Vertical and horizontal subspace

[math6]

where in the definition of vertical subspace we understand that the notion of canonical vertical vector: a vertical vector is a vector tangent to the fiber. ?

 

[arkajad]

Where did you find the term "canonical vertical vector"?

 

[math6]

i found it in he differential geometry book for Thierry Masson. why ayou are surprised ?

 

[arkajad]

"Soit P(M,G) un fibré principal. Sur la variété P, nous avons la notion canonique de vecteur vertical"

Here it is to be understood as a general term: "canonical concept of a vertical vector" not "a concept of canonical vertical vector".

 

[math6]

oh yes I'm so sorry when i translate i didn't pay attention .now can you help me to answer the question ?

 

[arkajad]

"un vecteur vertical est un vecteur tangent à la fibre"

So, yes, "a vertical vector is a vector tangent to the fiber."

 

[math6]

why ? how you can understand these from the definition ?

 

[arkajad]

""un vecteur vertical est un vecteur tangent à la fibre" - This is a definition!

It can be written also as: We will call a vector tangent to P at p a vertical vector if it is tangent to the fiber through p. It is clear then, from the definition that $$X\in T_p P$$ is vertical if and only if $$T_p\pi X=0$$ which can also be written as $$(d\pi)_p(X)=0.$$

 

[math6]

we know that if we take a function f, the partial derivative of f is tangent to f? that's true? is not it?
So we can do the same analysis taking into account that (dP) is the derivative of P?
I do not know? I wanted to give meaning to this definition.

 

[arkajad]

we know that if we take a function f, the partial derivative of f is tangent to f? that's true? is not it?

You are mixing different concepts here. Thus: neither yes nor no. Your question is messy.

So we can do the same analysis taking into account that (dP) is the derivative of P?
I do not know? I wanted to give meaning to this definition.

There is no such thing as dP. P is a set and a derivative of a set is a rather strange concept.

 

[math6]

ok thnxxx very much . me too i doubt for this meaning :)

 

[arkajad]

By the way: in Masson's book $$T_pP$$ stands for the tangent space to P at p, while $$T_p f$$ stands for the tangent map (derivative) of some map f, taken at p.

 

[math6]

yes i know :) in some books thet denote derivative of function f like these .