Lagmental Vicfred

Pullback Moves Covectors in the Opposite Direction by Vicfred

tags: mathematics, smooth manifolds, tangent spaces, differential forms

A smooth map pulls differential forms on the target back to forms on the source. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

The mathematical object

A smooth \(n\)-manifold is locally modeled on \(\mathbf R^n\) with smooth transition maps. A smooth map \(F:M\to N\) differentiates to linear maps between tangent spaces.

$$ F^\ast:T^\ast_{F(p)}N\to T^\ast_pM $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ (F^\ast\alpha)_p(v)=\alpha_{F(p)}(dF_pv) $$

One explicit computation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ F^\ast\!\left(\sum_i a_i(y)\,dy^i\right)=\sum_{i,j}a_i(F(x))\frac{\partial F^i}{\partial x^j}\,dx^j $$

Why the identity matters

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad F^\ast:T^\ast_{F(p)}N\to T^\ast_pM,\\[5pt] \mathsf{C}\;&:\quad (F^\ast\alpha)_p(v)=\alpha_{F(p)}(dF_pv). \end{aligned} $$

Where it can fail

Coordinates are computational tools, not intrinsic data. Tensorial formulas must transform correctly on chart overlaps.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] (F^\ast\alpha)_p(v)=\alpha_{F(p)}(dF_pv) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sun 28 May 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.