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').