- #1

- 43

- 0

## Main Question or Discussion Point

I've encountered a definition of the tangent vector via the notion of derivations on the manifold and I have some problems with it. I would actually like to show that every derivation can be expressed as a directional derivative, but I'm not ver successful in doing so.

I have this definition of the derivation:

A derivation X in the point p on the manifold M is a linear functional on [tex]C^{\infty}(M)[/tex] such that the Jacobi rule holds: [tex]X(fg) = X(f)g(p)+f(p)X(g)[/tex], where [tex]f,g \in C^{\infty}(M)[/tex]. (The tangent space in p is then the set of all derivations in p.)

Now we know that each directional derivative, i.e. each operator [tex]v_i\frac{\partial }{\partial x_i}[/tex] (in some local coordinates [tex]x_i[/tex]) is a derivation. Thus there is a map [tex]h: DD \rightarrow D [/tex] between the set of directional derivatives (DD) and the set of the derivations (D) and it is easy to see that this map is linear. Now I'd like to show that this map is bijective, i.e. that every derivation can be uniquely expressed by a directional derivative.

I'm able to show the injectivity of this map, by showing that the kernel of h is {0}. To do this, I take any directional derivative [tex]V=v_i\frac{\partial }{\partial x_i}[/tex] (in some coordinate chart U, with local coordinates [tex]x_i[/tex]) and plug in a function [tex]f_j=\chi x_j[/tex], where [tex]\chi[/tex] is a smooth bump function around the point p with support in the chart U. This gives us that all [tex]v_j=0[/tex] and thus V=0.

But how do I show surjectivity? Note that the the functions are defined globally on the whole manifold, not locally (i.e. not via germs). How can I do this?

I have this definition of the derivation:

A derivation X in the point p on the manifold M is a linear functional on [tex]C^{\infty}(M)[/tex] such that the Jacobi rule holds: [tex]X(fg) = X(f)g(p)+f(p)X(g)[/tex], where [tex]f,g \in C^{\infty}(M)[/tex]. (The tangent space in p is then the set of all derivations in p.)

Now we know that each directional derivative, i.e. each operator [tex]v_i\frac{\partial }{\partial x_i}[/tex] (in some local coordinates [tex]x_i[/tex]) is a derivation. Thus there is a map [tex]h: DD \rightarrow D [/tex] between the set of directional derivatives (DD) and the set of the derivations (D) and it is easy to see that this map is linear. Now I'd like to show that this map is bijective, i.e. that every derivation can be uniquely expressed by a directional derivative.

I'm able to show the injectivity of this map, by showing that the kernel of h is {0}. To do this, I take any directional derivative [tex]V=v_i\frac{\partial }{\partial x_i}[/tex] (in some coordinate chart U, with local coordinates [tex]x_i[/tex]) and plug in a function [tex]f_j=\chi x_j[/tex], where [tex]\chi[/tex] is a smooth bump function around the point p with support in the chart U. This gives us that all [tex]v_j=0[/tex] and thus V=0.

But how do I show surjectivity? Note that the the functions are defined globally on the whole manifold, not locally (i.e. not via germs). How can I do this?

Last edited: