# Changes

## Connection is splitting of first-order symbol sequence

, 17:20, 6 January 2012
no edit summary
Suppose $E$ is a [[vector bundle]] over a [[differential manifold]] $M$. Denote by $\mathcal{E}$ the [[sheaf of sections of a vector bundle|sheaf of sections]] of $E$. Consider the first-order symbol sequence for $E$, given by:
$0 \to \operatorname{Hom}(\mathcal{E},\mathcal{A}) \to \mathcal{D}^1(\mathcal{E},\mathcal{A}) \to \operatorname{Hom}(\mathcal{E},\mathcal{T}) \to 0$
Here $\mathcal{A}$ is the [[sheaf of infinitely differentiable functions]] on $M$, $\mathcal{D}^1$ denotes the space of first-order differential operators from $\mathcal{E}$ to $\mathcal{A}$, and $\mathcal{T}$ denotes the [[sheaf of derivations]] of $M$.
A splitting of the above sequence is equivalent to a section map:
$\operatorname{Hom}(\mathcal{E},\mathcal{T}) \to \mathcal{D}^1(\mathcal{E},A)$
which is equivalent to a map (satisfying some additional conditions):