Direct sum of connections

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Definition

Suppose M is a differential manifold and E,E' are vector bundles on M. Suppose \nabla,\nabla' are connections on E and E' respectively. Then, we define \nabla \oplus \nabla' as a connection on E \oplus E' given by:

(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s').

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 E,E',E'' are vector bundles over a differential manifold M, with connections \nabla,\nabla',\nabla'' respectively. Then, under the natural isomorphism:

E \otimes (E' \oplus E'') \to (E \otimes E') \oplus (E \otimes E'')

we have an identification between \nabla \otimes (\nabla' \oplus \nabla'') and (\nabla \otimes \nabla') \oplus \nabla \otimes \nabla''. Here, \oplus is the direct sum of connections.

An analogous distributivity law identifies (\nabla \oplus \nabla') \otimes \nabla'' and (\nabla \otimes \nabla'') \oplus (\nabla' \otimes \nabla'').

Commutes with dual connection operation

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