First Order Differential Operators
II. Dierential Operators
We will now proceed to develop the formal machinery necessary to carry the notions
of dierential calculus in the Euclidean space over to arbitrary manifolds. The rst step
in this programme is to dene dierential operators on manifolds. We deal only with
linear dierential operators even if we do not say so explicitly each time. We will start
with rst order operators.
1 First Order Dierential Operators
Let M be a dierential manifold. We rst dene homogeneous rst order operators
on the algebra of dierentiable functions, taking as characteristic example, an operator
like
Df =
φ
i
∂f
∂x
i
where φ
i
are some dierentiable functions on R
n
or on an open subset U of R
n
. One of
the basic properties of such an operator (sometimes called the Leibniz property) is
1.1 D(fg) = (Df)g + f(Dg)
for any two dierentiable functions f, g on U . A k-linear homomorphism of a k-algebra
into itself satisfying the Leibniz property, is for this reason called a derivation. Notice
that a consequence of the denition is that D(1) = 0, for D(1) = D(1.1) = D(1).1 +
1.D(1) = 2D(1). It follows that D(λ) = 0 for all λ ∈ k.
We propose to take the purely algebraic property 1.1 as the denition of a linear
homogeneous dierential operator of order 1 on an arbitrary manifold. We will now
provide the justication for doing so.
Firstly, the algebraic condition implies that the operator D is local in the following
sense.
1.2. Proposition. If D is a linear operator on the R-algebra of dierentiable functions
satisfying the Leibniz property, then the value of Df in any open set V depends only
on the restriction of f to V .
Proof. In fact, if f = g in an open neighbourhood N of a point x in V , consider
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Chapter II: Differential Operators
a dierentiable function φ which is 1 in a smaller neighbourhood of x and vanishes
outside N . Then we clearly have φ(f − g) = 0. Applying D and using ?? (�����������
label � equ:2.1.leibniz) we get (Dφ)(f − g) + φD(f − g) = 0. In particular, we see that
φD(f − g) = 0 on N , proving that D(f − g) = Df − Dg = 0 in a neighbourhood of x.
Since x is an arbitrary point of V , our assertion is proved.
Consequently, any map A(M ) → A(M ) which satises the Leibniz property (in
particular, an operator of the form
φ
i
∂
∂x
i
) induces a sheaf homomorphism of A into
itself, the homomorphism being one of R-vector spaces.
1.3. Proposition. Let U be an open submanifold of R
n
. If D : A
U
→ A
U
is a sheaf
homomorphism of R-vector spaces, satisfying the Leibniz rule, then D is an operator of
the form f 7→
φ
i
∂f
∂x
i
for some dierentiable functions φ
i
on U .
Proof. Clearly, if D is to be of the form
φ
i
∂
∂x
i
, then applying D to the functions x
i
,
we see that φ
i
ought to be Dx
i
. Replacing D by D −
(Dx
i
)
∂
∂x
i
we deduce that it
is enough to prove the following. If D is a sheaf derivation such that Dx
i
= 0 for all
i, then D is the zero homomorphism. Let then a = (a
1
, . . . , a
n
) be a point of U and
f any dierentiable function in a neighbourhood of a. Then f can be written as the
sum of the constant function f(a) and
(x
i
− a
i
)g
i
in some neighbourhood of a. Let
us assume this for the moment. Now Df =
(x
i
− a
i
)Dg
i
in view of our assumption.
But (x
i
− a
i
) vanishes at a, implying that (Df)(a) = 0. Since a is any point in U , it
follows that Df = 0, as was to be proved.
It remains to prove our assertion about the decomposition of f. Indeed we have
f(x) − f(a) =
1
0
d
dt
{f(tx + (1 − t)a)}dt
=
1
0
∂
∂x
i
f(tx + (1 − t)a) ·
d
dt
(tx
i
+ (1 − t)a
i
)dt
=
(x
i
− a
i
)
1
0
∂
∂x
i
f(tx + (1 − t)a)dt.
The above considerations motivate the following denition.
1.4. Denition. A homogeneous rst order dierential operator on a dierential
manifold M is a linear operator on A(M ) satisfying the Leibniz rule. Equivalently, it
is a sheaf homomorphism A → A which satises the Leibniz rule. A linear dierential
operator of order at most one on functions is of the form f 7→ Df + φ · f where D is a
homogeneous operator as above.
From the geometric point of view, a homogeneous rst order operator is called a vec-
tor eld or an innitesimal transformation. We will presently explain this terminology.
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