Graduate Proving the Differential Map (Pushforward) is Well-Defined

[Fgard]

I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined.

The map
$$f:M\rightarrow N$$ is a smooth map, where $M, N$ are two smooth manifolds. If $(U,x)\in A_M$ and $(V,y)\in A_N$ are two charts in their respective atlas then the map $x\circ f\circ y^{-1}$ is also smooth. Then what remains to prove is that $f$ is surjective. Is this correct? Or do I need to prove something else as well?

 

[martinbn]

No, what you have is the map, the differential of this map is another map, that takes tangent vectors on $M$ to tangent vectors on $N$. You have to prove that whatever definition was given in the lectures (something we can only guess) is well defined.

 

[fresh_42]

I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined.

The map
$$f:M\rightarrow N$$ is a smooth map, where $M, N$ are two smooth manifolds. If $(U,x)\in A_M$ and $(V,y)\in A_N$ are two charts in their respective atlas then the map $x\circ f\circ y^{-1}$ is also smooth. Then what remains to prove is that $f$ is surjective. Is this correct? Or do I need to prove something else as well?

Shouldn't it be $y\circ f\circ x^{-1}$ which is smooth? But besides this, the differentiability is given by definition. If the chart mappings weren't smooth, we wouldn't call the manifolds smooth. Also surjectivity isn't needed except you only want to consider $N=f(M)$. Well-definition means, that no two elements of $N$ can be the image of one point in $M$. What does this mean in the given situation? We don't have an element in $M$ that could end up in two different ways under $f$. But we do have eventually two different charts for this element $x \in M$. Could it end up in two different charts of $N$ where they represent different points?

 

[Fgard]

Okej, so I have to choose a definition for the differential map and show that map dose not depend on a certain choice of chart. Thanks.

 

[martinbn]

Okej, so I have to choose a definition for the differential map and show that map dose not depend on a certain choice of chart. Thanks.

Well, you should choose the one your professor told you to think about. And you need to show it doesn't depend on whatever choice was made, it could be a chart, but it could be something else. For example given a vector on M, choose a curve, whose tangent vector is the given one, then map the curve to N, take the tangent vector. If your definition is something along these lines, then you need to show that it doesn't depend on the choice of the curve.