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Connections Nicolson, Robert Alexander

Abstract

The main purpose of this exposition is to explore the relations between the notions of covariant derivative, connection, and spray. We begin by introducing the basic definitions and then use a method of Gromoll, Klingenberg and Meyer to show that covariant derivatives and connections on vector bundles are in a natural one-to-one correspondence. We conclude by showing that on the tangent bundle of a manifold, sprays and "symmetric" connections are in a natural one-to-one correspondence. Although we use a different method, this re-establishes a result of Ambrose, Palais, and Singer.

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