Vertical and horizontal subspace

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Discussion Overview

The discussion revolves around the concept of vertical subspaces in differential geometry, particularly focusing on the definition and understanding of vertical vectors as tangent to fibers. Participants explore terminology, definitions, and the implications of these concepts in the context of principal bundles.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about the term "canonical vertical vector" and its origin, with one referencing a differential geometry book by Thierry Masson.
  • A participant clarifies that the term should be understood as a general concept of a vertical vector being tangent to the fiber, rather than a specific "canonical vertical vector."
  • There is a definition presented that states a vertical vector is a vector tangent to the fiber, which is supported by a mathematical expression involving tangent spaces.
  • One participant questions the understanding of the definition, asking how it can be interpreted, while another attempts to relate it to the concept of partial derivatives and their tangency to functions.
  • A participant expresses confusion about mixing concepts and notes that the derivative of a set is a strange concept, indicating a lack of clarity in the discussion.
  • Clarifications are made regarding notation, specifically that T_pP refers to the tangent space at point p, while T_p f denotes the tangent map of a function at p.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the definition of vertical vectors, with some expressing uncertainty and confusion about the concepts discussed. Multiple viewpoints and interpretations remain present throughout the discussion.

Contextual Notes

Limitations include potential misunderstandings of terminology, the mixing of different mathematical concepts, and the unclear relationship between derivatives and the definitions being discussed.

math6
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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. ?
 
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Where did you find the term "canonical vertical vector"?
 
i found it in he differential geometry book for Thierry Masson. why ayou are surprised ?
 
"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".
 
oh yes I'm so sorry when i translate i didn't pay attention .now can you help me to answer the question ?
 
"un vecteur vertical est un vecteur tangent à la fibre"

So, yes, "a vertical vector is a vector tangent to the fiber."
 
why ? how you can understand these from the definition ?
 
""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.
 
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.
 
  • #10
math6 said:
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.
 
  • #11
ok thnxxx very much . me too i doubt for this meaning :)
 
  • #12
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.
 
  • #13
yes i know :) in some books thet denote derivative of function f like these .
 

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