Tangent Space and Cotangent Space on a Surface.

[geordief]

I have read that the Tangent space and the Cotangent Spaces are Duals of each other.

Why is this so?

I  can understand that both  are vector spaces and so "qualify" on that account but are they uniquely qualified to be Duals of each other ?

Is the fact that they have a vector in common  (the point p on the surface) important*?

Can the Tangent space be Dual to any other vector spaces or is the Cotangent Space the only possibility?

 

*important in making them Dual Spaces

Edited February 1, 2020 by geordief

[taeto]

When you have a manifold and a particular point on it, there is a particular tangent space to the manifold located at that point. Now usually the cotangent space to the manifold located at the same point is defined to be the dual space of that tangent space. Your question suggests that you have in mind some other definition of the cotangent space, and if so, then which?

Edited February 1, 2020 by taeto

[geordief]

33 minutes ago, taeto said:

When you have a manifold and a particular point on it, there is a particular tangent space to the manifold located at that point. Now usually the cotangent space to the manifold located at the same point is defined to be the dual space of that tangent space. Your question suggests that you have in mind some other definition of the cotangent space, and if so, then which?

Well I have come across the geometrical representation of the Tangent space as a kind of group of all the vectors at a point of a Surface that are tangential to the surface

Visually they look  a bicycle wheel with very many spokes lying flat on the surface with the centre of the wheel situated at the point on the surface

 

Now ,the Cotangent Planes I view visually as an open book with many,many pages and with the spine of the book following a direction that is perpendicular to the point on the surface of the surface.

 

So I picture the first plane (the Tangent plane) as geometrically perpendicular to the Cotangent Planes

 

I see them as very similar but mutually perpendicular.

 

Have I got a correct picture of the Cotangent plane(s)?

[taeto]

Well, if we look at an actual example, then the real line \(\mathbb{R}\) itself is quite a respectable example of a real manifold, with which I hope you will agree. At any point \(x\) the tangent space \(T_x\) is again isomorphic to \(\mathbb{R}\), right? How would you explain the cotangent space at \(x\), and the geometric picture that you will use to describe it? Is there some sense in which it is perpendicular to \(T_x\)?  

[geordief]

32 minutes ago, taeto said:

Well, if we look at an actual example.........

.

Would the Tangent space at x be   a plane above and along the line   and the the Tangent space any one of any number of  those vertical "revolving doors"  you go through going into e.g. a hospital?

That "revolving door" is centred at x.

If you take the Tangent Plane and "swing  it" 90 degrees  ( about a fulcrum at x) you also get one of the Cotangent planes(set those planes "revolving" and you get  any number of them.)

 

Does that make sense?

 

 

[taeto]

4 minutes ago, geordief said:

Would the Tangent space at x be   a plane above and along the line   and the the Tangent space any one of any number of  those vertical "revolving doors"  you go through going into e.g. a hospital?

That "revolving door" is centred at x.

If you take the Tangent Plane and "swing  it" 90 degrees  ( about a fulcrum at x) you also get one of the Cotangent planes(set those planes "revolving" and you get  any number of them.)

Does that make sense?

First of all, since \(\mathbb{R}\) is a 1-dimensional real manifold, the tangent space at any point is a 1-dimensional space, and so it is a line, not a plane.

And in addition, in usual understanding there is only one cotangent space at every point of the manifold. If you have a different understanding which allows for several cotangents at a point, then you have to provide that definition of "cotangent" which you find suitable to allow for this.  

[studiot]

These pictures might help.

Tangent covectors, (cotangents) are also called one-forms.

https://en.wikipedia.org/wiki/One-form

[geordief]

18 minutes ago, taeto said:

First of all, since R is a 1-dimensional real manifold, the tangent space at any point is a 1-dimensional space, and so it is a line, not a plane.

And in addition, in usual understanding there is only one cotangent space at every point of the manifold. If you have a different understanding which allows for several cotangents at a point, then you have to provide that definition of "cotangent" which you find suitable to allow for this.  

Is the Cotangent Space not perpendicular to the Tangent space then ?

If it is not then it seems I have my wires crossed 

 

I thought it was a space defined by/including  a vector that was erected normal to the point p on a manifold (I was thinking of a 2 dimensional surface)

 

Perhaps you are telling me that it  is simply the dual space of the Tangent space and there is  no visually geometric representation of it (as there seems to be with the Tangent space)

 

I am not really familiar with the concept of the Tangent space of a line .....

[taeto]

25 minutes ago, geordief said:

Is the Cotangent Space not perpendicular to the Tangent space then ?

If it is not then it seems I have my wires crossed 

I thought it was a space defined by/including  a vector that was erected normal to the point p on a manifold (I was thinking of a 2 dimensional surface)

Perhaps you are telling me that it  is simply the dual space of the Tangent space and there is  no visually geometric representation of it (as there seems to be with the Tangent space)

I am not really familiar with the concept of the Tangent space of a line .....

But you are okay with 2-dimensional examples of manifolds embedded in 3-dimensional space, and in 4-dimensional space?

In any case you have to distinguish between abstract manifolds, which do not always have to exist as a part of any larger spaces, and 2-dimensional surfaces that can be considered to be part of 3-dimensional space. The abstract manifolds do not really have or need any space around them to wriggle around in, and neither do their tangent spaces nor their cotangent spaces.

Then instead we can consider the 2-dimensional manifold \(\mathbb{R}^2\) embedded in our familiar 3-space, as a fairly boring, yet for the purpose still illustrative example. 

For a point \(x\) in  \(\mathbb{R}^2\) do you see what is the tangent space \(T_x\) and the cotangent space \(T_x^*\)? What are their respective dimensions? Is either of them naturally contained as a subspace of the same 3-dimensional space in which  the original \(\mathbb{R}^2\) lives?

Edited February 1, 2020 by taeto

[geordief]

35 minutes ago, taeto said:

But you are okay with 2-dimensional examples of manifolds embedded in 3-dimensional space, and in 4-dimensional space?

In any case you have to distinguish between abstract manifolds, which do not always have to exist as a part of any larger spaces, and 2-dimensional surfaces that can be considered to be part of 3-dimensional space. The abstract manifolds do not really have or need any space around them to wriggle around in, and neither do their tangent spaces nor their cotangent spaces.

Then instead we can consider the 2-dimensional manifold \(\mathbb{R}^2\) embedded in our familiar 3-space, as a fairly boring, yet for the purpose still illustrative example. 

For a point \(x\) in  \(\mathbb{R}^2\) do you see what is the tangent space \(T_x\) and the cotangent space \(T_x^*\)? What are their respective dimensions? Is either of them naturally contained as a subspace of the same 3-dimensional space in which  the original \(\mathbb{R}^2\) lives?

Well to go with this 2 dimensional surface embedded in the 3-space I  think I have a  very clear picture of the Tangent space.To describe it it would be as if I was one of those circus performers spinning plates at the end of a stick.

I,the performer am vertical (normal) to the surface  and the plates ,hopefully are spinning in the Tangent Plane.

 

And ,yes I think I would  describe that tangent plane as a  2d subset of the 3-space

 

But where is the Cotangent plane,? Not representative   by one of the other  circus performers doing cartwheels from what you seem to be saying  (and as I  was imagining it)

 

I thought it was a plane orthogonal to the Tangent Plane, very much like the z=0 plane is orthogonal to the x=0  plane

 

If that is not the correct plane then I am stumped.

 

Perhaps I am getting mixed up with the terms "plane" and "space"

Is the "Tangent plane" the same as the "Tangent space"?

1 hour ago, studiot said:

These pictures might help.

Tangent covectors, (cotangents) are also called one-forms.

https://en.wikipedia.org/wiki/One-form

I am grasping at this.

Is it ,with covectors as if , instead of running x,=,y = and z = lines  across the 3d space  and measuring the vectors against them ,with covectors we are ,instead drawing parallel surfaces (e.g. x=0,×=1....y=0,y=1.....z=0,z=1     ...z=n) and measuring how many surfaces the covector passes through?

 

With the distance between surfaces being perhaps arbitrary or perhaps  corresponding to that used on the ordinary framework used by the normal vectors.

Is that why I am reading about parameterization  and linear functionals at the moment (either reading around the subject or going in circles )

[geordief]

Suppose we have two hotels in an earthquake zone. Each hotel has an observer inside that uses the hotel he/she is in as their local coordinate system 

ie Observer A uses the different  room levels to indicate  the  value of z and  points along  the  sides of the walls to indicate the values of x or y .

 

After an earthquake , one of the hotels is pushed up on its foundations with the result that  Observer B's  coordinate system is tilted by ,say exactly 45 degrees in each of the x,y,z directions .... (Faulty Towers )

 

Is Observer B's coordinate system (vector space) now the Dual of A's ? Vectors in his coordinate system do indeed  "pierce" the coordinate planes of  those of Observer A.

 

Do I now understand the idea behind the Dual Space?

 

 

Edited February 2, 2020 by geordief

[studiot]

17 hours ago, geordief said:

Is the "Tangent plane" the same as the "Tangent space"?

Well sort of.

A mathematical space is a more general classification than a plane.

A 'space' is contains enough sets of objects to define all the mathematical structure you want to display.

So the space you are talking about contains not only a set of all the elements of your space, another set of all the numbers (scalars) you want to use but also a set of all the things you can do with them according to a fourth set of all the rules and axioms you want to assert. I have highlighted the all since it is the property which guarantees that any operation will yield a known result.
Your space would usually be called a vector space and the elements vectors.
There are only a few rules which are about about combining vectors with each other and with scalars from the scalar set. Other axioms  give you a zero and unit vectors.

A plane is a restricted set of vectors which have all the properties so is a 'subspace' in 3Dimensions, but a space in its own right in 2Dimensions.
There is no combination of plane vectors which can produce any vector in the 3rd dimension, but any pair of non parallel vectors in the plane can be combined in such a way as to create any other vector in the plane.

 

[uncool]

On 2/1/2020 at 2:32 PM, geordief said:

Well I have come across the geometrical representation of the Tangent space as a kind of group of all the vectors at a point of a Surface that are tangential to the surface

Visually they look  a bicycle wheel with very many spokes lying flat on the surface with the centre of the wheel situated at the point on the surface

 

Now ,the Cotangent Planes I view visually as an open book with many,many pages and with the spine of the book following a direction that is perpendicular to the point on the surface of the surface.

 

So I picture the first plane (the Tangent plane) as geometrically perpendicular to the Cotangent Planes

 

I see them as very similar but mutually perpendicular.

 

Have I got a correct picture of the Cotangent plane(s)?

I think you are thinking of "normal vectors", which are not cotangent vectors, no. For one thing, normal vectors depend on how you have embedded your manifold into a larger Euclidean space, whereas tangent and cotangent vectors do not. 

[geordief]

44 minutes ago, uncool said:

I think you are thinking of "normal vectors", which are not cotangent vectors, no. For one thing, normal vectors depend on how you have embedded your manifold into a larger Euclidean space, whereas tangent and cotangent vectors do not. 

Yes,that was what I was thinking of (the mistake I was making,)

So what relation does the Cotangent Space have to the Tangent Space?

I appreciate that it is its Dual but is it  related in any other way?

 

Is there any significance in the fact that the Tangent and the Cotangent are inverse of each other in regular geometry?

 

Are these two spaces somehow also inverses of each other?

 

It is not the case,is it that they are related by means of a 90 degree  rotation about a common (0,0) vector?**

**edit: I think I  have learned elsewhere just now  that this can not  be the case

Edited February 3, 2020 by geordief

[joigus]

On 2/1/2020 at 8:06 PM, geordief said:

I have read that the Tangent space and the Cotangent Spaces are Duals of each other.

Why is this so?

I  can understand that both  are vector spaces and so "qualify" on that account but are they uniquely qualified to be Duals of each other ?

Is the fact that they have a vector in common  (the point p on the surface) important*?

Can the Tangent space be Dual to any other vector spaces or is the Cotangent Space the only possibility?

 

*important in making them Dual Spaces

If you are a future mathematician, I would advise you not to try to think of the cotangent space as something embedded in the space you're starting from. In fact, I bet your problem is very much like mine when I started studying differential geometry: You're picturing in your mind a curved surface in a 3D embedding space, the tangent space as a plane tangentially touching one point on the surface, and then trying to picture in your mind another plane that fits the role of cotangent in some geometric sense. Maybe perpendicular? No, that's incorrect! First of all try to think in terms of intrinsic geometry: there is no external space embedding your surface. Your surface (or n-surface) is all there is. It locally looks to insiders like a plane (or a flat space). What's the other plane? Where is it? It's just a clone of your tangent plane if you wish, that allows you to obtain numbers from your vectors (projections) in the tangent plane. It's the set of all the vectors you may want to project your vector against, therefore, some kind of auxiliary copy of you tangent space. That's more or less all there is to it.

Sometimes there are subtleties involved in forms/vectors related to covariant/contravariant coordinates if you wish to go a step further and completely identify forms with vectors when your basis is not orthogonal. That's why mathematicians have invented a separate concept. Also because mathematicians sometimes need to consider a space of functions and the forms as a bunch of integrals (very different objects). In the less exotic case, the basis of forms identifies completely with the basis of contravariant vectors. I will go into more detail if you're curious about it or send you references.

I hope that helps.

Edited May 3, 2020 by joigus
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