- #1

- 37

- 4

He starts off in Chapter 1 by introducing some definitions on ##\mathbb{R}^n## that will carry across to general manifolds.

In Chapter 1, 2.2, he defines germs of functions as a certain equivalence class of smooth functions ##C^\infty_p##. I have not come across this concept previously.

In Chapter 1, 2.3, he then defines, incompletely, derivations as a mapping ##D_v : C^\infty_p \to \mathbb{R}## where ##D_v## is a previously defined directional derivative. (I say "incompletely" as ##D_v## was previously defined as a mapping on a smooth function to the reals. I assume that the full definition applies ##D_v## to any representative of ##C^\infty_p## and then shows that this is well-defined.)

He then shows that there is a vector space isomorphism between tangent vectors in ##\mathbb{R}^n## and derivations.

However, it is not clear to me what is the advantage of defining a derivation as a mapping from the set of germs, as opposed to the set of underlying functions. Nothing in his results seem to depend crucially on germs, unless I am confused; it seems that it would work as well, and be simpler, merely to work with derivations ##D_v : C^\infty \to \mathbb{R}## i.e. from the set of smooth functions.

What am I missing here? The only possible advantage I can think of is that it is a bit of mathematical "cleanliness", in the sense that it stops someone complaining that a given tangent vector at a point can be associated with derivations of two different functions ##f,g## at that point, if the functions agree in a neighbourhood of p. However, this doesn't strike me as a major problem, and not really enough to make worthwhile the definition involving germs.

Do germs become unavoidable at some in manifold theory, and he is merely introducing them early to save doing so later?