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It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they are, involving sections of smooth sections of vector bundles.

Recall that if $E, F \to M$ are smooth vector bundles over a smooth manifold $M$, a differential operator $E \to F$ is a morphism between the sheaves of smooth sections of $E$ and $F$ such that over an affine chart $U$ trivialising both $E,F$, say with $E|_U \cong U \times \mathbb{R}^n$ and $F|_U \cong U \times \mathbb{R}^m$ , $C^{\infty}(U, \mathbb{R}^n) \cong \Gamma(U, E) \to \Gamma(U, F) \cong C^{\infty}(U, \mathbb{R}^m)$ is given by a classical differential operator.

Question: why is this is the right generalisation of differential operators to manifolds -- and what is the (geometric/physical) significance of the vector bundles here?

After-thought: Ah. I believe I was confused by the fact that differential operators over manifolds are labelled as differential operators on manifolds. Not being pedantic here. It's plain why one might want to consider how smooth assignments of directions vary in certain directions (that is, why one would want to partially differentiate sections of vector bundles). However, I was under the misconception that vector bundles were brought into the picture to facilitate the generalisation of partial derivatives directly on manifolds (which incidentally are just given by the theory over the trivial bundle). A bit silly on reflection.

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Naively, I would've figured a starting point to defining differential operators on a manifold $M$ would be to look at derivations $C^{\infty}(M,\mathbb{R}) \to C^{\infty}(M,\mathbb{R})$. –  Joshua Seaton May 15 at 1:58
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Well, that's fine if all you want to differentiate is functions, but sometimes you want to differentiate sections of vector bundles! E.g. it's interesting to talk about the Laplacian on $k$-forms in Hodge theory. –  Qiaochu Yuan May 15 at 2:02
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First of all, note that your definition of a differential operator includes as a special case the case when both $E$ and $F$ are both the trivial rank one bundle $M \times \mathbb{R}$, in which case sections of $E$ and $F$ are just real-valued functions on $M$.

Also note that on $\mathbb{R}^n$, one might be interested in differential operators that act on vector-valued functions. Vector-valued functions on $\mathbb{R}^n$ are really, in fancy terminology, sections of a trivial vector bundle over $\mathbb{R}^n$. So it is natural to generalize this to sections of more general (possibly nontrivial) vector bundles on manifolds.

In my (limited) experience, one reason that differential operators between vector bundles (e.g., the exterior bundle of differential forms) are important is because some of them (e.g., Laplace- and Dirac-type operators) are closely connected to the geometry and topology of $M$.

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Consider the following operators:

  1. Gradient - Acts on functions and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla \colon \Gamma(M,M \times \mathbb{R}) \rightarrow \Gamma(M,TM)$.
  2. Directional derivative of some vector field with respect to a fixed vector field $X$. Acts on vector fields and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
  3. Divergence - Acts on vector field and return functions. On $M = \mathbb{R}^3$, it has the signature $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M \times \mathbb{R})$.

They are all classical first order differential operators with different domains and codomains that can be interpreted as acting on and returning sections of vector bundles. On $M = \mathbb{R}^3$, all the vector bundles involved (and in general) are trivial and so you can just pick some global isomorphism $TM \cong M \times \mathbb{R}^3$ and think of the operators as operators with signature $D \colon C^{\infty}(\mathbb{R}^3,\mathbb{R}^n) \rightarrow C^{\infty}(\mathbb{R}^3,\mathbb{R}^m)$ for appropriate $n$ and $m$ (they act on vector valued functions and return vector valued functions). However, working with arbitrary manifolds, the notion of a vector field which was before (identified with) just a tuple of smooth function (an element of $C^{\infty}(\mathbb{R}^3,\mathbb{R}^3)$) now generalizes to a section of a possibly non-trivial vector bundle $TM$ and so the operators now should be generalized to operators that act on sections of (possibly non-trivial) vector bundles. And indeed,

  1. Gradient is replaced/generalized with the differential of a function $d \colon \Gamma(M, M \times \mathbb{R}) \rightarrow \Gamma(M, T^{*}M)$ (function $\mapsto$ covector field) or, in the presence of a Riemannian metric, by a gradient operator $\nabla \colon \Gamma(M,M\times \mathbb{R}) \rightarrow \Gamma(M,TM)$ (function $\mapsto$ vector field).
  2. The directional derivative is generalized by the notion of a covariant derivative (also called an affine connection) denoted by $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
  3. Divergence is generalized in the presence of a Riemannian metric to an operator $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M\times \mathbb{R})$.
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