0
$\begingroup$

I'm struggling with a very basic question...could someone help?

Take a smooth manifold $M$, and a curve on it, $\gamma:I \rightarrow M$, where $I$ is an interval of the real numbers. Consider a function on the curve, $f(\gamma)$.

What happens to $f$ if we change the manifold itself? And what if we change the coordinate charts on the manifold?

  • 1
    what do you mean by change the manifold? – 2017-01-19
  • 0
    E.g., let's say the original $M$ is a sphere. What happens (qualitatively), if we "morph" it in a cylinder, to the curve $\gamma$ and the function $f$? (I'm pretty sure of the answer, but I need someone else's opinion). – 2017-01-19
  • 0
    What is the codomain of f? – 2017-01-19
  • 0
    You're right, I've omitted it. Think of $f$ as $M \rightarrow \mathbb R$ – 2017-01-19
  • 1
    If you transform your manifold by a map $T: M \to N$, then your curve simply becomes $T \circ \gamma$, and your function $f$ may not be well defined on that new curve. – 2017-01-19

1 Answers 1

1

A curve on a manifold is defined independently of charts. If you have defined it within a chart, you can just use transition maps to go to a different chart.

If we change the (differentiable) manifold, you will need a way to describe the change. This can be done via a smooth map $\phi: N \rightarrow M$. Then the new function will be the pullback of $f$ by $\phi$, defined as $\phi^*f = f \circ \phi$. Now, if $\phi$ is a diffeomorphism, $f \circ \gamma = \phi^*f \circ \phi^{-1} \circ \gamma$. But if there is no diffeomorphism, I don't really see a way to describe the curve in the manifold $N$.

  • 0
    Sorry if I say so roughly, still my idea was that if I switch the manifold, the curve $\gamma$ will change (i.e. there will be a different curve), the function $f$ will not necessary change, and $f(\gamma)$ will change – 2017-01-19
  • 0
    If we change the coordinates on the manifold, $\gamma$ wouldn't change (even if its representation in coordinates will), the representation of the function too, but $f(\gamma)$ will not necessarily change...where I am wrong? – 2017-01-19
  • 1
    I think the whole thing has an answer in the "Active and passive transformations" subject, which I wasn't aware of. Thank you indeed for the details – 2017-01-21
  • 0
    Oh yes, the two ways of looking at change of coordinates. Diffeomorphisms vs change of charts in the differential geometry language. Good luck. – 2017-01-21