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

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.

Tensor product of connections viewed as splittings

Fill this in later

Facts

Formula for Riemann curvature tensor

Further information: Formula for curvature of tensor product of connections