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