Connection is module structure over connection algebra

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Statement

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

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 M, a vector bundle E over M, a connection \nabla on E. B is the algebra of smooth fiber-preserving maps from \Gamma(E) to \Gamma(E). \mathcal{D}^1(M) is the Lie algebra of first-order differential operators on M and \mathcal{C}(M) is the connection algebra on M.

To prove: \nabla gives rise to a homomorphism from \mathcal{C}(M) to B.

Proof: \nabla gives rise to a map:

f_\nabla: D^1(M) \to B

as follows:

f_\nabla(X+m(g)) = s \mapsto \nabla_X(s) + (gs).

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

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

  • f_\nabla is \R-bilinear: This is obvious.
  • Left module map property: For any element X + m(g) in \mathcal{D}^1(M) and any h \in C^\infty(M), we have f_\nabla(m(g) \cdot (X + m(h))(s) = (m(g) \circ f_\nabla(X + m(h)))(s). This essentially follows from the fact that a connection is tensorial in the direction of differentiation:

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).

  • For any element X + m(g) in \mathcal{D}^1(M) and any h \in C^\infty(M), we have (f_\nabla((X + m(h)) \cdot m(g))(s) = (f_\nabla(X + m(h)) \circ m(g))(s). This essentially follows from the Leibniz rule property.

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).

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

From a module structure to a connection

Given: A manifold M, a vector bundle E over M. B is the algebra of smooth fiber-preserving maps from \Gamma(E) to \Gamma(E). \mathcal{D}^1(M) is the Lie algebra of first-order differential operators on M and \mathcal{C}(M) is the connection algebra on M. A module structure of \Gamma(E) over \mathcal{C}(M).

References

Textbook references