Tensor product of connections

Definition

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

$(\mathrm{\nabla }\otimes {\mathrm{\nabla }}^{\prime }{)}_X(s\otimes s^{\prime })={\mathrm{\nabla }}_X(s)\otimes s^{\prime }+s\otimes \mathrm{\nabla }{\text{'}}_X(s^{\prime })$.

Facts

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 $E,E^{\prime },E^{″}$ are vector bundles over a differential manifold $M$, with connections $\mathrm{\nabla },{\mathrm{\nabla }}^{\prime },{\mathrm{\nabla }}^{″}$ respectively. Then, under the natural isomorphism:

$(E\otimes E^{\prime })\otimes E^{″}\to E\otimes (E^{\prime }\otimes E^{″})$,

the connections $(\mathrm{\nabla }\otimes {\mathrm{\nabla }}^{\prime })\otimes {\mathrm{\nabla }}^{″}$ and $\mathrm{\nabla }\otimes ({\mathrm{\nabla }}^{\prime }\otimes {\mathrm{\nabla }}^{″})$ get identified.

Commutativity

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

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

$E\otimes E^{\prime }\to E^{\prime }\otimes E$

the connections $\mathrm{\nabla }\otimes {\mathrm{\nabla }}^{\prime }$ and ${\mathrm{\nabla }}^{\prime }\otimes \mathrm{\nabla }$ get identified.

Distributivity with direct sum

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

Suppose $E,E^{\prime },E^{″}$ are vector bundles over a differential manifold $M$, with connections $\mathrm{\nabla },{\mathrm{\nabla }}^{\prime },{\mathrm{\nabla }}^{″}$ respectively. Then, under the natural isomorphism:

$E\otimes (E^{\prime }\oplus E^{″})\to (E\otimes E^{\prime })\oplus (E\otimes E^{″})$

we have an identification between $\mathrm{\nabla }\otimes ({\mathrm{\nabla }}^{\prime }\oplus {\mathrm{\nabla }}^{″})$ and $(\mathrm{\nabla }\otimes {\mathrm{\nabla }}^{\prime })\oplus \mathrm{\nabla }\otimes {\mathrm{\nabla }}^{″}$. Here, $\oplus$ is the direct sum of connections.

An analogous distributivity law identifies $(\mathrm{\nabla }\oplus {\mathrm{\nabla }}^{\prime })\otimes {\mathrm{\nabla }}^{″}$ and $(\mathrm{\nabla }\otimes {\mathrm{\nabla }}^{″})\oplus ({\mathrm{\nabla }}^{\prime }\otimes {\mathrm{\nabla }}^{″})$.

Commutes with dual connection operation

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