View source for Connection is module structure over connection algebra

==Statement==

Let <math>E</math> be a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Then, a [[connection]] on <math>E</math> is equivalent to giving <math>\Gamma(E)</math> (the vector space of sections of <math>E</math>) the structure of a module over the [[connection algebra]] of <math>M</math>. Equivalently, it gives <math>\mathcal{E}</math> (the [[sheaf of sections of a vector bundle|sheaf of sections]] of <math>E</math>) the structure of a module over the [[sheaf of connection algebras]] over <math>M</math>.

==Definitions used==

===Connection===
{{further|[[Connection]]}}

===Connection algebra===
{{further|[[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 <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>.

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

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

as follows:

<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 that the map <math>\nabla \mapsto f_\nabla</math> is a <math>C^\infty(M)</math>-bimodule map from <math>D^1(M)</math> to <math>B</math>, i.e., left and right multiplication by <math>m(g)</math> can be ''pulled out'' of the <math>f_\nabla</math>:

* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.
* Left module map property: 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) \cdot (X + m(h))(s) = (m(g) \circ 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) \cdot (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) = m(g)f_\nabla(X + m(h))(s)</math>.

* 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)) \cdot m(g))(s) = (f_\nabla(X + m(h)) \circ m(g))(s)</math>. This essentially follows from the Leibniz rule property.

<math>f_\nabla((X + m(h)) \cdot 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) = (f_\nabla(X + m(h)) \cdot m(g))(s)</math>.

Since <math>\nabla \mapsto f_\nabla</math> is a <math>C^\infty</math>-bimodule map, it extends to a unique <math>C^\infty</math>-bimodule map from the <math>C^\infty</math>-tensor algebra over <math>\mathcal{D}^1(M)</math>. By definition, the element <math>m(1) - 1</math> induces the zero map on <math>\Gamma(E)</math>, so the map descends to a homomorphism <math>\mathcal{C}(M) \to B</math>, as desired.

===From a module structure to a connection===

'''Given''': A manifold <math>M</math>, a vector bundle <math>E</math> over <math>M</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>. A module structure of <math>\Gamma(E)</math> over <math>\mathcal{C}(M)</math>.

==References==

===Textbook references===

* {{booklink|Globalcalculus}}, Page 64