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Bookmark or Share- Intuition about pullbacks in differential geometryI think the best intuition for push-forwards and pull-backs is offered from the case M =N M = N and both are embedded in Rn R n. the pull-back amounts to taking the total differential of the coordinate formulas, or equivalently, applying the chain-rule to swap differentials in one coordinate system for another.
- category theory - Meaning of 'pullback of a pullback square ...The intended exercise is, as Zhen Lin points out: Take 8 objects, and form a cube, with all arrows directed towards the back-right-bottom corner. Suppose the back, top, bottom, left, and right faces are pullback squares. Show the front is also a pullback square. (You actually only need some of the faces to be pullbacks.)
- What is a pullback of a metric, and how does it work?This post is over 4 years old by now, so I'm guessing the OP has moved on with their life. But seeing as this post was the first result when I googled "pullback metric", and it has amassed over 8k views, I figure an answer may still be useful to someone else.
- Pullback commutes with dual for locally free sheaf of finite rankA bunch of tiny facts come together here - pullback commutes with direct sums, pullback takes the structure sheaf to the structure sheaf, pullback on functions is the original map on sheaves in the definition of morphism of ringed space...
- Definition of pullback. - Mathematics Stack ExchangeFor differential forms the pull-back is well-defined even if the function is not injective. That said, the formula on page 163 concerns pulling back a form (not a vector field). Again, f ∗ ω(x) = (dfx) ∗ ω[f(x)] Let's analyze this again, dfx is a N -vector-valued function of TxM and ω[f(x)] is a p -form at f(x).
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