# Tensor product of connections

## Contents

## 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:

.

## Facts

### 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 **Failed to parse (unknown function "\optimes"): (\nabla \otimes \nabla'') \oplus (\nabla' \optimes \nabla'')**
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