Logical foundations of smooth manifolds

[Avatrin]

Hi

I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This question is about whether or not there is a more "foundational" subject under smooth manifolds as well. Is there a book or online resource that discusses the logic and methodology used in the study of smooth manifolds?

 

[fresh_42]

Hi

I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This question is about whether or not there is a more "foundational" subject under smooth manifolds as well. Is there a book or online resource that discusses the logic and methodology used in the study of smooth manifolds?

Not that I'm aware of. The foundation is ordinary analysis, not much has changed. Continuity and derivatives are all local conceptions, and this means in return, that we can do analysis on manifolds as soon as we have a local neighborhood which behaves like the Euclidean space we are used to. I'm sure the course started with an atlas, so what's wrong with the comparison of a real atlas or a street map? The real world isn't flat, but we can perfectly cope with it looking into a flat street map. Therefore the logical foundation is completely given by the two properties of an atlas. It is only more work to do, if we consider global properties, in which case the charts will have to be clued.

 

[lavinia]

I agree with @fresh_42 that there is probably not a more general subject that is foundational. There is no way to get around calculus on manifolds but once this is under your belt things should go smoothly (so to speak). Can you describe what you are having trouble with?

In my mind there are several layers to manifold theory - geometrical, differentiable, combinatorial, and topological. Which are you focussing on?

 

[Avatrin]

Well, I guess I was just reading a book that was a bit too dense (Loring Tus Intro to Manifolds). I have started supplementing it with Lees intro to smooth manifolds and I think my grasp of the subject is finally improving.

 

[WWGD]

Lee's book on Smooth Manifolds have a section at the end of the book dealing with the foundations: Linear Algebra, Topology, etc.

 

[Avatrin]

Hmm, I think I've finally gotten where my problem lies; I start struggling with tangent spaces after a while and lose track of what is going on. The different notations in the field do not help, and I think it would be great to get a slower introduction to it. I haven't checked out Lees chapter on it yet, but Tus chapter is a bit too dense.

Also, a lot of definitions are completely unmotivated. It's hard to understand why certain things have to be defined the way they are (like compatible charts). What are some good resources that provide better motivation for the definitions?

 

[fresh_42]

What are some good resources that provide better motivation for the definitions?

I have and find this one a good introduction:
https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20

If you want to have a cheap overview of the jungle (5 parts):
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
or an important example in physics:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/

 

[WWGD]

Well, it is not a simple topic. You can do it, but it may take time. And I agree, notation is a nightmare, but must be dealt with.

 

[mathwonk]

as far as compatible charts, think of how to do calculus on the circle. Since the circle is one dimensional we want to use one coordinate near each point on the circle. e.g. we could project the circle onto the x-axis and use the x coordinate. but two things arise:

1) we cannot use this projection at either of the two points where the circle meets the x axis, since there the projection is not locally injective, so we should only use the x coordinate away from those two points.

2) at all points except the two intersections with the y axis, we could also use projection on the y-axis and use the y coordinate. since there is no way to prefer one coordinate over the other where both are usable, does it matter which coordinate we use?

The answer "no", is called "compatibility of charts", i.e. at all points where both x and y may be used as coordinate, they give the same answer to basic questions, such as "is our function differentiable? I.e. in this case, we show that at the interesting points, y is a smooth function of x and x is smooth function of y, hence any function that is a smooth function of either one is also a smooth function of the other.

 

[mathwonk]

in general, the more foundational subjects for calculus on manifolds are indeed calculus of several variables in euclidean space, hence even more basically, topology and linear algebra. I agree that Loring's discussion is a little terse. But did you read the whole book? The chapter on tangent spces assumes you are well acquainted with the earlier chapters on tangent vectors as derivations. But still I agree it is a little succinct.

In my own career, I became comfortable when I worked carefully first through Spivak's Calculus, and then his Calculus on Manifolds

 

[lavinia]

Hmm, I think I've finally gotten where my problem lies; I start struggling with tangent spaces after a while and lose track of what is going on. The different notations in the field do not help, and I think it would be great to get a slower introduction to it. I haven't checked out Lees chapter on it yet, but Tus chapter is a bit too dense.

Also, a lot of definitions are completely unmotivated. It's hard to understand why certain things have to be defined the way they are (like compatible charts). What are some good resources that provide better motivation for the definitions?

For tangent spaces -which are the heart of smooth manifold theory - a rock solid foundation can start with parameterizations of surfaces in Euclidean space. I used Struik's book on Classical Differential Geometry but there are many good books. Another is by Barret O'Neill.

A surface in 3 space is locally described by a differentiable three dimensional vector valued function of two variables $X(u,v)$. This function is meant to label each point on the surface by the parameters $(u,v)$ and so must be $1-1$ - one parameter pair for each point on the surface. The inverse of this function assigns the "coordinates" $(u,v)$ to the point $X(u,v)$ and is called a coordinate patch. If one has two parameterizations whose images overlap then one would like to be able to compare them using calculus. This is possible if the composition of one with the inverse of the other - the other's coordinate patch - is a smooth function from a domain in $R^2$ into $R^2$.

Tangent to each point $X(u,v)$ are the derivative vectors $∂X/∂u$ and $∂X/∂v$. These are tangent to the surface at the point $X(u,v)$. These tangent vectors are linearly independndent and will span a plane called the tangent plane at $X(u,v)$.

A surface embedded $R^3$ is then any subset that can be described by "compatible" local parameterizations - or coordinate patches. Usually one also assumes that a coordinate patch can be extended to a smooth function in a neighborhood of each point $X(u,v)$.

This definition generalizes to manifolds in $R^n$ with no change except for the number of parameters.

The tangent bundle of the manifold is then the collection of all tangent planes and may be described as a topological space as the subset of $M×R^{n}$ of ponts $(p,v)$ where $v$ is tangent of $M$ at $p$.

The next step is to observe that embeddings in Euclidean space are not necessary to define a smooth manifold but only coordinate charts and their overlaps. With this abstraction it gets harder to define tangent spaces and one must fight through the various ways to define them without the geometric picture of a vector in Euclidean space.

 

[WWGD]

For tangent spaces -which are the heart of smooth manifold theory - a rock solid foundation can start with parameterizations of surfaces in Euclidean space. I used Struik's book on Classical Differential Geometry but there are many good books. Another is by Barret O'Neill.

A surface in 3 space is locally described by a differentiable three dimensional vector valued function of two variables $X(u,v)$. This function is meant to label each point on the surface by the parameters $(u,v)$ and so must be $1-1$ - one parameter pair for each point on the surface. The inverse of this function assigns the "coordinates" $(u,v)$ to the point $X(u,v)$ and is called a coordinate patch. If one has two parameterizations whose images overlap then one would like to be able to compare them using calculus. This is possible if the composition of one with the inverse of the other - the other's coordinate patch - is a smooth function from a domain in $R^2$ into $R^2$.

Tangent to each point $X(u,v)$ are the derivative vectors $∂X/∂u$ and $∂X/∂v$. These are tangent to the surface at the point $X(u,v)$ for each pair of parameter values $u$ and $v$. These tangent vectors are linearly independndent and will span a plane called the tangent plane at $X(u,v)$.

A surface in embedded $R^3$ is then any subset that can be described by "compatible" local parameterizations - or coordinate patches. Usually one assumes that a coordinate patch can be extended to a smooth function in a neighborhood of each point $X(u,v)$.

This definition generalizes to manifolds in $R^n$ with no change except fo the number of parameters.

The tangent bundle of the manifold is then the collection of all tangent planes and may be described as a topological space as the subset of $M×R^{n}$ of ponts $(p,v)$ where $v$ is tangent of $M$ at $p$.

The next step is to observe that embeddings in Euclidean space are not necessary to define a smooth manifold but only coordinate charts and their overlaps. With this abstraction it gets harder to define tangent spaces and one must fight through the various ways to define them without the geometric picture of a vector in Euclidean space.

One thing I am confused about is what the tangent space of, say a line or surface is when they are embedded in a codimension-2 or higher Euclidean space. If I have , say a curve ( a 1-manifold) embedded in , say, $\mathbb R^3$ or a surface embedded in $\mathbb R^4$ or higher, it is not clear to me what the tangent space is. For the line, there are many directions of tangency. Do you know how this is addressed?

 

[WWGD]

@Avatrin :
The book by William Boothby goes into a lot of details too, which you may like.

 

[lavinia]

One thing I am confused about is what the tangent space of, say a line or surface is when they are embedded in a codimension-2 or higher Euclidean space. If I have , say a curve ( a 1-manifold) embedded in , say, $\mathbb R^3$ or a surface embedded in $\mathbb R^4$ or higher, it is not clear to me what the tangent space is. For the line, there are many directions of tangency. Do you know how this is addressed?

The definition of the tangent space is always the same in the case of an embedding in Euclidean space. It is the span of the vectors $∂X/∂u_{i}$.

For example suppose one parameterizes a torus in 4 space by the function

$X(u,v) = (cos(u),sin(u),cos(v),sin(v))$.

This is a two dimensional manifold in four dimensional space. Its tangent space at $X(u,v)$ is all vectors $a∂X/∂u + b∂X/∂v$ which is all linear combinations $a(-sin(u),cos(u),0,0) + b(0,0,-sin(v),cos(v))$. This is a two dimensional subplane of $R^4$.

If $X(t)$ is a curve in 3 space $X(t) = (x(t),y(t),z(t))$ then its tangent space is the line spanned by the vector $(dx/dt,dy/dt,dz/dt)$. The other directions that you are thinking of are not tangent. There is for instance the normal direction which can be found by parameterizing the curve by arc length $s$ and taking the second derivative $k(s)N(s) = (d^2x/ds^2,d^2y/ds^2,d^2z/ds^2)$ where k(s) is the curvature. The unit tangent vector $T(s)$ and the unit normal $N(s)$ span a plane. There is still one dimension to account for and usually this is obtained as the cross product $B(s) = T(s)×N(s)$.

 

[WWGD]

The definition os the tangent space is always the same in the case of embedding in Euclidean space. It is the span of the vectors $ϑX/ϑu_{i}$.

For example suppose one parameterizes a torus in 4 spaces by the function

$X(u,v) = (cos(u),sin(u),cos(v),sin(v))$.

This is a two dimensional manifold in four dimensional space. It tangent space at $X(u,v)$ is all vectors $aϑX/ϑu + bϑX/ϑv$ which is all linear combinations $a(-sin(u),cos(u),0,0) + b(0,0,-sin(v),cos(v))$. This is a two dimensional subplane of $R^4$.

If $X(t)$ is a curve in 3 space $X(t) = (x(t),y(t),z(t)$ then its tangent space is the line spanned by the vector $dx/dt,dy/dt,dz/dt)$. The other direction that you are thinking of are not tangent. There is the normal direction which can be found by parameterizing the curve by arc length $s$ and taking the second derivative $N(s) = (d^2x/ds^2,d^2y/dy^2,d^2z/dz^2)$. The unit tangent vector $T(s)$ and the unit normal $N(s)$ span a plane. There is still one dimension to account for and usually this is obtained as the cross product $B(s) = T(s)×N(s)$.

The definition os the tangent space is always the same in the case of embedding in Euclidean space. It is the span of the vectors $ϑX/ϑu_{i}$.

For example suppose one parameterizes a torus in 4 spaces by the function

$X(u,v) = (cos(u),sin(u),cos(v),sin(v))$.

This is a two dimensional manifold in four dimensional space. It tangent space at $X(u,v)$ is all vectors $aϑX/ϑu + bϑX/ϑv$ which is all linear combinations $a(-sin(u),cos(u),0,0) + b(0,0,-sin(v),cos(v))$. This is a two dimensional subplane of $R^4$.

If $X(t)$ is a curve in 3 space $X(t) = (x(t),y(t),z(t)$ then its tangent space is the line spanned by the vector $dx/dt,dy/dt,dz/dt)$. The other direction that you are thinking of are not tangent. There is the normal direction which can be found by parameterizing the curve by arc length $s$ and taking the second derivative $N(s) = (d^2x/ds^2,d^2y/dy^2,d^2z/dz^2)$. The unit tangent vector $T(s)$ and the unit normal $N(s)$ span a plane. There is still one dimension to account for and usually this is obtained as the cross product $B(s) = T(s)×N(s)$.

Ah, yes, the Frenet-Serret system, I am refreshing my knowledge. Thanks.

 

[mathwonk]

as i recall, those frenet serret formulas just reflect the fact that the tangent space to the space of orthonormal frames is the space of skew symmetric matrices. but this is not my speciality, just a vague memory from teaching the course once in 1972.

 

[WWGD]

Yes, it was confusing to see 3 vectors in a tangent space which is supposed to be 1d, but then I realized they were be all a function of a single variable, making it 1d.