Direct sum of connections

Definition

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

$(\nabla \oplus \nabla ')(s,s')=\nabla (s)\oplus \nabla '(s')$.

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 $E,E',E''$ 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: Direct sum of dual connections equals dual connection to direct sum