# Vertical and horizontal subspace

• math6
In summary, the conversation discusses the concept of canonical vertical vector in the context of differential geometry. The definition of a vertical vector is given as a vector tangent to the fiber, and it is clarified that this is a general term, not specific to a canonical vertical vector. The conversation then delves into the understanding of this definition, with the speaker questioning the relationship between partial derivatives and tangent vectors. The expert concludes by mentioning the notation used for derivatives in the book.
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. ?

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.

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.

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

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.

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

## 1. What is the difference between vertical and horizontal subspace?

Vertical subspace refers to the space along the y-axis, while horizontal subspace refers to the space along the x-axis. In other words, vertical subspace is the height dimension, while horizontal subspace is the width dimension.

## 2. How are vertical and horizontal subspaces used in data analysis?

Vertical and horizontal subspaces are commonly used in data analysis to organize and visualize data. Vertical subspaces are often used to represent categories or groups, while horizontal subspaces are used to display the data values associated with those categories.

## 3. Can vertical and horizontal subspaces be applied in different fields of science?

Yes, vertical and horizontal subspaces can be applied in various fields of science, such as physics, chemistry, biology, and economics. They are particularly useful in analyzing and interpreting data in these fields.

## 4. How do data points in vertical and horizontal subspaces relate to each other?

Data points in vertical and horizontal subspaces are related to each other through their coordinates. Each data point has a specific x-coordinate on the horizontal subspace and a y-coordinate on the vertical subspace, which determine its exact location in the subspace.

## 5. Is there a limit to the number of dimensions in vertical and horizontal subspaces?

Technically, there is no limit to the number of dimensions in vertical and horizontal subspaces. However, in practical data analysis, it is more common to work with two or three dimensions in these subspaces, as it becomes difficult to visualize and interpret data in higher dimensions.

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