1
$\begingroup$

In Lee's Introduction to Smooth manifolds, he asserts the following:

If $X$ is a smooth vector field on $M$, and $f$ is a smooth real-valued function defined on an open subset $U \subset M$, we obtain a new function $Xf:U \to \mathbb R$ by defining $Xf(p) = X_pf$. This bothers me since $X_p$ is a derivation at point $p$ of $M$, it should take functions which is defined on the whole $M$, rather than an open subset of $M$. Clearly we can take advantage of the extension lemma to extend $f$ to $\tilde f$ defined on the whole $M$ and define $Xf(p) = X \tilde f(p)$.

Am I right about this? Why bother to start from an open subset of $M$ instead of $M$ itself from the very beginning?

$\endgroup$
3
$\begingroup$

Many natural functions on a manifold $M$ are defined only on an open subset $U$ of $M$ (the most basic example being coordinate systems). The extension lemma is a useful technical tool but it is not constructive so you want to understand in advance which concepts are local and which concepts are global. Given $f \colon U \rightarrow \mathbb{R}$, you can define the directional derivative of $f$ at a point $p \in U$ in the direction $v_p \in T_pM$ by considering $v_p$ as a global derivation of $C^{\infty}(M)$, extending $f$ to a global function $\tilde{f}$ and then computing $v_p(\tilde{f})$ but this is quite pointless because the number $v_p(\tilde{f})$ will be independent of the extension chosen and you won't be able to compute anything with this definition because you won't be able to construct a single extension $\tilde{f}$ to the whole of $M$.

If you consider $v_p \in T_pM$ as an equivalence class of curves, you can always find a representative of $v$ as a curve $\alpha \colon I \rightarrow M$ such that $\alpha(I) \subseteq U$ and then $v_p(f)$ will be just

$$ \frac{d}{dt} f(\alpha(t))|_{t = 0} $$

and this is a definition which can actually be applied to compute something.

From a higher point of view, a tangent vector $v \in T_pM$ is a derivation of the ring $C^{\infty}_p(M)$ consisting of germs of smooth functions at $p$ and so you can apply it to any smooth function that is defined on a neighborhood of $p$ (and in particular, global functions).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.