# Connection is splitting of first-order symbol sequence

## 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