# Parallel Transport: Uses & Benefits

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• kent davidge
In summary: I'll take a look and see if I can figure out what you're referring to, or if I have any of it written down anywhere. In summary, Parallel transport is a useful tool in comparing vectors at different points in spacetime, as vectors are "attached" to particular points and may require comparison at other points. This is achieved through a connection, which is a rule for transporting vectors between tangent spaces at different points. The most natural rule is to use an orthogonal basis and move it along a geodesic between the two points. The Ehresmann connection is a specific type of connection, different from the Levi Civita connection, and allows for the use of orthogonal frames to compare vectors.
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?

kent davidge said:
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 and kent davidge
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kent davidge said:
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?

kent davidge said:
@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.

kent davidge
kent davidge said:
@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##.

kent davidge
kent davidge said:
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.

kent davidge
fresh_42 said:
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.

WWGD said:
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
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.

Last edited:
fresh_42 said:
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

Let's see if I got this right.
fresh_42 said:
##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?

kent davidge said:
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.

kent davidge said:
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/

WWGD said:
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.

fresh_42 said:
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.

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.

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

WWGD said:
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.

## 1. What is parallel transport?

Parallel transport is a mathematical concept used to describe the movement of objects along a curved surface or manifold. It involves the transportation of an object along a path while keeping the object's orientation constant.

## 2. What are some uses of parallel transport?

Parallel transport has many applications in various fields, such as physics, engineering, and computer graphics. It is commonly used in the study of curved spaces, such as in general relativity, to describe the motion of particles and light. It is also used in navigation and robotics to calculate the movement of objects along curved paths.

## 3. How is parallel transport different from regular transport?

Regular transport involves moving an object from one point to another without any restrictions on its orientation. In contrast, parallel transport ensures that the object maintains the same orientation throughout its movement, even on curved surfaces. This makes it a more accurate method for describing the movement of objects in curved spaces.

## 4. What are the benefits of using parallel transport?

One of the main benefits of parallel transport is its ability to accurately describe the movement of objects in curved spaces. This makes it a valuable tool in fields like physics and engineering. Additionally, parallel transport can also be used to calculate geodesics, which are the shortest paths between two points on a curved surface.

## 5. Are there any limitations to parallel transport?

One limitation of parallel transport is that it can only be applied to objects that can be smoothly deformed, such as vectors or tensors. It also requires a well-defined curved surface or manifold to work properly. In some cases, parallel transport may also be mathematically complex and difficult to calculate, making it challenging to use in certain applications.

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