# Difference between revisions of "Connection is splitting of first-order symbol sequence"

(New page: ==Statement== Suppose <math>E</math> is a vector bundle over a differential manifold <math>M</math>.) |
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==Statement== | ==Statement== | ||

− | Suppose <math>E</math> is a [[vector bundle]] over a [[differential manifold]] <math>M</math>. | + | 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 '''connection''' on <math>E</math> is equivalent to a choice of splitting for this sequence. | ||

+ | |||

+ | ==Proof== | ||

+ | |||

+ | 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): | ||

+ | |||

+ | <math>\mathcal{T} \to \mathcal{E} \otimes \mathcal{D}^1(\mathcal{E},\mathcal{A})</math> | ||

+ | |||

+ | The right side is equivalent to <math>\mathcal{D}^1(\mathcal{E},\mathcal{E})</math>, so a splitting of the sequence is equivalent to a map: | ||

+ | |||

+ | <math>\mathcal{T} \to \mathcal{D}^1(\mathcal{E},\mathcal{E})</math> | ||

+ | |||

+ | satisfying some additional conditions. Clearly, a '''connection''' is also a map of the above form, so it remains to check that the additional condition that comes from it being a splitting, is equivalent to the Leibniz rule for the connection. | ||

+ | |||

+ | ==References== | ||

+ | |||

+ | ===Textbook references=== | ||

+ | |||

+ | * {{booklink|Globalcalculus}}, Page 125-127 |

## Latest revision as of 17:20, 6 January 2012

## Statement

Suppose is a vector bundle over a differential manifold . Denote by the sheaf of sections of . Consider the first-order symbol sequence for , given by:

Here is the sheaf of infinitely differentiable functions on , denotes the space of first-order differential operators from to , and denotes the sheaf of derivations of .

A **connection** on is equivalent to a choice of splitting for this sequence.

## Proof

A splitting of the above sequence is equivalent to a section map:

which is equivalent to a map (satisfying some additional conditions):

The right side is equivalent to , so a splitting of the sequence is equivalent to a map:

satisfying some additional conditions. Clearly, a **connection** is also a map of the above form, so it remains to check that the additional condition that comes from it being a splitting, is equivalent to the Leibniz rule for the connection.

## References

### Textbook references

- Book:Globalcalculus
^{More info}, Page 125-127