Differential operators as sections of a vector bundle

Most of the books which need to use differential operators in a hands on way adopt some variation of the following definition (here for simplicity I am avoiding discussion of differential operators between bundles):

Defintion 1.

A differential operator of order $\leq k$ on a smooth manifold $M$ is an $\mathbb{R}$-linear map $D:C^{\infty}(M) \to C^{\infty} (M)$ such that in local coordinates it looks like $D f (p)= \sum_{i_1+...+i_n \leq k} A_{i_1 ... i_n}(p) \frac{\partial^{i_1+...+i_n} f}{\partial x_1^{i_1} ...x_n^{i_n}}|_p$ for some smooth functions $A_{i_1 ... i_n}$.

I have been searching for a different definition that didn't employ coordinates right away, and I found the following definition in the "Lectures on the geometry of manifolds" by Liviu Nicolaescu and lecture notes of Misha Verbitsky (which are in Russian), which is, I believe, due to Grothendieck:

Definition 2.

Define differential operator of order zero to be the multiplication by a smooth function $m_f (g) = f \cdot g$, or alternatively an operator $D$ such that the commutator $[D, m_g]=D \circ m_g - m_g \circ D$ is zero for any smooth function g.

Define inductively differential operator of order $\leq k$ to be an operator $D$ such that $[D, m_g]$ is a differential operator of order $\leq k-1$ for any smooth $g$. Denote the set of all differential operators of order $\leq k$ by $Diff^k (M)$.

Luviu's book than proves that differential operators are local in the sense that supp $D(f) \subset$ supp $f$, which means that we can restrict those operators to open subsets of $M$, as is usually done to show that differential forms form a sheaf if one starts with top-down approach thinking of differential forms as alternating $C^{\infty}(M)$ multilinear maps of vector fields.

This naturally brings me to the first question:

Shouldn't the second definition be modified so that a differential operator is a sheaf homomorphism instead of just being a linear map between globally defined functions?

Misha later on in his notes shows in a series of exercises, that if one defines the symbol algebra $\oplus S^i := \oplus \frac{Diff^i(M)}{Diff^{i-1}(M)}$, which is an algebra over $C^{\infty}(M)$, then it is isomorphic to $Sym^{\bullet} \mathfrak{X}$ - symmetric algebra over the vector fields.

Finally he asks the reader to prove that for the case $M=\mathbb{R}^n$ we have an isomorphism of algebras $Diff^k(\mathbb{R}^n) \cong \oplus_{i \leq k} Sym^i \mathfrak{X(\mathbb{R}^n)}$, which is basically the local form of the first definition. The latter algebra is just the space of global sections of the bundle $\oplus_{i \leq k} Sym^i (T \mathbb{R}^n)$, which brings me to my second question:

Can a differential operator of order $\leq k$ on M be thought of as a global section of some bundle of "differential operators"?

By the usual correspondence, that would imply that differential operators of order $\leq k$ form a locally trivial sheaf of $C^{\infty}(M)$ modules.

Misha says that this is indeed the case in the last problem of his problem set, but I can't make a rigorous proof of it and I am asking for a reference where this approach is presented in detail.

I am also puzzled by the fact that it is not usually mentioned in most of the references I've looked at that differential operators can be thought of as sections of some bundle, which seems to be very fundamental from a differential-geometric perspective. Moreover, this approach can further be developed into a very elegant treatment of the symbol.

Solution 1:

There is a standard way to obtain differential operators, even those acting on sections of a vector bundle, as sections of a vector bundle. This goes via jet bundles. Given a vector bundle $E\to M$, the $k$th jet prolongation $J^kE\to M$ is again a vector bundle. Its fiber over $x\in M$ consist of all equivalence classes of local sections defined in a neighborhood of $x$ under the equivalence relation that their Taylor series in $x$ (in some local chart for $M$ and some vector bundle chart of $E$) agree up to order $k$. Denoting the class of $s$ as $j^k_xs$, one defines a tautological map from smooth sections of $E$ to smooth sections of $J^kE$. This associates to $s\in\Gamma(E)$ the map $x\mapsto j^k_xs$. More or less by definition, for a $k$th order differential operator $D$, the value of $D(s)$ in $x$ only depends on $j^k_xs$. Then $D$ defines a bundle map $\tilde D:J^kE\to F$ such that $D(s)(x)=\tilde D(j^k_xs)$ and vice versa. For smooth functions, you can similarly use the jet spaces of smooth functions.

Solution 2:

The answer to both answers is positive.

Sketch of an answer to the first question:

Using definition 2 one can inductively (on $k$) show that any differential operator $D \in Diff^k(M)$ is local, that is $supp(Df) \subset supp(f)$, and by a well-known principle, this means that $D$ extends to a morphism of sheaves $\tilde{D}:C^{\infty}_M \to C^{\infty}_M$, so that $\tilde{D}(M)=D$ on globally defined functions.

Sketch of an answer to the second question:

We "sheafify" definition 2: let a sheaf of zero-order differential operators be $Diff^0_M:=C^{\infty}_M$; then inductively define $$Diff^k_M(U):=\{D: C^{\infty}_M(U) \to C^{\infty}_M(U) \ | \ D \ \text{is} \ \mathbb{R}-\text{linear and for any function} \ f \ \text{on} \ U \ \text{we have} \ [D, m_f] \in Diff^{k-1}_M(U) \}$$ and check that this objects satisfies the sheaf axioms.

Finally one checks that on $U \sim \mathbb{R}^n$ it is locally free, so we get a bundle of differential operators.