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Connection is splitting of first-order symbol sequence

45 bytes added, 17:20, 6 January 2012
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Suppose <math>E</math> is a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Denote by <math>\mathcal{E}</math> the [[sheaf of sections of a vector bundle|sheaf of sections]] of <math>E</math>. Consider the first-order symbol sequence for <math>E</math>, given by:
<math>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</math>
Here <math>\mathcal{A}</math> is the [[sheaf of infinitely differentiable functions]] on <math>M</math>, <math>\mathcal{D}^1</math> denotes the space of first-order differential operators from <math>\mathcal{E}</math> to <math>\mathcal{A}</math>, and <math>\mathcal{T}</math> denotes the [[sheaf of derivations]] of <math>M</math>.
A splitting of the above sequence is equivalent to a section map:
<math>\operatorname{Hom}(\mathcal{E},\mathcal{T}) \to \mathcal{D}^1(\mathcal{E},A)</math>
which is equivalent to a map (satisfying some additional conditions):
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