# Difference between revisions of "Direct sum of connections"

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<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>. | <math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>. | ||

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+ | ==Facts== | ||

+ | |||

+ | ===Associativity=== | ||

+ | |||

+ | {{further|[[Direct sum of connections is associative upto natural isomorphism]]}} | ||

+ | |||

+ | ===Commutativity=== | ||

+ | |||

+ | {{further|[[Direct sum of connections is commutative upto natural isomorphism]]}} | ||

+ | |||

+ | ===Distributivity relation with tensor product=== | ||

+ | |||

+ | {{further|[[Distributivity relation between direct sum and tensor product of connections]]}} | ||

+ | |||

+ | Suppose <math>E,E',E''</math> are vector bundles over a differential manifold <math>M</math>, with connections <math>\nabla,\nabla',\nabla''</math> respectively. Then, under the natural isomorphism: | ||

+ | |||

+ | <math>E \otimes (E' \oplus E'') \to (E \otimes E') \oplus (E \otimes E'')</math> | ||

+ | |||

+ | we have an identification between <math>\nabla \otimes (\nabla' \oplus \nabla'')</math> and <math>(\nabla \otimes \nabla') \oplus \nabla \otimes \nabla''</math>. Here, <math>\oplus</math> is the [[direct sum of connections]]. | ||

+ | |||

+ | An analogous distributivity law identifies <math>(\nabla \oplus \nabla') \otimes \nabla''</math> and <math>(\nabla \otimes \nabla'') \oplus (\nabla' \otimes \nabla'')</math>. | ||

+ | |||

+ | ===Commutes with dual connection operation=== | ||

+ | |||

+ | {{further|[[Direct sum of dual connections equals dual connection to direct sum]]}} |

## Latest revision as of 21:24, 24 July 2009

## 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`