§ 4. Differential Manifolds
251
Relation (14.1) suggests a generalization to manifolds of (linear) classical
differential operators. For simplicity’s sake, on a manifold X of class C
∞ ,
a differential operator of order ≤ p and class C
∞ defines, for every open
subset U , a linear map L : C
∞ (U ) −→ C
∞ (U ) satisfying the following two
conditions : (i) it is compatible with the restriction maps C
∞ (U ) −→ C
∞ (V )
for V ⊂ U , so that for f ∈ C
∞ (U ), the value of the function Lf ∈ C
∞ (U ) at
x ∈ U only depends on the behaviour of f in the neighbourhood of x, (ii) in
each chart (U, ϕ) of X, there is a relation of the form
Lf (x) =
p
k=0
i1,...,i k
L
i1...i k (ξ)D i1 . . . D i k F (ξ) ,
(14.2)
where the functions L
i1...i k only depend on the chart considered and where
ξ, F and the D i are defined as in (1).
Obvious operations can be defined on these operators: sum L+M , product
LM , product fL by a function. If L and M are of order p and q respectively,
their product LM : f → L(Mf) is generally of order p + q, but their Jacobi
bracket
[L, M ] = LM − ML
(14.3)
is of order ≤ p + q − 1. It suffices to check this in R
n ; we can then suppose
that
L = ϕD i1 . . . D ip = ϕD (i) , M = ψD j1 . . . D jq = γD (j)
with given functions ϕ and ψ, and it all amounts to checking that, if we
calculate
LM f − MLf = ϕD (i)
ψD (j) F
− ψD (j)
ϕD (i) F
,
then the terms ϕψD (i) D (j) f and ψϕD (j) D (i) f cancel each other.
In particular, if L and M are defined by vector fields, the same holds for
[L, M ] ; using Einstein’s convention, it can be immediately seen that , in any
chart,
L = L
i D i , M = M
i D i =⇒ [L, M ] = N
i D i
(14.4)
with
N
i = L
j .D j
M
i
− M
j .D j
L
i
.
(14.5)
Hence, if D L denotes the differential operator f → Lf , the vector field [L, M ]
satisfies
D [L,M ] = D L D M − D M D L .
Exercise 1. Verify by a direct calculation that the vector field defined by
(5) does not depend on the chart used.
251
Relation (14.1) suggests a generalization to manifolds of (linear) classical
differential operators. For simplicity’s sake, on a manifold X of class C
∞ ,
a differential operator of order ≤ p and class C
∞ defines, for every open
subset U , a linear map L : C
∞ (U ) −→ C
∞ (U ) satisfying the following two
conditions : (i) it is compatible with the restriction maps C
∞ (U ) −→ C
∞ (V )
for V ⊂ U , so that for f ∈ C
∞ (U ), the value of the function Lf ∈ C
∞ (U ) at
x ∈ U only depends on the behaviour of f in the neighbourhood of x, (ii) in
each chart (U, ϕ) of X, there is a relation of the form
Lf (x) =
p
k=0
i1,...,i k
L
i1...i k (ξ)D i1 . . . D i k F (ξ) ,
(14.2)
where the functions L
i1...i k only depend on the chart considered and where
ξ, F and the D i are defined as in (1).
Obvious operations can be defined on these operators: sum L+M , product
LM , product fL by a function. If L and M are of order p and q respectively,
their product LM : f → L(Mf) is generally of order p + q, but their Jacobi
bracket
[L, M ] = LM − ML
(14.3)
is of order ≤ p + q − 1. It suffices to check this in R
n ; we can then suppose
that
L = ϕD i1 . . . D ip = ϕD (i) , M = ψD j1 . . . D jq = γD (j)
with given functions ϕ and ψ, and it all amounts to checking that, if we
calculate
LM f − MLf = ϕD (i)
ψD (j) F
− ψD (j)
ϕD (i) F
,
then the terms ϕψD (i) D (j) f and ψϕD (j) D (i) f cancel each other.
In particular, if L and M are defined by vector fields, the same holds for
[L, M ] ; using Einstein’s convention, it can be immediately seen that , in any
chart,
L = L
i D i , M = M
i D i =⇒ [L, M ] = N
i D i
(14.4)
with
N
i = L
j .D j
M
i
− M
j .D j
L
i
.
(14.5)
Hence, if D L denotes the differential operator f → Lf , the vector field [L, M ]
satisfies
D [L,M ] = D L D M − D M D L .
Exercise 1. Verify by a direct calculation that the vector field defined by
(5) does not depend on the chart used.