2

Pick a point $p$ on a smooth manifold $M$ and consider two $C^{\infty}$ functions $f: U \to \mathbb{R}$ and $g: V \to \mathbb{R}$, where $U,V$ are neighborhoods of $p$. We say that $(f,U)$ and $(g,V)$ are equivalent, if there exists an open $W \subset U \cap V$, such that $p \in W$ and $f|_{W} = g|_{W}$. The equivalence class of $(f,U)$ is then called the germ of $f$ at $p$. The set of all such germs at $p$ is denoted by $C^{\infty}_p(M)$.

I want to prove that $C^{\infty}_p(M)$ is a vector space over $\mathbb{R}$. This is a basic fact that is stated in almost every book on differential geometry, but I have never encountered a proof. I'm also a little bit confused of how to carry through such a proof because of the involved equivalence relation. I'd be very thankful if someone could help me formalize my arguments. I want to restrict myself here to my main problems of understanding:

Let us define the multiplication with an element $\lambda \in \mathbb{R}$ as $\lambda[(f,U)]_p := [(\lambda f, U)]_p$. How can I show that $C^{\infty}_p(M)$ is closed under this operation, i.e. how to show that $[(\lambda f, U)] \in C^{\infty}_p(M)$? I feel like there's almost nothing to show, because of course $\lambda f$ is also $C^{\infty}$ on $U$. How can I establish the existence of its equivalence class? Is there a subtlety I missed, or something else that needs to be shown for this closure property?

As two germs can have different domains we need to define the addition operation on their intersection, i.e. as $[(f, U)]_p + [(g, V)]_p := [(f+g, U\cap V)]_p$. Again $f + g$ is $C^{\infty}$. Is this all that needs to be shown? This feels really unsatisfactory!

We also need to show that the operations defined above are well defined. Let $(f,U) \sim (g,V)$, that is $f$ and $g$ agree in a neighborhood $W \subset U \cap V$ of $p$. Then we need to show that $\lambda [(f,W)]_p = \lambda [(g,W)]_p = [(\lambda f,W)]_p$. Since $f$ and $g$ are equivalent we have $f|_{W} = g|_{W}$. From this it follows that $\lambda f|_{W} = \lambda g|_{W}$ and thus $[(\lambda f,U)]_p = [(\lambda g,V)]_p$. Is this correct?

jazzinsilhouette
  • 1,218
  • 1
    Yes. Also the ring of germs is the quotient of the ring and $\Bbb{R}$-algebra of smooth functions by the ideal of functions vanishing on some neighborhood of the point, so it is automatically a $\Bbb{R}$-algebra – reuns May 28 '19 at 23:30
  • 1
    Yes. You could write a longer proof, but essentially this is the argument. Rigorously what you have shown is that the relations you have defined are functions (they are "well defined") and that they are binary operations (ie they are of the kind $ \ C^{\infty}_p (M) \times C^{\infty}_p (M) \to C^{\infty}_p (M) \ $), its ranges are contained in $C^{\infty}_p (M)$. It remains to show that $C^{\infty}_p (M)$ is actually a $\mathbb{R}$-vector space. For this you need to prove that all eight axioms of vector spaces are satisfied. – Gustavo May 29 '19 at 03:37

0 Answers0