Properties of Derivations and of Tangent Vectors

[Bacle]

Hi, everyone:

I am going over J.Lee's Smooth Manifolds, Chapter 3; specifically, Lemmas

3.1, 3.4, in which he states properties of derivations. Lee calls linear maps L with the

Leibniz property (i.e L(fg)(a)=f(a)L(g)+g(a)L(f) ) derivations, when these maps are

defined in a subset of R^n, and he calls these same maps tangent vectors when these

maps are defined on the tangent space of a general (abstract) manifold. So far, so good.


**Now** , what I find confusing is this: that the lemmas are cited separately,

as if the two lemmas need different proofs; I don't see why different proofs

are necessary.

The two properties cited (same properties cited in lemmas 3.1, 3.4 respectively;

a),b) below are in lemma 3.1, and a'), b') are in lemma 3.4) )

are : (f is in C^oo(M) , a is a point in R^n , p is any point on the manifold M)

a) If f is a constant function, then Lf=0


b) If f(a)=g(a) , then L(fg)(a)=0

a') If f is a constant function, then Xf=0


b') If f(p)=g(p)=0 , then X(fg)(p)=0

The proof for a),b) are straightforward :

a) L(c)=X(1c)=1X(c)+cX(1)=cX(1)+cX(1)

b) L(fg)(p)=f(p)L(g)+g(p)L(f)


Now, why do we need separate proofs of the same facts for a') and b').?

Thanks.

 

[quasar987]

Well, you said it yourself: the first lemma talks about derivations of R^n, and the second one talks about derivations of a manifolds. Not every manifold is an R^n, so you need a proof for the case of a general manifold.

 

[Bacle]

Hi, Quasar:

I don't see any difference between the two cases; in both cases we end up with

an expression f(a)Xg+g(a)Xf , with X linear and both f,g real-valued. I don't see

how both cases are not identical. Any hints, please.?

 

[quasar987]

Well, it is just that in the second lemma, X is a derivation at p on the space of (germs of) smooth maps on the manifold. And those are defined as the (germs of) maps whose composition with the chart maps are smooth, whatever the smooth atlas may be. So, they are more complicated objects than the derivations at p on R^n, who are just the (germs of) smooth maps in the usual sense, which is a special case of the above in the case where the atlas is the one chart atlas (R^n, id).

So clearly the second lemma is a generalization of the first.

 

[Bacle]

Thanks, Quasar:

I was considering working with maps f composed with chart maps, but , AFAIK,
the derivation of a composition is not defined; only the derivation on a product
is defined.