In this section we give detailed proofs of some of the theorems formulated in previous sections. In addition, we clarify the notion of vector fields along maps and their induced covariant derivatives, because this is used at various places in the book. For a convenient formulation we introduce the tangent bundle of a manifold, the prototype of a vector bundle. Applications to variations of curves will illustrate the usefulness of the concepts.
KeywordsVector Bundle Tangent Space Covariant Derivative Tangential Vector Tangent Bundle
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