Direct sum of connections

Definition

Suppose $M$ is a differential manifold and $E,E^{\prime }$ are vector bundles on $M$. Suppose $\mathrm{\nabla },{\mathrm{\nabla }}^{\prime }$ are connections on $E$ and $E^{\prime }$ respectively. Then, we define $\mathrm{\nabla }\oplus {\mathrm{\nabla }}^{\prime }$ as a connection on $E\oplus E^{\prime }$ given by:

$(\mathrm{\nabla }\oplus {\mathrm{\nabla }}^{\prime })(s,s^{\prime })=\mathrm{\nabla }(s)\oplus {\mathrm{\nabla }}^{\prime }(s^{\prime })$.

Facts

Associativity

Further information: Direct sum of connections is associative upto natural isomorphism

Commutativity

Further information: Direct sum of connections is commutative upto natural isomorphism

Distributivity relation with tensor product

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: Direct sum of dual connections equals dual connection to direct sum