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Properties of Derivations and of Tangent Vectors

  1. Aug 24, 2010 #1
    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').?

  2. jcsd
  3. Aug 24, 2010 #2


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    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.
  4. Aug 24, 2010 #3
    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.?
  5. Aug 24, 2010 #4


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    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.
  6. Aug 24, 2010 #5
    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.
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