Pullback of Connections (covariant Derivatives)
If is a connection (or covariant derivative) on a vector bundle E over N and φ is a smooth map from M to N, then there is a pullback connection on φ*E over M, determined uniquely by the condition that
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“Imagination is an almost divine faculty which, without recourse to any philosophical method, immediately perceives everything: the secret and intimate connections between things, correspondences and analogies.”
—Charles Baudelaire (1821–1867)