# Direct sum of connections

## Contents

## Definition

Suppose is a differential manifold and are vector bundles on . Suppose are connections on and respectively. Then, we define as a connection on given by:

.

## 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 are vector bundles over a differential manifold , with connections respectively. Then, under the natural isomorphism:

we have an identification between and . Here, is the direct sum of connections.

An analogous distributivity law identifies and .

### Commutes with dual connection operation

`Further information: Direct sum of dual connections equals dual connection to direct sum`