Tensor product of connections

Definition

Suppose $E,E'$ are vector bundles over a differential manifold $M$ . Suppose $\nabla$ is a connection on $E$ and $\nabla'$ is a connection on $E'$ . The tensor product $\nabla \otimes \nabla'$ is defined as the unique connection on $E \otimes E'$ such that the following is satisfied for all sections $s,s'$ of $E,E'$ respectively:

$(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')$ .

Properties

Well-definedness

Further information: Tensor product of connections is well-defined

It is not completely clear from the definition that the tensor product of connections is well-defined. What needs to be shown is that the definition given above for pure tensor products of sections can be extended to all sections consistently, while maintaining the property of being a connection.

Associativity

Further information: Tensor product of connections is associative upto natural isomorphism

Suppose $E,E',E''$ are vector bundles over a differential manifold $M$ , with connections $\nabla,\nabla',\nabla''$ respectively. Then, under the natural isomorphism:

$(E \otimes E') \otimes E'' \to E \otimes (E' \otimes E'')$ ,

the connections $(\nabla \otimes \nabla') \otimes \nabla''$ and $\nabla \otimes (\nabla' \otimes \nabla'')$ get identified.

Commutativity

Further information: Tensor product of connections is commutative upto natural isomorphism

Suppose $E,E'$ are vector bundles over a differential manifold $M$ , with connections $\nabla,\nabla'$ respectively. Then, under the natural isomorphism:

$E \otimes E' \to E' \otimes E$

the connections $\nabla \otimes \nabla'$ and $\nabla' \otimes \nabla$ get identified.

Distributivity with direct sum

Further information: Distributivity relation between direct sum and tensor product of connections

Suppose $E,E',E''$ are vector bundles over a differential manifold $M$ , with connections $\nabla,\nabla',\nabla''$ respectively. Then, under the natural isomorphism:

$E \otimes (E' \oplus E'') \to (E \otimes E') \oplus (E \otimes E'')$

we have an identification between $\nabla \otimes (\nabla' \oplus \nabla'')$ and $(\nabla \otimes \nabla') \oplus \nabla \otimes \nabla''$ . Here, $\oplus$ is the direct sum of connections.

An analogous distributivity law identifies $(\nabla \oplus \nabla') \otimes \nabla''$ and $(\nabla \otimes \nabla'') \oplus (\nabla' \otimes \nabla'')$ .

Commutes with dual connection operation

Further information: Dual connection to tensor product equals tensor product of dual connections

Relation with other interpretations of connection

Tensor product of module structures

Further information: Connection is module structure over connection algebra, Tensor product of connections corresponds to tensor product of modules over connection algebra

A connection on a vector bundle can be thought of as an interpretation of its global sections as a module over the connection algebra. With this interpretation, a tensor product of connections corresponds to the tensor product of these modules over the connection algebra.