The Differential On The Tangent Bundle
The differential of a smooth map φ induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of M to the tangent bundle of N, denoted by dφ or φ*, which fits into the following commutative diagram:
where πM and πN denote the bundle projections of the tangent bundles of M and N respectively.
Equivalently (see bundle map), φ* = dφ is a bundle map from TM to the pullback bundle φ*TN over M, which may in turn be viewed as a section of the vector bundle Hom(TM,φ*TN) over M.
Read more about this topic: Pushforward (differential)
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