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r r r r
m n m,n m,n
Every local di¬eomorphism f : M ’ N induces a map P r f : P r M ’ P r N
by P r f (j0 •) = j0 (f —¦ •). Since Gr acts on the right on both P r M and P r N ,
r r
P r f is a local principal ¬ber bundle isomorphism. Hence P r is a functor from
Mfm into the category PB(Gr ). m
Given a left action of Gr on a manifold S, we have an induced map

{P r f, idS } : P r M [S] ’ P r N [S]

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
12. Jets 123

between the associated ¬ber bundles with standard ¬ber S, see 10.9. The rule
M ’ P r M [S], f ’ {P r f, idS } is a bundle functor on Mfm as de¬ned in 14.1. A
very interesting result is that every bundle functor on Mfm is of this type. This
will be proved in section 22, but the proof involves some rather hard analytical
12.13. For every Lie group G, Tk G is also a Lie group with multiplication
(j0 f (u))(j0 g(u)) = j0 (f (u)g(u)), u ∈ Rk , where f (u)g(u) is the product in
r r r

G. Clearly, if we consider the multiplication map µ : G — G ’ G, then the
r r r r r
multiplication map of Tk G is Tk µ : Tk G — Tk G ’ Tk G. The jet projections
r r s
πs : Tk G ’ Tk G are group homomorphisms. For s = 0, there is a splitting
r r r r
ι : G ’ Tk G of π0 = β : Tk G ’ G de¬ned by ι(g) = j0 g , where g means the
ˆ ˆ
k r
constant map of R into g ∈ G. Hence Tk G is a semidirect product of G and of
the kernel of β : Tk G ’ G.
r r
If G acts on the left on a manifold M , then Tk G acts on Tk M by
r r r
(j0 f (u))(j0 g(u)) = j0 f (u)(g(u)) ,

where f (u)(g(u)) means the action of f (u) ∈ G on g(u) ∈ M . If we consider
the action map : G — M ’ M , then the action map of the induced action is
r r r r
Tk : Tk G — Tk M ’ Tk M . The same is true for right actions.
12.14. r-th order tangent vectors. In general, consider the dual vector
bundle Tk M = (Tk M )— of the (k, r)-covelocities bundle on M . For every map
r r—
r r—
f : M ’ N the jet composition A ’ A —¦ (jx f ), x ∈ M , A ∈ (Tk N )f (x) de¬nes
a linear map »(jx f ) : (Tk N )f (x) ’ (Tk M )x . The dual map (»(jx f ))— =:
r r— r— r
r r r r
(Tk f )x : (Tk M )x ’ (Tk N )f (x) determines a functor Tk on Mf with values
in the category of vector bundles. For r > 1 these functors do not preserve
products by the dimension argument. In the most important case k = 1 we shall
write T1 = T (r) (in order to distinguish from the r-th iteration of T ). The

elements of T (r) M are called r-th order tangent vectors on M . We remark that
for r = 1 the formula T M = (T — M )— can be used for introducing the vector
bundle structure on T M .
Dualizing the exact sequence 12.10.(1), we obtain

0 ’ T (r’1) M ’ T (r) M ’ S r T M ’ 0.
’ ’ ’ ’

This shows that there is a natural injection of the (r ’1)-st order tangent vectors
into the r-th order ones. Analyzing the proof of 12.10.(1), one ¬nds easily that
(1) has functorial character, i.e. for every map f : M ’ N the following diagram

wT wT w S TM w0
(r’1) (r) r
0 M M

u u u
T (r’1) f T (r) f
(2) S Tf

wT wT w S TN w0
(r’1) (r) r
0 N N

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
124 Chapter IV. Jets and natural bundles

12.15. Contact elements. Let N be an n-dimensional submanifold of a man-
ifold M . For every local chart • : N ’ Rn , the rule x ’ •’1 (x) considered as a
map Rn ’ M is called a local parametrization of N . The concept of the contact
of submanifolds of the same dimension can be reduced to the concept of r-jets.
De¬nition. Two n-dimensional submanifolds N and N of M are said to have
r-th order contact at a common point x, if there exist local parametrizations
¯ ¯ r¯
ψ : Rn ’ M of N and ψ : Rn ’ M of N , ψ(0) = x = ψ(0), such that j0 ψ = j0 ψ.

An equivalence class of n-dimensional submanifolds of M will be called an
n-dimensional contact element of order r on M , in short a contact (n, r)-element
on M . We denote by Kn M the set of all contact (n, r)-elements on M . We have
a canonical projection ˜point of contact™ Kn M ’ M .
An (n, r)-velocity A ∈ (Tn M )x is called regular, if its underlying 1-jet corre-
sponds to a linear map Rn ’ Tx M of rank n. For every local parametrization
ψ of an n-dimensional submanifold, j0 ψ is a regular (n, r)-velocity. Since in
the above de¬nition we can reparametrize ψ and ψ in the same way (i.e. we
compose them with the same origin preserving di¬eomorphism of Rm ), every
contact (n, r)-element on M can be identi¬ed with a class A —¦ Gr , where A is
a regular (n, r)-velocity on M . There is a unique structure of a smooth ¬bered
manifold on Kn M ’ M with the property that the factor projection from the
r r r
subbundle regTn M ‚ Tn M of all regular (n, r)-velocities into Kn M is a surjec-
tive submersion. (The simplest way how to check it is to use the identi¬cation
of an open subset in Kn Rm with the r-th jet prolongation of ¬bered manifold

Rn — Rm’n ’ Rn , which will be described in the end of 12.16.)
Every local di¬eomorphism f : M ’ M preserves the contact of submanifolds.

r r
This induces a map Kn f : Kn M ’ Kn M , which is a ¬bered manifold morphism
r 1
over f . Hence Kn is a bundle functor on Mfm . For r = 1 each ¬ber (Kn M )x
coincides with the Grassmann manifold of n-planes in Tx M , see 10.5. That is
why Kn M is also called the Grassmannian n-bundle of M .
12.16. Jet prolongations of ¬bered manifolds. Let p : Y ’ M be a ¬bered
manifold, dim M = m, dim Y = m+n. The set J r Y (also written as J r (Y ’ M )
or J r (p : Y ’ M ), if we intend to stress the base or the bundle projection) of
all r-jets of the local sections of Y will be called the r-th jet prolongation of Y .
Using polynomial representatives we ¬nd easily that an element X ∈ Jx (M, Y )
belongs to J r Y if and only if (jβX p) —¦ X = jx (idM ). Hence J r Y ‚ J r (M, Y ) is a
r r

closed submanifold. For every section s of Y ’ M , j r s is a section of J r Y ’ M .
Let xi or y p be the canonical coordinates on Rm or Rn , respectively. Every
local ¬ber chart • : U ’ Rm+n on Y identi¬es (π0 )’1 (U ) with J r (Rm , Rn ). This

de¬nes the induced local coordinates y± on J r Y , 1 ¤ |±| ¤ r, where ± is any

multi index of range m.
Let q : Z ’ N be another ¬bered manifold and f : Y ’ Z be an FM-
morphism with the property that the base map f0 : M ’ N is a local dif-
feomorphism. Then the map J r (f, f0 ) : J r (M, Y ) ’ J r (N, Z) constructed in
12.4 transforms J r Y into J r Z. Indeed, X ∈ J r Y , βX = y is characterized
r r r r
by (jy p) —¦ X = jx idM , x = p(y), and q —¦ f = f0 —¦ p implies jf (y) q —¦ (jy f ) —¦
’1 ’1
r r r r r
X —¦ (jf0 (x) f0 ) = (jx f0 ) —¦ (jy p) —¦ X —¦ jf0 (x) f0 = jf0 (x) idN . The restricted

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
12. Jets 125

map will be denoted by J r f : J r Y ’ J r Z and called the r-th jet prolongation
of f . Let FMm denote the category of ¬bered manifolds with m-dimensional
bases and their morphisms with the additional property that the base maps are
local di¬eomorphisms. Then the construction of the r-th jet prolongations can
be interpreted as a functor J r : FMm ’ FM. (If there will be a danger of
confusion with the bifunctor J r of spaces of r-jets between pairs of manifolds,
we shall write J¬b for the ¬bered manifolds case.)
By proposition 12.11, πr’1 : J r (M, Y ) ’ J r’1 (M, Y ) is an a¬ne bundle,

the associated vector bundle of which is the pullback of T Y — S r T — M over
J r’1 (M, Y ). Taking into account the local trivializations of Y , we ¬nd that
πr’1 : J r Y ’ J r’1 Y is an a¬ne subbundle of J r (M, Y ) and its modelling vector

bundle is the pullback of V Y — S r T — M over J r’1 Y , where V Y denotes the
vertical tangent bundle of Y . For r = 1 it is useful to give a direct description
of the a¬ne bundle structure on J 1 Y ’ Y because of its great importance in
the theory of connections. The space J 1 (M, Y ) coincides with the vector bundle
T Y — T — M = L(T M, T Y ). A 1-jet X : Tx M ’ Ty Y , x = p(y), belongs to J 1 Y
if and only if T p —¦ X = idTx M . The kernel of such a projection induced by T p is
— —
Vy Y — Tx M , so that the pre-image of idTx M in Ty Y — Tx M is an a¬ne subspace

with modelling vector space Vy Y — Tx M .
If we specialize corollary 12.11 to the case of a ¬bered manifold Y , we deduce
that for every X ∈ J r Y the kernel of the restriction of T πr’1 : T J r Y ’ T J r’1 Y

to TX J r Y is VβX Y — S r T±X M .
In conclusion we describe the relation between the contact (n, r)-elements
on a manifold M and the elements of the r-th jet prolongation of a suitable
local ¬bration on M . In a su¬ciently small neighborhood U of an arbitrary
x ∈ M there exists a ¬bration p : U ’ N over an n-dimensional manifold N .
By the de¬nition of contact elements, every X ∈ Kn M transversal to p (i.e.
the underlying contact 1-element of X is transversal to p) is identi¬ed with an
element of J r (U ’ N ) and vice versa. In particular, if we take U ∼ Rn — Rm’n ,
then the latter identi¬cation induces some simple local coordinates on Kn M .
12.17. If E ’ M is a vector bundle, then J r E ’ M is also a vector bundle,
r r r
provided we de¬ne jx s1 (u) + jx s2 (u) = jx (s1 (u) + s2 (u)), where u belongs to a
r r
neighborhood of x ∈ M , and kjx s(u) = jx ks(u), k ∈ R.
Let Z ’ M be an a¬ne bundle with the modelling vector bundle E ’ M .
Then J r Z ’ M is an a¬ne bundle with the modelling vector bundle J r E ’ M .
Given jx s ∈ J r Z and jx σ ∈ J r E, we set jx s(u)+jx σ(u) = jx (s(u)+σ(u)), where
r r r r r

the sum s(u) + σ(u) is de¬ned by the canonical map Z —M E ’ Z.
12.18. In¬nite jets. Consider an in¬nite sequence
(1) A1 , A2 , . . . , Ar , . . .
of jets Ai ∈ J i (M, N ) satisfying Ai = πi (Ai+1 ) for all i = 1, . . . . Such a
sequence is called a jet of order ∞ or an in¬nite jet of M into N . Hence the set
J ∞ (M, N ) of all in¬nite jets of M into N is the projective limit of the sequence
π2 π3 π r+1
J 1 (M, N ) ←1 J 2 (M, N ) ←2 . . . ← ’ J r (M, N ) ←r ’ . . .

’ ’
’ ’’ ’’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
126 Chapter IV. Jets and natural bundles

We denote by πr : J ∞ (M, N ) ’ J r (M, N ) the projection transforming the

sequence (1) into its r-th term. In this book we usually treat J ∞ (M, N ) as a
set only, i.e. we consider no topological or smooth structure on J ∞ (M, N ). (For
the latter subject the reader can consult e.g. [Michor, 80].)
Given a smooth map f : M ’ N , the sequence

1 2 r
jx f ← jx f ← · · · ← jx f ← . . .

x ∈ M , which is denoted by jx f or j ∞ f (x), is called the in¬nite jet of f at

x. The classical Borel theorem, see 19.4, implies directly that every element of
J ∞ (M, N ) is the in¬nite jet of a smooth map of M into N , see also 19.4.

The spaces Tk M of all k-dimensional velocities of in¬nite order and the in¬-
nite di¬erential group G∞ in dimension m are de¬ned in the same way. Having
a ¬bered manifold Y ’ M , the in¬nite jets of its sections form the in¬nite jet
prolongation J ∞ Y of Y .

12.19. Jets of ¬bered manifold morphisms. If we consider the jets of mor-
phisms of ¬bered manifolds, we can formulate additional conditions concerning
the restrictions to the ¬bers or the induced base maps. In the ¬rst place, if we
have two maps f , g of a ¬bered manifold Y into another manifold, we say they
determine the same (r, s)-jet at y ∈ Y , s ≥ r, if

r r s s
jy f = jy g and jy (f |Yx ) = jy (g|Yx ),

where Yx is the ¬ber passing through y. The corresponding equivalence class will
be denoted by jy f . Clearly (r, s)-jets of FM-morphisms form a category, and
the bundle projection determines a functor from this category into the category
of r-jets. We denote by J r,s (Y, Y ) the space of all (r, s)-jets of the ¬bered
manifold morphisms of Y into another ¬bered manifold Y .
Moreover, let q ≥ r be another integer. We say that two FM-morphisms
f, g : Y ’ Y determine the same (r, s, q)-jet at y, if it holds (1) and

q q
(2) jx Bf = jx Bg,

where Bf and Bg are the induced base maps and x is the projection of y to the
base BY of Y . We denote by jy f such an equivalence class and by J r,s,q (Y, Y )

the space of all (r, s, q)-jets of the ¬bered manifold morphisms between Y and
Y . The bundle projection determines a functor from the category of (r, s, q)-jets
of FM-morphisms into the category of q-jets. Obviously, it holds

¯ ¯ ¯
J r,s,q (Y, Y ) = J r,s (Y, Y ) —J r (BY,B Y ) J q (BY, B Y )
(3) ¯

¯ ¯
where we consider the above mentioned projection J r,s (Y, Y ) ’ J r (BY, B Y )
¯ ¯
and the jet projection πr : J q (BY, B Y ) ’ J r (BY, B Y ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
12. Jets 127

12.20. An abstract characterization of the jet spaces. We remark that
[Kol´ˇ, to appear c] has recently deduced that the r-th order jets can be charac-
terized as homomorphic images of germs of smooth maps in the following way.
According to 12.3, the rule j r de¬ned by

j r (germx f ) = jx f

transforms germs of smooth maps into r-jets and preserves the compositions.
By 12.6, J r (M, N ) is a ¬bered manifold over M — N for every pair of manifolds
M , N . So if we denote by G(M, N ) the set of all germs of smooth maps of M
into N , j r can be interpreted as a map

j r = jM,N : G(M, N ) ’ J r (M, N ).

More generally, consider a rule F transforming every pair M , N of mani-
folds into a ¬bered manifold F (M, N ) over M — N and a system • of maps
•M,N : G(M, N ) ’ F (M, N ) commuting with the projections G(M, N ) ’ M —
N and F (M, N ) ’ M — N for all M , N . Let us formulate the following require-
ments I“IV.
I. Every •M,N : G(M, N ) ’ F (M, N ) is surjective.
¯¯ ¯
II. For every pairs of composable germs B1 , B2 and B1 , B2 , •(B1 ) = •(B1 )
¯ ¯ ¯
and •(B2 ) = •(B2 ) imply •(B2 —¦ B1 ) = •(B2 —¦ B1 ).
By I and II we have a well de¬ned composition (denoted by the same symbol
as the composition of germs and maps)

X2 —¦ X1 = •(B2 —¦ B1 )

for every X1 = •(B1 ) ∈ Fx (M, N )y and X2 = •(B2 ) ∈ Fy (N, P )z . Every local
¯ ¯
di¬eomorphism f : M ’ M and every smooth map g : N ’ N induces a map
F (f, g) : F (M, N ) ’ F (M , N ) de¬ned by

F (f, g)(X) = •(germy g) —¦ X —¦ •((germx f )’1 ), X ∈ Fx (M, N )y .

III. Each map F (f, g) is smooth.
p1 p2
Consider the product N1 ← N1 — N2 ’ N2 of two manifolds. Then
’ ’
we have the induced maps F (idM , p1 ) : F (M, N1 — N2 ) ’ F (M, N1 ) and
F (idM , p2 ) : F (M, N1 — N2 ) ’ F (M, N2 ). Both F (M, N1 ) and F (M, N2 ) are
¬bered manifolds over M .
IV. F (M, N1 —N2 ) coincides with the ¬bered product F (M, N1 )—M F (M, N2 )
and F (idM , p1 ), F (idM , p2 ) are the induced projections.

Then it holds: For every pair (F, •) satisfying I“IV there exists an integer
r ≥ 0 such that (F, •) = (J r , j r ). (The proof is heavily based on the theory of
Weil functors presented in chapter VIII below.)

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
128 Chapter IV. Jets and natural bundles

13. Jet groups
In spite of the fact that the jet groups lie at the core of considerations concern-
ing geometric objects and operations, they have not been studied very exten-
sively. The paper [Terng, 78] is one of the exceptions and many results presented
in this section appeared there for the ¬rst time.
13.1. Let us recall the jet groups Gk = invJ0 (Rm , Rm )0 with the multiplication
de¬ned by the composition of jets, cf. 12.6. The jet projections πl de¬ne the
Gk ’ Gk’1 ’ · · · ’ G1 ’ 1
(1) m m m
k k
and the normal subgroups Bl = ker πl (or Bl if more suitable) form the ¬ltration
Gk = B0 ⊃ B1 ⊃ · · · ⊃ Bk’1 ⊃ Bk = 1.
(2) m
Since we identify J0 (Rm , Rm ) with the space of polynomial maps Rm ’ Rm of

degree less then or equal to k, we can write Gk = {f = f1 + f2 + · · · + fk ; fi ∈
Li (Rm , Rm ), 1 ¤ i ¤ k, and f1 ∈ GL(m) = G1 }, where Li (Rm , Rn ) is the
sym m sym
space of all homogeneous polynomial maps R ’ R of degree i. Hence Gk is
m n
identi¬ed with an open subset of an Euclidean space consisting of two connected
components. The connected component of the unit, i.e. the space of all invertible

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