Definition of tangent space: why germs?

[strauser]

I am reading "An introduction to manifolds" by Tu.

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?

 

[fresh_42]

As I see it, it is necessary to compare two different atlases and to define a unique maximal atlas. It also allows a clean definition of $\mathbb{R}-$algebras $C^r(M)$ and a reasonable calculus on $M$, i.e. analytic continuations, Taylor expansion etc. Otherwise one always would have to deal with different functions although they are locally identical. However, the entire field is about local behavior so it's natural to concentrate on germs. In the end germs close the gap between a purely analytical approach and the algebraic properties of function spaces. The latter requires uniqueness.

 

[martinbn]

You can ask a similar question for any object that has elements that are equivalence classes. For example why factor groups? Why not just work with cosets?

 

[rpr]

I see that the question is one year old. Nevertheless, for the sake of future visitors, I found a text that explains clearly (IMHO) the technical advantage of germs: they allow you to define a "vector space of function defined locally at p" (that is, in a open set that contains p). The problem is that the set of functions locally defined at p is not a vector space since you do not know how to combine linearly two functions defined on different neighbors of p. Moving to germs solve this technical problem.

http://maths.adelaide.edu.au/peter.hochs/Tangent_spaces.pdf (see pages 26 and following)

 

[lavinia]

@strauser

I rather agree that using germs of smooth functions is overkill for introducing calculus on manifolds. Many people learn calculus without germs.

I have not read your book but unless it introduces sheaves of germs you can probably just think of a germ as represented by a locally defined function.

The analytic point is that the differential of the function at a point $p$ depends only on its values in an arbitrarily small domain around $p$. So all of the stuff outside of any small domain can be thought of as fat that can be trimmed off.

 

[jdc6284]

There are several equivalent definitions of the tangent space, many of which do not rely on the definition of a germ. Lee's Introduction to Smooth Manifolds includes a section in Chapter 3 comparing these definitions. Personally, i hate the definition in Warner, which also uses germs. Lee or Guillemin & Pollack have better treatments, in my opinion.

 

[mathwonk]

The point of germs, is that the tangent space to a manifold at a point p, should be determined just by looking at any open neighborhood of p. Hence one needs a concept that is "infinitesimally local". I.e. if U is an open nbhd of the point p in the manifold M, there should be a natural isomorophism between the tangent spaces Tp(U) and Tp(M). This is easy using germs, but not so much using globally defined function algebras.

Notice that Guillemin and Pollack do not even define abstract manifolds, but only deal with embedded manifolds, so their job is much easier. Warner on the other hand defines abstract manifolds, hence needs a more general definition of tangent space. You may agree that it is preferable at first encounter to restrict to embedded manifolds, as G and P do, but if you want to consider tangent spaces to general manifolds, their approach is not sufficient. I do not have Lee at hand, but I agree it is helpful to compare the abstract with the embedded approach.

There is really nothing sophisticated about germs beyond the ideas of freshman calculus. Think about the definition of a tangent line to a graph at a point (p,f(p)). It is the limit of the secants to that graph passing through that point. Now if we chop off part of that graph, i.e. restrict the function f to an open interval about p, we get a different, smaller family of secants. But if we get close enough to p, the two families of secants become the same. This is exactly the idea of a germ, namely something defined on an open nbhd which is identified with its restriction to a smaller open nbhd.