies ξ i = bi (x)y j y k . Then the coordinate form of
jk
A(X) is
dy i = bi (x)y j dxk .
jk



32. Jet functors

32.1. By 12.4, the construction of r-jets of smooth maps can be viewed as
a bundle functor J r on the product category Mfm — Mf . We are going to
determine all natural transformations of J r into itself. Denote by y : M ’ N
ˆ
r
the constant map of M into y ∈ N . Obviously, the assignment X ’ j±X βX
is a natural transformation of J r into itself called the contraction. For r = 1,
J 1 (M, N ) coincides with Hom(T M, T N ), which is a vector bundle over M — N .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
260 Chapter VII. Further applications


Proposition. For r ≥ 2 the only natural transformations J r ’ J r are the
identity and the contraction. For r = 1, all natural transformations J 1 ’ J 1
form the one-parametric family of homotheties X ’ cX, c ∈ R.
Proof. Consider ¬rst the subcategory Mfm —Mfn ‚ Mfm —Mf . The standard
¬ber S = J0 (Rm , Rn )0 is a Gr — Gr -space and the action of (A, B) ∈ Gr — Gr
r
m n m n
on X ∈ S is given by the jet composition
¯
X = B —¦ X —¦ A’1 .
(1)

According to the general theory, the natural transformations J r ’ J r are in
bijection with the Gr — Gr -equivariant maps f : S ’ S.
m n
p p
Write A’1 = (˜i , . . . , ai 1 ...jr ), B = (bp , . . . , bp1 ...qr ), X = (Xi , . . . , Xi1 ...ir ) =
aj ˜j q q
(X1 , . . . , Xr ). Consider the equivariance of f = (f1 , . . . , fr ) with respect to the
homotheties in GL(m) ‚ Gr . This gives the homogeneity conditions
m

kf1 (X1 , . . . , Xs , . . . , Xr ) = f1 (kX1 , . . . , k s Xs , . . . , k r Xr )
.
.
.
k s fs (X1 , . . . , Xs , . . . , Xr ) = fs (kX1 , . . . , k s Xs , . . . , k r Xr )
(2)
.
.
.
k r fr (X1 , . . . , Xs , . . . , Xr ) = fr (kX1 , . . . , k s Xs , . . . , k r Xr ).

Taking into account the homotheties in GL(n), we further ¬nd

kf1 (X1 , . . . , Xr ) = f1 (kX1 , . . . , kXr )
.
.
(3) .
kfr (X1 , . . . , Xr ) = fr (kX1 , . . . , kXr ).

Applying the homogeneous function theorem to both (2) and (3), we deduce that
fs is linear in Xs and independent of the remaining coordinates, s = 1, . . . , r.
Consider furthemore the equivariance with respect to the subgroup GL(m) —

<< . .

. 42
( : 71)



. . >>