- #1

Korybut

- 60

- 2

- TL;DR Summary
- Need some clarifying commentary on the proof

Hello!

According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##.

Cited here another proposition (1.4.5) states the following

1. For constant function ##D_m(f)=0##

2. If ##f\vert_U=g\vert_U## for some neighborhood ##U## of ##m##, then ##D_m(f)=D_m(g)##.

I don't get the following.

1. Where ##C^\infty## is important? I think that this is important due to algebraic reasons. In Teylor's formula all terms should of the same ##C^k## and this happens only for ##C^\infty##. Am I right?

2. I don't get this ##h##-thing completely. Even without it I can apply derivation ##D_{m_0}## to both sides of (1.4.19) and get statement of the proposition. What am I missing?

According to the attached proposition on ##C^\infty## manifold space of derivations ##D_m M## is isomorphic to Tangent space ##T_m M##.

Cited here another proposition (1.4.5) states the following

1. For constant function ##D_m(f)=0##

2. If ##f\vert_U=g\vert_U## for some neighborhood ##U## of ##m##, then ##D_m(f)=D_m(g)##.

I don't get the following.

1. Where ##C^\infty## is important? I think that this is important due to algebraic reasons. In Teylor's formula all terms should of the same ##C^k## and this happens only for ##C^\infty##. Am I right?

2. I don't get this ##h##-thing completely. Even without it I can apply derivation ##D_{m_0}## to both sides of (1.4.19) and get statement of the proposition. What am I missing?