Parallel Transport: Uses & Benefits

[kent davidge]

What is the usefulness of parallel transporting a vector? Of course, you can use it to determine whether a curve is a geodesic, but aside from that, what can it be used for?

 

[PeterDonis]

What is the usefulness of parallel transporting a vector?

To compare vectors at different points in spacetime. Vectors in spacetime are "attached" to particular points (more precisely, they lie in the tangent space at some point--each point has its own distinct tangent space). But many problems require you to compare vectors at different points, and to do that, you have to transport one of the vectors to the point where the other one is. And to do that, you have to have some rule that tells you how to do the transport. Parallel transport is the rule that makes the most sense physically.

 

[WWGD]

N

What is the usefulness of parallel transporting a vector? Of course, you can use it to determine whether a curve is a geodesic, but aside from that, what can it be used for?

Notice this is not necessary in Euclidean space, where vector spaces are naturally isomorphic, i.e., the isomorphism is independent of the choice of basis, so you may compare vectors at different spaces without the need for a connection. And notice the ( standard) metric is trivial, i.e., given by the identity.

 

[kent davidge]

@WWGD , @PeterDonis can we say that a connection is also a rule for how to transport the vectors?

 

[WWGD]

@WWGD , @PeterDonis can we say that a connection is also a rule for how to transport the vectors?

Yes; a choice of vector space ( tangent space) isomorphism. Maybe Peter Donis can expand on this.

 

[fresh_42]

@WWGD , @PeterDonis can we say that a connection is also a rule for how to transport the vectors?

That's what it is meant for.

I think the misunderstandings already take place at school, and I don't mean specifically you. E.g. we say $x \longmapsto 2x$ is the derivative of $f(x)=x^2$ and continue to call it: derivative, sometimes differential, slope, tangent, later covariant derivative and whatever more. And all are wrong, i.e. not precise enough. In general we consider $D_p f(x)$ which equals $2p$ in our case, and not $2x$ as used at school. Now $D_p f(x)$ is an expression which has four input parameters: location $p$, function $f$, differentiation $D$ and manifold $x$. So depending on which of these is variable and which are fixed, we get completely different objects out of $D_pf(x)$.

In your question, it is $p$ which varies. The tangents at $p$ are different from the tangents at $q$. They don't even share the same vector space, as one has $p$, the other one has $q$ as its origin. But tangents are e.g. velocities and we want to compare velocities at different locations. Hence we have to connect the tangent space $T_pM$ at $p$ with the tangent space $T_qM$ at $q$. A connection is a rule for this comparison. The only one which is to some extend natural, is to take a orthogonal basis of $T_pM$ and move it along a geodesic into $T_qM$, so the velocities at $q$ can be expressed in the coordinates which formerly have been those of $T_pM$.

 

[PeterDonis]

can we say that a connection is also a rule for how to transport the vectors?

Defining a connection is (I believe) equivalent to defining a parallel transport rule, so I think they're just two different ways of describing the same thing.

 

[WWGD]

That's what it is meant for.

I think the misunderstandings already take place at school, and I don't mean specifically you. E.g. we say $x \longmapsto 2x$ is the derivative of $f(x)=x^2$ and continue to call it: derivative, sometimes differential, slope, tangent, later covariant derivative and whatever more. And all are wrong, i.e. not precise enough. In general we consider $D_p f(x)$ which equals $2p$ in our case, and not $2x$ as used at school. Now $D_p f(x)$ is an expression which has four input parameters: location $p$, function $f$, differentiation $D$ and manifold $x$. So depending on which of these is variable and which are fixed, we get completely different objects out of $D_pf(x)$.

In your question, it is $p$ which varies. The tangents at $p$ are different from the tangents at $q$. They don't even share the same vector space, as one has $p$, the other one has $q$ as its origin. But tangents are e.g. velocities and we want to compare velocities at different locations. Hence we have to connect the tangent space $T_pM$ at $p$ with the tangent space $T_qM$ at $q$. A connection is a rule for this comparison. The only one which is to some extend natural, is to take a orthogonal basis of $T_pM$ and move it along a geodesic into $T_qM$, so the velocities at $q$ can be expressed in the coordinates which formerly have been those of $T_pM$.

Good description, but it may be better to give an example for e.g., the sphere or circle, some manifold with non-zero curvature, where connection is not trivial.

 

[fresh_42]

Good description, but it may be better to give an example for e.g., the sphere or circle, some manifold with non-zero curvature, where connection is not trivial.

https://en.wikipedia.org/wiki/Affine_connection

https://en.wikipedia.org/wiki/Parallel_transport

I had planned to use my two chapter insight about $SU(2)$
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/as example and write a third chapter with explicit calculations rather than just $\nabla_Xf$.
Edit: $\nabla_f X$

I wanted to answer questions like:
Why is the Ehresmann connection a connection?
What makes it different from the Levi Civita connection?
What are orthogonal frames?
I even have already a couple of pages, but as this isn't my home ground, things became a bit complicated.

 

[WWGD]

https://en.wikipedia.org/wiki/Affine_connectionView attachment 251086
https://en.wikipedia.org/wiki/Parallel_transportView attachment 251087I had planned to use my two chapter insight about $SU(2)$
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/as example and write a third chapter with explicit calculations rather than just $\nabla_Xf$.

I wanted to answer questions like:
Why is the Ehresmann connection a connection?
What makes it different from the Levi Civita connection?
What are orthogonal frames?
I even have already a couple of pages, but as this isn't my home ground, things became a bit complicated.

I had written something on the Ehresmann connection too, but I am moving soon and everything I have is packed ( I had not written it down to my PC). EDIT: More examples from one of freshmeister/heimer paysans: https://people.mpim-bonn.mpg.de/hwbllmnn/archiv/dg1cag02.pdf

 

[kent davidge]

Let's see if I got this right.

$D_p f(x)$ is an expression which has four input parameters: location $p$, function $f$, differentiation $D$ and manifold $x$. So depending on which of these is variable and which are fixed, we get completely different objects out of $D_pf(x)$.

Your $D_p f(x) \equiv 2p$ can be called "derivative of $f$" only when $p = x$ and $f(x) = x^2$. If say, $f(x) = x^3$, then $D_x x^3 = 2x$ would have another name?

 

[WWGD]

Let's see if I got this right.

Your $D_p f(x) \equiv 2p$ can be called "derivative of $f$" only when $p = x$ and $f(x) = x^2$. If say, $f(x) = x^3$, then $D_x x^3 = 2x$ would have another name?

Remember to define which of the basic input parameters you are considering.

 

[WWGD]

@kent davidge :
Maybe this is th type of thing you're looking for: https://khudian.net/Teaching/Geometry/Geom19/geom19.html
Has homeworks with solutions.
This one computed the curvature and geodesics of the Torus.
http://www.rdrop.com/~half/math/torus/torus.geodesics.pdf

 

[fresh_42]

Let's see if I got this right.

Your $D_p f(x) \equiv 2p$ can be called "derivative of $f$" only when $p = x$ and $f(x) = x^2$. If say, $f(x) = x^3$, then $D_x x^3 = 2x$ would have another name?

The full specification without any variables left is, i.e. all variables have a specific value, for $f(x)=x^2$
$$
D_pf(x) =\lim_{h\to 0}\dfrac{f(p+h)-f(p)}{h}=\left. \dfrac{d}{dx}\right|_{x=p} f(x) = f\,'(p) = 2p
$$
is a real number.

Note the ambiguity of $x$. It is the variable of the function $f(x)$ and at the same time often the location $p$ where we evaluate the derivative, because people write $f\,'(x)$ when they should write $f\,'(p)\,.$

Of course $p \longmapsto f\,'(p)$ is again a function and the domain is the same as that of $f(x)$ so there is no difference in writing $x \longmapsto f\,'(x)$ instead. But do they mean this function, the derivative, or do they mean the function value, the slope? This isn't the same thing. If we say $f(x)$ we are used to associate the function, not its value at $x$. But if we say slope or tangent, then this makes only sense at a certain point, hence the function value $f\,'(p)$ and not the function $x \longmapsto f\,'(x)$ itself, which is the derivative.

If you like to get more confused, have a look at:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

 

[fresh_42]

More examples from one of freshmeister/heimer paysans: https://people.mpim-bonn.mpg.de/hwbllmnn/archiv/dg1cag02.pdf

Thanks. Now I know that there is such a thing called Wiedersehen manifold. Now I'm torn between curiosity, spirit and opportunity. On a first attempt I only found the classification (basically spheres) not the definition.

 

[WWGD]

The full specification without any variables left is, i.e. all variables have a specific value, for $f(x)=x^2$
$$
D_pf(x) =\lim_{h\to 0}\dfrac{f(p+h)-f(p)}{h}=\left. \dfrac{d}{dx}\right|_{x=p} f(x) = f\,'(p) = 2p
$$
is a real number.

Note the ambiguity of $x$. It is the variable of the function $f(x)$ and at the same time often the location $p$ where we evaluate the derivative, because people write $f\,'(x)$ when they should write $f\,'(p)\,.$

Of course $p \longmapsto f\,'(p)$ is again a function and the domain is the same as that of $f(x)$ so there is no difference in writing $x \longmapsto f\,'(x)$ instead. But do they mean this function, the derivative, or do they mean the function value, the slope? This isn't the same thing. If we say $f(x)$ we are used to associate the function, not its value at $x$. But if we say slope or tangent, then this makes only sense at a certain point, hence the function value $f\,'(p)$ and not the function $x \longmapsto f\,'(x)$ itself, which is the derivative.

If you like to get more confused, have a look at:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

The functional expression , here f'(x)=2x is the differential, i.e., the general linear map that describes/defines the local . Or is 2xdx the differential? I know if f is differentiable with differential f'(x) then f'(x)dx is a differential form.

 

[fresh_42]

I'd say $2x\,dx$ is the differential, $2x$ the slope, $\text{ multiply by two }$ the derivative, $p \stackrel{dx}{\longmapsto} D_p = \left. \dfrac{d}{dx}\right|_{p}$ the differential form, $f(\gamma (t))'(0)$ with a path $\gamma \, , \,\gamma(0)=p$ is the covariant derivative, the Koszul connection, $(2p,p)$ the tangent, $\{\,(2p,p)\,|\,p\,\}$ the tangent bundle, $\dfrac{\partial f}{\partial x}(p)$ the cotangent, ...

... and again I realize why I like the Weierstraß formula: $f(p+v)=f(p) + \left( D_pf(x) \right)v + o(||v||)$ where I can see everything.

 

[WWGD]

Yes, different books have different definitions. Mine uses 2xdx as the differential 1-form.

 

[fresh_42]

Yes, different books have different definitions. Mine uses 2xdx as the differential 1-form.

I stretched the things a little bit to get different objects for the many terms. And I have forgotten to mention that $D$ is a derivation.

This entire subject is cruel. Now write everything in coordinates and you have physics, call it bundles and sections and you have mathematics. And all we started with was a simple directional change of value.