Differential operator between vector bundles - Imported Knowlpedia Content

Definition

Let $E$ and $F$ be smooth vector bundles over the same smooth manifold $M$ . A linear differential operator of order at most $m$ is a linear map

D:\Gamma^\infty(E)\longrightarrow\Gamma^\infty(F)

such that, in every coordinate neighborhood and bundle trivialization, it has the form

Ds=\sum_{|\alpha|\leq m}a_\alpha\,\partial^\alpha s,

where each $a_\alpha$ is a smooth bundle-homomorphism-valued coefficient. The least such $m$ is the order of $D$ . Thus $D(s)(x)$ depends only on the $m$ -jet of $s$ at $x$ , not on values of $s$ away from $x$ .

Intrinsic characterization

For $f\in C^\infty(M)$ , let $M_f$ denote multiplication by $f$ and set $\operatorname{ad}_{M_f}(D)=[D,M_f]$ . The operator $D$ has order at most $m$ exactly when every $(m+1)$ -fold iterated commutator vanishes:

\operatorname{ad}_{M_{f_m}}\cdots\operatorname{ad}_{M_{f_0}}(D)=0.

This formulation is independent of coordinates and trivializations. Order-zero operators are precisely smooth bundle homomorphisms acting pointwise. These equivalent descriptions are developed in Wells, chapter IV.

Structure and examples

Orders add under composition: if $D_1$ and $D_2$ have orders at most $m_1$ and $m_2$ , then $D_2D_1$ has order at most $m_1+m_2$ . A covariant derivative is first order, while a connection Laplacian is second order. The leading-order coefficients assemble into the principal symbol, which controls ellipticity.

Conventions and scope

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Publisher record. Relevant: chapter IV, differential operators and symbols.
H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: chapter III, differential operators on vector bundles.