# Tensor product of connections

## Definition

Suppose are vector bundles over a differential manifold . Suppose is a connection on and is a connection on . The **tensor product** is defined as the unique connection on such that the following is satisfied for all sections of respectively:

.

## Properties

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

,

the connections and get identified.

### Commutativity

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

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

the connections and get identified.

### Distributivity with direct sum

`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: Dual connection to tensor product equals tensor product of dual connections`

## Relation with other interpretations of connection

### Tensor product of module structures

`Further information: Connection is module structure over connection algebra, Tensor product of connections corresponds to tensor product of modules over connection algebra`

A connection on a vector bundle can be thought of as an interpretation of its global sections as a module over the connection algebra. With this interpretation, a tensor product of connections corresponds to the tensor product of these modules over the connection algebra.

### Tensor product of connections viewed as splittings

*Fill this in later*

## Facts

### Formula for Riemann curvature tensor

`Further information: Formula for curvature of tensor product of connections`