# Difference between revisions of "Tensor product of connections"

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<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>. | <math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>. | ||

− | == | + | ==Properties== |

===Well-definedness=== | ===Well-definedness=== | ||

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{{further|[[Dual connection to tensor product equals tensor product of dual connections]]}} | {{further|[[Dual connection to tensor product equals tensor product of dual connections]]}} | ||

+ | |||

+ | ==Relation with other interpretations of connection== | ||

+ | |||

+ | ===Tensor product of module structures=== | ||

+ | |||

+ | {{further|[[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=== | ||

+ | |||

+ | {{fillin}} | ||

+ | |||

+ | ==Facts== | ||

+ | |||

+ | ===Formula for Riemann curvature tensor=== | ||

+ | |||

+ | {{further|[[Formula for curvature of tensor product of connections]]} | ||

+ | |||

+ | ===Preservation of properties=== | ||

+ | |||

+ | * [[Tensor product of flat connections is flat]] | ||

+ | * [[Tensor product of metric connections is metric]] |

## Revision as of 21:46, 24 July 2009

## 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|Formula for curvature of tensor product of connections}