# Difference between revisions of "Connection is module structure over connection algebra"

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==Proof== | ==Proof== | ||

− | + | ===From a connection to a module structure=== | |

− | + | The outline of the proof is as follows: | |

− | + | * We first show that a connection gives an action of the first-order differentiable operators on the space of sections. | |

+ | * Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra. | ||

− | <math>D^1(M) \ | + | '''Given''': A manifold <math>M</math>, a vector bundle <math>E</math> over <math>M</math>, a connection <math>\nabla</math> on <math>E</math>. <math>B</math> is the algebra of smooth fiber-preserving maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>. <math>\mathcal{D}^1(M)</math> is the Lie algebra of first-order differential operators on <math>M</math> and <math>\mathcal{C}(M)</math> is the connection algebra on <math>M</math>. |

− | + | '''To prove''': <math>\nabla</math> gives rise to a homomorphism from <math>\mathcal{C}(M)</math> to <math>B</math>. | |

− | <math> | + | '''Proof''': <math>\nabla</math> gives rise to a map: |

− | + | <math>f_\nabla: D^1(M) \to B</math> | |

− | + | as follows: | |

− | <math>\ | + | <math>f_\nabla(X+m(g)) = s \mapsto \nabla_X(s) + (gs)</math>. |

− | + | First observe that the map sends <math>C^\infty(M) \subset \mathcal{D}^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>. | |

− | + | We now prove some basic results about <math>f_\nabla</math>: | |

− | <math>D^1(M) \ | + | * <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious. |

+ | * For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \dot (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map|tensorial]] in the direction of differentiation: | ||

− | + | <math>f_\nabla(m(g) \dot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs)</math>. | |

− | <math>\mathcal{C | + | * For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property. |

− | + | <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s)</math>. | |

==References== | ==References== |

## Revision as of 00:18, 24 July 2009

## Contents

## Statement

Let be a vector bundle over a differential manifold . Then, a connection on is equivalent to giving (the vector space of sections of ) the structure of a module over the connection algebra of . Equivalently, it gives (the sheaf of sections of ) the structure of a module over the sheaf of connection algebras over .

## Definitions used

### Connection

`Further information: Connection`

### Connection algebra

`Further information: Connection algebra`

## Proof

### From a connection to a module structure

The outline of the proof is as follows:

- We first show that a connection gives an action of the first-order differentiable operators on the space of sections.
- Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra.

**Given**: A manifold , a vector bundle over , a connection on . is the algebra of smooth fiber-preserving maps from to . is the Lie algebra of first-order differential operators on and is the connection algebra on .

**To prove**: gives rise to a homomorphism from to .

**Proof**: gives rise to a map:

as follows:

.

First observe that the map sends to , and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function , goes to the operator of multiplication by the function .

We now prove some basic results about :

- is -bilinear: This is obvious.
- For any element in and any , we have . This essentially follows from the fact that a connection is tensorial in the direction of differentiation:

.

- For any element in and any , we have . This essentially follows from the Leibniz rule property.

.

## References

### Textbook references

- Book:Globalcalculus
^{More info}, Page 64