I Definition of tangent space: why germs?

  • Thread starter strauser
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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

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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

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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

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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

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@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.
 
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