Vertical and horizontal subspace of a vector space T_pP.

In summary, a principle fiber bundle P at a point p \in P has a decomposition T_pP=V_pP + H_pP, where the vertical subspace V_pP is uniquely defined while the horizontal subspace H_pP is not. This is a basic fact in linear algebra and means that the complement to a unique subspace is also unique. In the context of a principle fiber bundle, this can be seen in the determination of the horizontal subspace for the horizontal lift of a curve in the base manifold M to a curve in the bundle P, where some connect is needed to determine the horizontal subspace. This problem can be extended to the problem of finding subspaces in general, where many choices may be isomorphic but
  • #1
wdlang
307
0
suppose we have a principle fiber bundle P

at a point p \in P

we have the decomposition T_pP=V_pP + H_pP

it is said that the vertical subspace V_pP is uniquely defined while H_pP is not

i cannot understand this point

the complement to a unique subspace is surely unique, i think.

it is a basic fact in linear algebra.
 
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  • #2
I assume you mean a vector bundle, right.? (or maybe you use a different name for it),
and I guess you want a direct sum decomposition and uniqueness up to isomorphism.?

Where did you read that the complement was not unique (and unique up to what)?.

Then, you are correct: given an n-dimensional subspace S of an m-dim
vector space V , we can always define a subspace S' (e.g., by extending the
basis B_S of S to a basis B_V for V , so that S' is the subspace with basis
B_V-B_S )so that

V=S(+)S'

And dimensions add up, so DimS'=m-n . so S' is unique up to vector space isomorphism.

And all
 
  • #3
Bacle said:
I assume you mean a vector bundle, right.? (or maybe you use a different name for it),
and I guess you want a direct sum decomposition and uniqueness up to isomorphism.?

Where did you read that the complement was not unique (and unique up to what)?.

Then, you are correct: given an n-dimensional subspace S of an m-dim
vector space V , we can always define a subspace S' (e.g., by extending the
basis B_S of S to a basis B_V for V , so that S' is the subspace with basis
B_V-B_S )so that

V=S(+)S'

And dimensions add up, so DimS'=m-n . so S' is unique up to vector space isomorphism.

And all

no, it is about principle fiber bundle

it is in the context of horizontal lift of a curve in the base manifold M to a curve in the bundle P

it is said that some connect is needed to determine the horizontal subspace.
 
  • #4
wdlang said:
the complement to a unique subspace is surely unique, i think.

The orthogonal complement is unique, but that needs an inner product. In general, given a vector space V with a subspace U, there are many choices of subspaces W such that V is the direct sum of U and W.
 
  • #5
But there is only one such W up to isomorphism, by invariance of dimension.

I have always been curious about "solving" for quotients, or direct sums, i.e.,

If we know V=S(+)S' , and we know S, how do we find S' up to isomorphism;

similarly, if we know that a group G is the quotient of two groups H,K, i.e.,

G~ H/K , and we only know either H or K, but not both, can we find the other.?
 
  • #6
But the isomorphism problem for vector spaces is nearly trivial (two vector spaces are isomorphic iff they have the same dimension), making it uninteresting. If V = S ⊕ S', then S' is always isomorphic to V/S, but who cares?

Back to the original problem: It wasn't saying unique up to isomorphism; just unique.edit: I did some reading about the objects in the original problem. A fibre bundle p: E -> M gives a canonically defined (i.e. uniquely satisfies a certain property) linear map of vector bundles Tp: TE -> TM; the vertical bundle VE is defined as the kernel of Tp. A horizontal bundle is then an arbitrary subbundle HE of TE such that TE = VE ⊕ HE. Even though any two horizontal bundles may be isomorphic, they may still be different subbundles of TE.

Generally speaking, if you're given a fixed object, you aren't interested in the various subobjects only up to isomorphism.
 
Last edited:
  • #7
adriank said:
The orthogonal complement is unique, but that needs an inner product. In general, given a vector space V with a subspace U, there are many choices of subspaces W such that V is the direct sum of U and W.

to the point

thanks a lot!
 
  • #8
Spelling nitpick: It's principal, not principle.
 
  • #9


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. ?
 
  • #10
wdlang said:
suppose we have a principle fiber bundle P

at a point p \in P

we have the decomposition T_pP=V_pP + H_pP

it is said that the vertical subspace V_pP is uniquely defined while H_pP is not

i cannot understand this point

the complement to a unique subspace is surely unique, i think.

it is a basic fact in linear algebra.

The vertical space is tangent to the fiber. But a complementary subspace of the tangent space to the fiber is not uniquely determined.
 

1. What is a vector space T_pP?

A vector space T_pP is a mathematical concept that represents a set of vectors (or arrows) that can be added together and multiplied by a scalar to create new vectors. It is typically used to describe the properties of a point p in a larger space P, and is useful in fields such as linear algebra and differential geometry.

2. What is a subspace of a vector space T_pP?

A subspace of a vector space T_pP is a subset of T_pP that still maintains the properties of a vector space. This means that it must be closed under vector addition and scalar multiplication, and must contain the zero vector. In simpler terms, it is a smaller space within a larger space that still follows the same rules.

3. What is the difference between a vertical and horizontal subspace of a vector space T_pP?

A vertical subspace of a vector space T_pP is a subspace that is tangent to the space P at point p. This means that it is perpendicular to the tangent space of P at p, and can be thought of as the direction in which the space P is "vertical" at point p. A horizontal subspace, on the other hand, is a subspace that is tangent to both the space P at point p and the tangent space of P at p. This means that it is perpendicular to both the vertical subspace and the tangent space at p, and can be thought of as the direction in which the space P is "horizontal" at point p.

4. How are vertical and horizontal subspaces related to each other?

Vertical and horizontal subspaces are complementary to each other, meaning that they together span the entire tangent space of P at point p. In other words, any vector in the tangent space of P at p can be expressed as a combination of a vector in the vertical subspace and a vector in the horizontal subspace. This relationship is important in fields such as differential geometry, where it is used to study the curvature and properties of surfaces.

5. What are some applications of understanding vertical and horizontal subspaces of a vector space T_pP?

Understanding vertical and horizontal subspaces of a vector space T_pP is crucial in fields such as differential geometry, where it is used to study the properties of surfaces. It is also useful in fields such as physics and engineering, where it is used to describe and analyze the behavior of systems. Additionally, it has applications in computer graphics and computer vision, where it is used to manipulate and analyze images and 3D models.

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