coincides with the derivations of the sheaf. Thus we shall follow the convention,

that either the manifolds in question are smoothly regular, or that (as de¬ned

above) Der means the space of derivations of the corresponding sheaf also denoted

by C ∞ (M, R).

32.2. Lemma. On any manifold M the operational vector ¬elds correspond exactly

to the smooth sections of the operational tangent bundle. Moreover we have an

isomorphism of convenient vector spaces Der(C ∞ (M, R)) ∼ C ∞ (M ← DM ).

=

Proof. Every smooth section X ∈ C ∞ (M ← DM ) de¬nes an operational vector

¬eld by ‚U (f )(x) := X(x)(germx f ) = pr2 (Df (X(x))) for f ∈ C ∞ (U, R) and x ∈ U .

We have that ‚U (f ) = pr2 —¦Df —¦ X = df —¦ X ∈ C ∞ (U, R) by (28.15). Then ‚U

is obviously a derivation, since df (Xx ) = Xx (f ) by (28.15). The linear mapping

‚U : C ∞ (U, R) ’ C ∞ (U, R) is bounded if and only if evx —¦‚U : C ∞ (U, R) ’ R is

bounded, by the smooth uniform boundedness principle (5.26), and this is true by

(28.15), since (evx —¦X)(f ) = df (Xx ).

Moreover, the mapping

C ∞ (M ← DM ) ’ Der(C ∞ (M, R)) ’ L(C ∞ (U, R), C ∞ (U, R))

U

given by X ’ (‚U )U is linear and bounded, since by the uniform boundedness

principle (5.26) this is equivalent to the boundedness of X ’ ‚U (f )(x) = df (Xx )

for all open U ⊆ M , f ∈ C ∞ (U, R) and x ∈ X.

Now let conversely ‚ be an operational vector ¬eld on M . Then the family evx —¦‚U :

C ∞ (U, R) ’ R, where U runs through all open neighborhoods of x, de¬nes a unique

bounded derivation Xx : C ∞ (M ⊇ {x}, R) ’ R, i.e. an element of Dx M . We have

to show that x ’ Xx is smooth, which is a local question, so we assume that M is

a c∞ -open subset of a convenient vector space E. The mapping

X

M ’ DM ∼ M — D0 E ⊆ M — L(C ∞ (U, R), R)

’ =

U

is smooth if and only if for every neighborhood U of 0 in E the component M ’

L(C ∞ (U, R), R), given by ‚ ’ Xx (f ( ’x)) = ‚Ux (f ( ’x))(x) is smooth, where

Ux := U + x. By the smooth uniform boundedness principle (5.18) this is the case

if and only if its composition with evf is smooth for all f ∈ C ∞ (U, R). If t ’ x(t)

32.2

32.5 32. Vector ¬elds 323

is a smooth curve in M ⊆ E, then there is a δ > 0 and an open neighborhood W

of x(0) in M such that W ⊆ U + x(t) for all |t| < δ and hence Xx(t) (f ( ’x(t))) =

‚W (f ( ’x(t)))(x(t)), which is by the exponential law smooth in t.

Moreover, the mapping Der(C ∞ (M, R)) ’ C ∞ (M ← DM ) given by ‚ ’ X is

linear and bounded, since by the uniform boundedness principle in proposition

(30.1) this is equivalent to the boundedness of ‚ ’ Xx ∈ Dx M ’ U C ∞ (U, R)

for all x ∈ M , i.e. to that of ‚ ’ Xx (f ) = ‚U (f )(x) for all open neighborhoods U

of x and f ∈ C ∞ (U, R), which is obviously true.

32.3. Lemma. There is a natural embedding of convenient vector spaces

X(M ) = C ∞ (M ← T M ) ’ C ∞ (M ← DM ) ∼ Der(C ∞ (M, R)).

=

Proof. Since T M is a closed subbundle of DM this is obviously true.

32.4. Lemma. Let M be a smoothly regular manifold.

Then each bounded derivation X : C ∞ (M, R) ’ C ∞ (M, R) is already an opera-

tional vector ¬eld. Moreover, we have an isomorphism

C ∞ (M ← DM ) ∼ Der(C ∞ (M, R), C ∞ (M, R))

=

of convenient vector spaces.

Proof. Let ‚ be a bounded derivation of the algebra C ∞ (M, R). If f ∈ C ∞ (M, R)

vanishes on an open subset U ‚ M then also ‚(f ): For x ∈ U we take a bump

function gx,U ∈ C ∞ (M, R) at x, i.e. gx,U = 1 near x and supp(gx,U ) ‚ U . Then

‚(f ) = ‚((1 ’ gx,U )f ) = ‚(1 ’ gx,U )f + (1 ’ gx,U )‚(f ), and both summands are

zero near x. So ‚(f )|U = 0.

Now let f ∈ C ∞ (U, R) for a c∞ -open subset U of M . We have to show that we can

de¬ne ‚U (f ) ∈ C ∞ (U, R) in a unique manner. For x ∈ U let gx,U ∈ C ∞ (M, R) be

a bump function as before. Then gx,U f ∈ C ∞ (M, R), and ‚(gx,U f ) makes sense.

By the argument above, ‚(gf ) near x is independent of the choice of g. So let

‚U (f )(x) := ‚(gx,U f )(x). It has all the required properties since the topology on

C ∞ (U, R) is initial with respect to all mappings f ’ gx,U f for x ∈ U .

This mapping ‚ ’ ‚U is bounded, since by the uniform boundedness principles

(5.18) and (5.26) this is equivalent with the boundedness of ‚ ’ ‚U (f )(x) :=

‚(gx,U f )(x) for all f ∈ C ∞ (U, R) and all x ∈ U

32.5. The operational Lie bracket. Recall that operational vector ¬elds are

the bounded derivations of the sheaf C ∞ ( , R), see (32.1). This is a convenient

vector space by (32.2) and (30.1).

If X, Y are two operational vector ¬elds on M , then the mapping f ’ X(Y (f )) ’

Y (X(f )) is also a bounded derivation of the sheaf C ∞ ( , R), as a simple compu-

tation shows. We denote it by [X, Y ] ∈ Der(C ∞ ( , R)) ∼ C ∞ (M ← DM ).

=

32.5

324 Chapter VII. Calculus on in¬nite dimensional manifolds 32.6

The R-bilinear mapping

] : C ∞ (M ← DM ) — C ∞ (M ← DM ) ’ C ∞ (M ← DM )

[,

is called the Lie bracket. Note also that C ∞ (M ← DM ) is a module over the

algebra C ∞ (M, R) by pointwise multiplication (f, X) ’ f X, which is bounded.

Theorem. The Lie bracket [ , ] : C ∞ (M ← DM ) — C ∞ (M ← DM ) ’

C ∞ (M ← DM ) has the following properties:

[X, Y ] = ’[Y, X],

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]], the Jacobi identity,

[f X, Y ] = f [X, Y ] ’ (Y f )X,

[X, f Y ] = f [X, Y ] + (Xf )Y.

The form of the Jacobi identity we have chosen says that ad(X) = [X, ] is a

derivation for the Lie algebra (C ∞ (M ← DM ), [ , ]).

Proof. All these properties can be checked easily for the commutator [X, Y ] =

X —¦ Y ’ Y —¦ X in the space of bounded derivations of the algebra C ∞ (U, R).

32.6. Lemma. Let b : E1 — . . . — Ek ’ R be a bounded multilinear mapping on a

product of convenient vector spaces. Let f ∈ C ∞ (E, R), let fi : E ’ Ei be smooth

(1)

mappings, and let Xx ∈ E = Dx E.

Then we have

= df (x)—— .Xx

Xx (f ) = Xx , df (x) E

Xx (b—¦(f1 , . . . , fk )) = d(b —¦ (f1 , . . . , fk ))(x)—— .Xx

, fi+1 (x), . . . , fk (x))—— .dfi (x)—— .Xx

= b(f1 (x), . . . , fi’1 (x),

1¤i¤k

dfi (x)—— .Xx , b(f1 (x), . . . , fi’1 (x),

= , fi+1 (x), . . . , fk (x)) Ei .

1¤i¤k

If B : E1 — . . . — Ek ’ F is a vector valued bounded multilinear mapping, and if

g : E ’ F is a smooth mapping, then we have

Dx g.Xx = dg(x)—— .Xx ∈ F

(1)

(1)

Dx (B —¦ (f1 , . . . , fk )).Xx =

(1)

, fi+1 (x), . . . )—— .dfi (x)—— .Xx ∈ DB(f1 (x),...,fk (x)) F.

= B(. . . , fi’1 (x),

1¤i¤k

H : H — H ’ R is the duality pairing for any convenient vector space

Here ,

H. We will further denote by ιH : H ’ H the canonical embedding into the

bidual space.

32.6

32.6 32. Vector ¬elds 325

Proof. The ¬rst equation is immediate.

We have

k

d(b —¦ (f1 , . . . , fk ))(x) = , fj+1 (x), . . . , fk (x)) —¦ dfj (x)

b(f1 (x), . . . , fj’1 (x),

j=1

k

dfj (x)— b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x)) .

j=1

(1)

Thus for Xx ∈ Dx E we have

Xx (b—¦(f1 , . . . , fk )) = Xx d(b —¦ (f1 , . . . , fk ))(x)

k

dfj (x)— b(f1 (x), . . . , fj’1 (x),

= Xx , fj+1 (x), . . . , fk (x))

j=1

k

Xx dfj (x)— b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x))

j=1

k

dfj (x)—— (Xx ) b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x)) .

j=1

For the second assertion we choose a test germ

h ∈ C ∞ (F ⊇ {B(f1 (x), . . . , fk (x))}, R)

and proceed as follows:

(1)

(Dx g.Xx )(h) = Xx (h —¦ g) = Xx , d(h —¦ g)(x) E

—

= Xx , dh(g(x)) —¦ dg(x) = Xx , dg(x) .dh(g(x))

E E

= dg(x)—— .Xx , dh(g(x)) = (dg(x)—— .Xx )(h).

E

(1)

(Dx (B —¦ (f1 , . . . , fk ))Xx )(h) = Xx (h —¦ B —¦ (f1 , . . . , fk ))

= d(h —¦ B —¦ (f1 , . . . , fk ))(x)—— .Xx

——

k

dh(B(f1 (x), . . . )) —¦ , fi+1 (x), . . . ) —¦ dfi (x)

= B(. . . , fi’1 (x), .Xx

i=1

k

= dh(B(f1 (x), . . . ))—— . , fi+1 (x), . . . )—— .dfi (x)—— .Xx

B(. . . , fi’1 (x),

i=1

k

, fi+1 (x), . . . )—— .dfi (x)—— .Xx

= B(. . . , fi’1 (x), (h).

i=1 B(f1 (x),... )

32.6

326 Chapter VII. Calculus on in¬nite dimensional manifolds 32.7

32.7. The Lie bracket of operational vector ¬elds of order 1. One could

hope that the Lie bracket restricts to a Lie bracket on C ∞ (D(1) M ). But this is

not the case. We will see that for a c∞ -open set U in a convenient vector space E

and for X, Y ∈ C ∞ (U, E ) the bracket [X, Y ] has also components of order 2, in

general.

For a bounded linear mapping : F ’ G the transposed mapping t : G ’ F

is given by t := — —¦ ιG , where ιG : G ’ G is the canonical embedding into the

H : H — H ’ R is the duality pairing, then this may also be

bidual. If ,

described by (x), y G = t (y), x F .

For X, Y ∈ C ∞ (U, E ), for f ∈ C ∞ (U, E) and for x ∈ U we get:

X(f )(x) = Xx (f ) = Xx (df (x))

X(f ) = ev —¦(X, df )

Y (X(f ))(x) = Yx (X(f )) = Yx (ev —¦(X, df ))

= Yx dX(x)— ev( , df (x)) + d(df )(x)— ev(X(x), )

= Yx dX(x)— ι(df (x)) + d(df )(x)— (Xx )

= Yx ι(df (x)) —¦ dX(x) + Xx —¦ d(df )(x)

= Yx dX(x)t df (x) + Xx —¦ d(df )(x)

= Yx —¦ dX(x)t df (x) + Yx Xx —¦ d(df )(x) .

Here we used the equation:

ι(y) —¦ T = T t (y) for y ∈ F, T ∈ L(E, F ),

which is true since

ι(y) —¦ T (x) = ι(y)(T (x)) = T (x)(y) = T t (y)(x).

Note that for the symmetric bilinear form b := d(df )(x)§ : E — E ’ R a canonical

extension to a bilinear form ˜ on E is given by

b

˜ x , Yx ) := Xx (Yx —¦ b∨ )

b(X

However, this extension is not symmetric as the following remark shows: Let b :=

ev : E — E ’ R. Then ˜ : E — E ’ R is given by

b

˜

b(X, Y ) := X(Y —¦ b∨ ) = X(Y —¦ Id) = X(Y ) = ιE (Y )(X)

For b := ev —¦ ¬‚ip : E — E ’ R we have that ˜ : E — E ’ R is given by

b

˜ X) := Y (X —¦ b∨ ) = Y (X —¦ ιE ) = Y (ι— (X)) = (Y —¦ ι— )(X) = (ιE )—— (Y )(X).

b(Y, E E

Thus, ˜ is not symmetric in general, since ker(ι—— ’ ιE ) = ιE (E), at least for

b E

Banach spaces, see [Cigler, Losert, Michor, 1979, 1.15], applied to ιE .

32.7

32.8 32. Vector ¬elds 327

Lemma. For X ∈ C ∞ (T M ) and Y ∈ C ∞ (D(1) M ) we have [X, Y ] ∈ C ∞ (D(1) M ),

and the bracket is given by the following local formula for M = U , a c∞ -open subset

in a convenient vector space E:

[X, Y ](x) = Y (x) —¦ dX(x)— ’ dY (x).X(x) ∈ E .

Proof. From the computation above we get:

Y (X(f ))(x) = (d(ιE —¦ X)(x)t )— .Y (x), df (x) + d(df )(x)—— .Y (x), ιE .X(x)

E E

+ Y (x), d(df )(x)— .ιE .X(x)

= Y (x), (ιE —¦ dX(x))t .df (x) E E

= Y (x), dX(x)— .df (x) + Y (x), d(df )(x)t .X(x)

E E

= Y (x) —¦ dX(x)— , df (x) + Y (x), d(df )(x)t .X(x)

E E

X(Y (f ))(x) = (dY (x)t )— .ιE .X(x), df (x) + d(df )(x)—— .ιE .X(x), Y (x)

E E

+ ιE .X(x), d(df )(x)— .Y (x)

= ιE .X(x), dY (x)t .df (x) E E

+ d(df )(x)— .Y (x), X(x)

= dY (x)t .df (x), X(x) E E

= dY (x).X(x), df (x) + Y (x), d(df )(x).X(x)

E E

Since d(df )(x) : E ’ E is symmetric in the sense that d(df )(x)t = d(df )(x), the

result follows.

32.8. Theorem. The Lie bracket restricts to the following mappings between split-

ting subspaces

] : C ∞ (M ← D(k) M ) — C ∞ (M ← D( ) M ) ’ C ∞ (M ← D(k+ ) M ).

[ ,

The spaces X(M ) = C ∞ (M ← T M ) and C ∞ (D[1,∞) M ) := C ∞ (M ←

1¤i<∞

D(i) M ) are sub Lie algebras of C ∞ (M ← DM ).

] maps C ∞ (M ← D( ) M ) into

If X ∈ X(M ) is a kinematic vector ¬eld, then [X,

itself.

This suggests to introduce the notation D(0) := T , but here it does not indicate

the order of di¬erentiation present in the tangent vector.

Proof. All assertions can be checked locally, so we may assume that M = U is

open in a convenient vector space E.

We prove ¬rst that the kinematic vector ¬elds form a Lie subalgebra. For X,

Y ∈ C ∞ (U, E) we have then for the vector ¬eld ‚X |x (f ) = df (x)(X(x)), compare

the notation set up in (28.2)

[‚X , ‚Y ](f ) = ‚X (‚Y (f )) ’ ‚Y (‚X (f ))

= d(df.Y ).X ’ d(df.X).Y

= d2 f.(X, Y ) + df.(dY.X) ’ d2 f.(Y, X) ’ df.(dX.Y )

= ‚dY.X’dX.Y f.

32.8

328 Chapter VII. Calculus on in¬nite dimensional manifolds 32.9

k

Let ‚X ∈ C ∞ (U ← D(k) U ) for X = i=1 X [i] , where X [i] ∈ C ∞ (U, Li (E; R) )

sym

∞

vanishes on decomposable forms. Similarly, let ‚Y ∈ C (U ← D( ) U ), and

suppose that f : (U, x) ’ R is a (k + )-¬‚at germ at x. Since ‚Y (f )(y) =

1i

[i]

i=1 Y (y)( i! d f (y)) the germ ‚Y (f ) is still k-¬‚at at x, so ‚X (‚Y (f ))(x) = 0.

Thus, [‚X , ‚Y ](f )(x) = ‚X (‚Y (f ))(x) ’ ‚Y (‚X (f ))(x) = 0, and we conclude that

[‚X , ‚Y ] ∈ C ∞ (U ← D(k+ ) U ).

Now we suppose that X ∈ C ∞ (U, E) and Y ∈ C ∞ (U, Lsym (E; R) ). Let f :

(U, x) ’ R be an -¬‚at germ at x. Then we have

1

‚Y (‚X (f ))(x) = Y (x) !d df, X E (x)

’k

sym dk (df )(x), d

1

= Y (x) X(x) E

! k

k=0

1

= Y (x) ! d (df )(x), X(x) E + 0

(x), 1! d1+ f (x)( , X(x)) Lsym (E;R)

=Y

‚X (‚Y (f ))(x) = ‚X(x) Y, 1! d f Lsym (E;R)

= d Y, 1! d f Lsym (E;R) (x).X(x)

= dY (x).X(x), 1! d f (x) + Y (x), 1! d(d f )(x).X(x)

Lsym (E;R) Lsym (E;R)

+1

= 0 + Y (x), 1! d f (x)(X(x), ) Lsym (E;R)

So [‚X , ‚Y ](f )(x) = 0.

Remark. In the notation of (28.2) we have shown that on a convenient vector

space we have

k+

] : C ∞ (E ← D[k] E) — C ∞ (E ← D[ ] E) ’ C ∞ (E ← D[i] E).

[,

i=min(k, )

Thus, the space C ∞ (E ← D[k,∞) E) := k¤i<∞ C ∞ (E ← D[i] E) for k ≥ 1 is a sub

Lie algebra. The (possibly larger) space C ∞ (D[k,∞] E) of all operational tangent

¬elds which vanish on all polynomials of degree less than k is obviously a sub Lie

algebra. But beware, none of these spaces of vector ¬elds is invariant under the

action of di¬eomorphisms.

32.9. f -related vector ¬elds. Let D± be one of the following functors D, D(k) ,

T . If f : M ’ M is a di¬eomorphism, then for any vector ¬eld X ∈ C ∞ (M ←

D± M ) the mapping D± f ’1 —¦ X —¦ f is also a vector ¬eld, which we will denote by

f — X. Analogously, we put f— X := D± f —¦ X —¦ f ’1 = (f ’1 )— X.

But if f : M ’ N is a smooth mapping and Y ∈ C ∞ (N ← D± N ) is a vector

¬eld there may or may not exist a vector ¬eld X ∈ C ∞ (M ← D± M ) such that the

following diagram commutes:

wu