Stalks of Manifolds I am currently learning Algebraic Geometry and (at least in my very limited experience so far) stalks have proven to be very useful when discussing properties of locally ringed spaces and schemes. Manifolds can be considered as locally ringed spaces and in fact resemble schemes in some sense. Hence the question

Of what use are stalks to the discussion of properties of (topological or smooth) manifolds?

I haven't seen any references to stalks with respect to manifolds in classical literature on algebraic topology. Neither have I seen an extensive discussion of manifolds as examples of locally ringed spaces in standard literature on algebraic geometry (note however that this might stem from the fact that I have only read a semester's worth on algebraic geometry). A shallow search on the internet did not produce more information either, so I hope asking for reference is appropriate.
Thank you for your time!
 A: It's definitely true that smooth manifolds can be regarded as locally ringed spaces, but this is seldom the approach taken in any introduction on manifolds. It's a good question as to why this is not done. There are probably two reasons:


*

*It adds complications unnecessary to the development of the material; indeed one can escape saying the word locally ringed space when discussing smooth manifolds entirely, so why add the extra terminology?

*We actually don't need to be too concerned about local objects versus global objects when we deal with smooth manifolds, because of partitions of unity.
The first point does not need much explanation. The second point is important, though. If you read John Lee's book on Smooth Manifolds, you will notice that the tangent space is defined to be the space of pointed derivations at $p$ of $\mathscr{C}^\infty(M)$, the space of smooth functions on the manifold. That is, we define  $T_pM$ to be the set of maps $\partial:\mathscr{C}^\infty(M)\to \mathbb{R}$ that are $\mathbb{R}-$linear and satisfy the pointed Leibniz rule:
$$ \partial(fg)=g(p)\partial(f)+f(p)\partial(g).$$
This is well and good, and is a correct definition, but it does not generalize. If you look at Tu's book on the same subject, he defines $T_pM$ to be the set of pointed derivations at $p$ of the germs of smooth functions $\mathscr{C}^\infty_p(M)$, which is exactly the stalk of the structure sheaf at $p$. In particular, elements of $\mathscr{C}^\infty_p(M)$ are equivalence classes $\langle U,f\rangle$ so that $f$ is smooth on $U$ and $p\in U$.  The definition is the same, $\partial:\mathscr{C}^\infty_p(M)\to \mathbb{R}$ an $\mathbb{R}-$linear map satisfying the same Leibniz rule. 
So, why the difference? When we pass to more algebraic objects – like complex manifolds, we start to run into issues. If we look at a compact complex manifold $X$, it follows from the theory of holomorphic functions (Liouville's Theorem) that any global holomorphic function $f\in \mathcal{O}_X(X)$ is a constant. If we try the first definition of the tangent space above, we run quickly into issues. For instance, it isn't hard to see that a pointed derivation $\partial$ like above has $\partial(\lambda)=0$ for any scalar $\lambda \in \mathbb{R}$. So, it we try to define $T_pX$ to be the set of pointed derivations 
$$ \partial:\mathcal{O}_X(X)\to \mathbb{C}$$
we really just get a derivation $\partial:\mathbb{C}\to \mathbb{C}$. Actually, such a $\partial$ vanishes on constants, so by this definition, $T_pX=0$ no matter what compact complex manifold we choose. However, if we work instead with germs of functions, we get the right amount of functions, so that the tangent space could be defined as the set of pointed derivations 
$$ \partial:\mathcal{O}_{X,p}\to \mathbb{C}.$$
The moral of this story is that in the smooth category, the difference between local and global objects is not so different as in the complex or algebraic case. So, when we study complex manifolds, algebraic varieties, or schemes we really do need to use sheaves to keep track of local data. In the smooth case, we can escape without worrying so much.
By the way, the two definitions are equivalent in the smooth manifolds case because you can choose a bump function around $p$, call it $\rho$ such that $\rho\equiv 1$ in a neighborhood of $p$, and extend germs of $f_p$ to global functions by multiplying by $\rho$.
