From page $64$ of Heat Kernels and Dirac Operators:
Let $E$ be a vector bundle over $M$. The filtered algebra of differential operators on $E$, denoted by $D(M, E)$, is the subalgebra of $\mathrm{End}(\Gamma(M, E))$ generated by elements of $\Gamma(M, \mathrm{End}(E))$ and the covariant derivatives $\nabla_X$, where $\nabla$ is any covariant derivative on $E$ and $X$ ranges over all vector fields on $M$. The algebra $D(M, E)$ has a natural filtration, defined by letting \begin{equation} D_i(M, E) = \Gamma(M, \mathrm{End}(E))\cdot\mathrm{span}\{\nabla_{X_1}\cdots\nabla_{X_j}:j \leq i\}. \end{equation} We call an element of $D_i(M, E)$ an $i$-th differential order operator.
In view of the rest of the chapter I believe that the differential operators in the book correspond to the Partial differential operators in Nicolaescu's notes. But the definition seems a bit off:
- First of all, the definition is only correct for trivial bundles on trivial manifolds and otherwise the representation is only possible locally, isn't it?
- Given a function $U\times V\to W$ I would expect that $U\cdot V=\{u\cdot v:(u,v)\in U\times V\}$, but this doesnt yield the correct definition of $D_i(M,E)$, does it? I feel like we should move the "span" to the left. But even then the definition does not seem quite correct to me:
- Consider a trivial bundle $E= M\times V$ on a trivial manifold $M$. Let \begin{equation} \nabla=\mathrm{d}:C^\infty(M)\to\Omega^1(M) \end{equation} be the exterior differential, which induces a covariant derivative $\nabla:\Gamma(M,E)\to\Omega^1(M,E)$ and let $(X_1,\ldots,X_n)$ be the frame induced by a chart defined on all of $M$. Then it is well known that $D\in D_i(M,E)$ can be written as \begin{equation} D=\sum_{\alpha:|\alpha|\leq i}A_\alpha \nabla_{X_{\alpha_1}}\cdots \nabla_{X_{\alpha_{|\alpha|}}}e \end{equation} This is different from the definition in the book which says that \begin{equation} D=\sum_{j=0}^iA_j\nabla_{X_1}\cdots \nabla_{X_j} \end{equation} for some list of vector fields $(X_1,\ldots,X_i)$, right?