¯

rank over each component Pσ . Assume by induction P r’1 ‚ W r’3 — P r’1 is a

¯ ¯

r r’1

submanifold. Then P ‚ P — {0} — Wr’2 is characterized by the curvature

equations Er and by

n n

(3) Wiklm1 ...mr’2 Pnj + Wjklm1 ...mr’2 Pin = 0.

By the above argument we deduce that this is a system of a¬ne equations of

constant rank over each Pσ .

Consider a Gr+1 -map f : Tm P ’ F . Applying 28.10 to f —¦ p2 = Tm Q —

r r’1

m

Tm P ’ F , where p2 is the second product projection, we obtain a G1 -map

r

m

r

h : ZP ’ F satisfying

f —¦ p2 = h —¦ (C r’2 , DP ).

r

(4)

r r’1

Let »r : Tm P ’ Tm Q be the map determined by constructing the r-jets of the

r r’1 r

Levi-Civit` connection. Composing (4) with (»r , id) : Tm P ’ Tm Q — Tm P ,

a

we ¬nd

f = h —¦ (C r’2 , DP ) —¦ (»r , id).

r

(5)

¯

Let g be the restriction of h to P r . Since the Levi-Civit` connection is char-

a

acterized by the fact that the absolute di¬erential of the metric tensor van-

¯

ishes, the values of (C r’2 , DP ) —¦ (»r , id) lie in P r . Write Lr’2 = (C r’2 , DP ) —¦

r r

¯

(»r , id) : Tm P ’ P r . Then we can summarize by

r

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244 Chapter VI. Methods for ¬nding natural operators

Proposition. For every Gr+1 -map f : Tm P ’ F there exists a G1 -map

r

m m

¯

g : P r ’ F satisfying f = g —¦ Lr’2 .

This is the classical assertion that every r-th order natural operator on pseu-

doriemannian metrics with values in an arbitrary ¬rst order natural bundle fac-

torizes through the metric itself and through the absolute derivatives of the

curvature tensor of the Levi-Civit` connection up to order r ’ 2.

a

We remark that each component Pσ of P can be treated separately in course

of the proof of the above proposition. Hence the result holds for any kind of

pseudoriemannian metrics (in particular for the proper Riemannian metrics).

28.15. Pseudoriemannian metric and a general vector ¬eld. A simple

modi¬cation of 28.11 and 28.14 leads to a reduction theorem for the r-th order

natural operators transforming a pseudoriemannian metric and a general vector

¬eld into a section of a ¬rst order natural bundle. In the notation from 28.11 and

28.14, let f : Tm P — Tm V ’ F be a Gr+1 -map. Consider the product projection

r r

m

r’1 r r r r

p : Tm Q — Tm P — Tm V ’ Tm P — Tm V . Then we can apply 28.9 and 28.10 to

the product P — V . Hence there exists a G1 -map h : ZP —V ’ F satisfying

r

m

f —¦ p = h —¦ (C r’2 , DP —V ).

r

(1)

¯r

Denote by PV ‚ ZP P — V ‚ K r’2 — P r — V r the subspace determined by

r

¯r

0 ∈ P1 , . . . , 0 ∈ Pr . Analogously to 28.14 we deduce that PV is a submani-

¯r

fold. Write Lr’2 = (»r , idTm P ) — idTm V : Tm P — Tm V ’ PV , i.e. Lr’2 (u, v) =

r r

r r

V V

(C r’2 (»r (u)), u0 , 0, . . . , 0, DV (»r (u), v)), u ∈ Tm P , v ∈ Tm V , u0 = π0 (u). Then

r r r r

¯r

(1) implies f = h —¦ Lr’2 . If we denote by g the restriction of h to PV , we obtain

V

the following assertion.

Proposition. For every Gr+1 -map f : Tm P — Tm V ’ F there exists a G1 -map

r r

m m

¯r

g : PV ’ F satisfying f = g —¦ Lr’2 .

V

Hence every r-th order natural operator transforming a pseudoriemannian

metric and a general vector ¬eld into a section of a ¬rst order natural bundle

factorizes through the metric itself, through the absolute derivatives of the cur-

vature tensor of the Levi-Civit` connection up to the order r ’ 2 and through

a

the absolute derivatives with respect to the Levi-Civit` connection of the general

a

vector ¬eld up to the order r.

28.16. Remark. Since Q„ P 1 M ’ M is an a¬ne bundle, the standard ¬ber

r

Tm Q of its r-th jet prolongation is an a¬ne space by 12.17. In other words,

Gr+2 acts on Tm Q by a¬ne isomorphisms. Consider an a¬ne action of G1

r

m m

of F (with the linear action as a special case). Then we can introduce the

r

concept of a polynomial map Tm Q ’ F analogously to 24.10. Analyzing the

proof of theorem 28.6, we observe that all the maps ψr and •r are polynomial.

This implies that for every polynomial Gr+2 equivariant map f : Tm Q ’ F ,

r

m

the unique G1 -equivariant map g : K r’1 ’ F from the theorem 28.6 is the

m

restriction of a polynomial map g : W r’1 ’ F .

¯

1

Consider further a Gm -module V as in 28.8 or an invariant open subset of such

a module as in 28.10. Then we also have de¬ned the concept of a polynomial

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

29. The method of di¬erential equations 245

map of Tm Q — Tm V into an a¬ne G1 -space F . Quite similarly to the ¬rst

r’1 r

m

part of this remark we deduce that for every polynomial Gr+1 -equivariant map

m

f : Tm Q — Tm V ’ F the unique G1 -equivariant map g : ZV ’ F from the

r’1 r r

m

proposition 28.9 is the restriction of a polynomial map g : W r’2 — V r ’ F .

¯

29. The method of di¬erential equations

29.1. In chapter IV we have clari¬ed that the ¬nite order natural operators

between any two bundle functors are in a canonical bijection with the equivariant

maps between certain G-spaces. We recall that in 5.15 we deduced the following

in¬nitesimal characterization of G-equivariance. Given a connected Lie group G

and two G-spaces S and Z we construct the induced fundamental vector ¬eld

S Z

ζA and ζA on S and Z for every element A ∈ g of the Lie algebra of G. Then

Q

S

f : S ’ Z is a G-equivariant map if and only if vector ¬elds ζA and ζA are

f -related for every A ∈ g, i.e.

S Z

T f —¦ ζA = ζA —¦ f for all A ∈ g.

(1)

The coordinate expression of (1) is a system of partial di¬erential equations

for the coordinate components of f . If we can ¬nd the general solution of this

system, we obtain all G-equivariant maps. This procedure is sometimes called

the method of di¬erential equations.

29.2. Remark. If G is not connected and G+ denotes its connected component

of unity, then the solutions of 29.1.(1) determine all G+ -equivariant maps S ’ Z.

Obviously, there is an algebraic procedure how to decide which of these maps

are G-equivariant. We select one element ga in each connected component of

G and we check which solutions of 29.1.(1) are invariant with respect to all

ga . However, one usually interprets the solutions of 29.1.(1) geometrically. In

practice, if we succeed in ¬nding the geometrical constructions of all solutions

of 29.1.(1), it is clear that all of them determine the G-equivariant maps and we

are not obliged to discuss the individual connected components of G.

29.3. From 5.12 we have that for each left G-space S the map of the fundamental

S S

vector ¬elds A ’ ζA , A ∈ g, is a Lie algebra antihomomorphism, i.e. ζ[A,B] =

SS

’[ζA , ζB ] for all A, B ∈ g, where on the left-hand side is the Lie bracket in g

and on the right-hand side we have the bracket of vector ¬elds. Hence if some

vectors A± , ± = 1, . . . , q ¤ dim G generate g as a Lie algebra, i.e. A± with all

their iterated brackets generate g as a vector space, then the equations

S Z

T f —¦ ζ A± = ζ A± —¦ f ± = 1, . . . , q

imply T f —¦ ζA = ζA —¦ f for all A ∈ g. In particular, for the group Gr the

S Z

m

generators of its Lie algebra are described in 13.9 and 13.10.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

246 Chapter VI. Methods for ¬nding natural operators

29.4. The Levi-Civit` connection. We are going to determine all ¬rst order

a

natural operators transforming pseudoriemannian metrics into linear connec-

tions. We denote by RegS 2 T — M the bundle of all pseudoriemannian metrics

over an m-manifold M , so that the standard ¬ber of the corresponding natural

bundle over m-manifolds is the subset RegS 2 Rm— ‚ S 2 Rm— of all elements gij

satisfying det(gij ) = 0. Since the zero of S 2 Rm— does not lie in RegS 2 Rm— , the

homogeneous function theorem is of no use for our problem. (Of course, this

analytical fact is deeply re¬‚ected in the geometry of pseudoriemannian mani-

folds.) Hence we shall try to apply the method of di¬erential equations. In the

canonical coordinates gij = gji , gij,k on the standard ¬ber S = J0 RegS 2 T — Rm ,

1

the action of G2 has the following form

m

gij = gkl ak al

(1) ¯ ˜i ˜j

gij,k = glm,n al am an + glm (˜l am + al am ).

(2) ¯ ˜i ˜j ˜k aik ˜j ˜i ˜jk

Since we deal with a classical problem, we shall use the classical Christo¬el™s on

the standard ¬ber Z = (QP 1 Rm )0 . In this case we have the following action of

G2m

¯ jk

“i = ai “l am an + ai al

(3) l mn ˜j ˜k l ˜jk

see 17.15.

We shall not need all di¬erential equations of our problem, since we shall

proceed in another way in the ¬nal step. It is su¬cient to deduce the funda-

i i

mental vector ¬elds Sjk on S and Zjk on Z corresponding to the one-parameter

subgroups ai = δj , ai = t for j = k and ai = δj , ai = 2t. From (1)“(3) we

i i

˜jk

j j jj

deduce easily

‚ ‚

i

(4) Sjk = 2gil +

‚glj,k ‚glk,j

and

‚ ‚

i

(5) Zjk = +

i ‚“i

‚“jk kj

Hence the corresponding part of the di¬erential equations for a G2 -equivariant

m

i

map “ : S ’ Z with components “jk (glm , glm,n ) is

‚“i ‚“i

jk jk i mn nm

(6) 2glp + = δl δj δk + δj δk .

‚gpm,n ‚gpn,m

Multiplying by g lq and replacing q by l, we ¬nd

‚“i ‚“i 1

jk jk

= g il δj δk + δk δj .

mn mn

(7) +

‚glm,n ‚gln,m 2

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

29. The method of di¬erential equations 247

Let (7™) or (7”) be the equations derived from (7) by the permutation (l, m, n) ’

(m, n, l) or (l, m, n) ’ (n, l, m), respectively. Then the sum (7)+(7 )’(7 ) yields

‚“i 1 il m n

jk

g (δj δk + δk δj ) + g im (δj δk + δk δj )

mn nl nl

2 =

‚glm,n 2

(8)

’ g in (δj δk + δk δj ) .

lm lm

The right-hand sides are independent on gij,k . Since we meet such a situation

frequently, it is useful to formulate a simple lemma of general character.

29.5. Lemma. Let U be an open subset in Ra with coordinates z ± and let

f (z ± , w» ) be a smooth function on U —Rb , (w» ) ∈ Rb , satisfying ‚f (z,w) = g» (z).

‚w»

Then

b

g» (z)w» + h(z)

(1) f (z, w) =

»=1

where h(z) is a smooth function on U .

b »

Proof. Notice that the di¬erence F (z, w) = f (z, w) ’ »=1 g» (z)w satis¬es

‚F

= 0.

‚w»

Applying lemma 29.5 to 29.4.(8), we ¬nd

1 il

“i = i

g (glj,k + glk,j ’ gjk,l ) + γjk (glm ).

jk

2

i

For γjk = 0 we obtain the coordinate expression of the Levi-Civit` connection Λ,

a

which is natural by its standard geometric interpretation. Hence the di¬erence

“ ’ Λ is a GL(m)-equivariant map RegS 2 Rm— ’ Rm — Rm— — Rm— .

29.6. Lemma. The only GL(m)-equivariant map f : RegS 2 Rm— ’ Rm —Rm— —

Rm— is the zero map.

Proof. Let Is be the matrix gii = 1 for i ¤ s, gjj = ’1 for j > s and gij = 0 for

i = j. Since every g ∈ RegS 2 Rm— can be transformed into some Is , it su¬ces to

i

deduce fjk (Is ) = 0 for all i, j, k. If j = i = k or j = i = k, the equivariance with

i i

respect to the change of orientation on the i-th axis gives fjk (Is ) = ’fjk (Is ). If

j = i = k, we obtain the same result by changing the orientation on both the

i-th and k-th axes.

Lemma 29.6 implies “ ’ Λ = 0. This proves

29.7. Proposition. The only ¬rst order natural operator transforming pseu-

doriemannian metrics into linear connections is the Levi-Civit` operator.

a

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

248 Chapter VI. Methods for ¬nding natural operators

Remarks

The ¬rst version of our systematical approach to the problem of ¬nding nat-

ural operators was published in [Kol´ˇ, 87b]. In the same paper both geometric

ar

results from section 25 are deduced. The smooth version of the tensor evaluation

theorem is ¬rst presented in this book. Proposition 26.12 was proved by [Kol´ˇ, ar

Radziszewski, 88]. The generalized invariant tensor theorem was ¬rst used in

[Kol´ˇ, 87b]. We remark that the natural equivalence s : T T — ’ T — T from 26.11

ar

was ¬rst studied in [Tulczyjew, 74].

The reduction theorems for symmetric linear connections and pseudorieman-

nian metrics are classical, see e.g. [Schouten, 54]. Some extensions or reformula-

tions of them are presented in [Lubczonok, 72] and [Krupka, 82]. The method of

di¬erential equations is used systematically e.g. in the book [Krupka, Janyˇka,

s

90]. The complete version of proposition 29.7 was deduced in [Slov´k, 89].

a

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

249

CHAPTER VII.

FURTHER APPLICATIONS

In this chapter we discuss some further geometric problems about di¬erent

types of natural operators. First we deduce that all natural bilinear operators

transforming a vector ¬eld and a di¬erential k-form into a di¬erential k-form

form a 2-parameter family. This further clari¬es the well known relation be-

tween Lie derivatives and exterior derivatives of k-forms. From the technical

point of view this problem can be considered as a preparatory exercise to the

problem of ¬nding all bilinear natural operators of the type of the Fr¨licher-

o

Nijenhuis bracket. We deduce that in general case all such operators form a

10-parameter family. Then we prove that there is exactly one natural operator

transforming general connections on a ¬bered manifold Y ’ M into general con-

nections on its vertical tangent bundle V Y ’ M . Furthermore, starting from

some geometric problems in analytical mechanics, we deduce that all ¬rst-order

natural operators transforming second-order di¬erential equations on a manifold

M into general connections on its tangent bundle T M ’ M form a one param-

eter family. Further we study the natural transformations of the jet functors.

The construction of the bundle of all r-jets between any two manifolds can be

interpreted as a functor J r on the product category Mfm — Mf . We deduce

that for r ≥ 2 the only natural transformations of J r into itself are the identity

and the contraction, while for r = 1 we have a one-parameter family of homo-

theties. This implies easily that the only natural transformation of the functor

of the r-th jet prolongation of ¬bered manifolds into itself is the identity. For

the second iterated jet prolongation J 1 (J 1 Y ) of a ¬bered manifold Y we look

for an analogy of the canonical involution on the second iterated tangent bundle

T T M . We prove that such an exchange map depends on a linear connection on

the base manifold and we give a simple list of all natural transformations of this