Christoffel symbols and covariant derivative intuition

[physlosopher]

So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see $\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}$, I pretty easily see a partial that's corrected for a changing basis, which is how the covariant derivative is presented in the content I'm reading, which is all fine.

The Christoffel symbol $\Gamma^{\alpha}{}_{\gamma \beta}$ is presented in the derivative of a basis vector (in this case a from the coordinate tangents): $$\frac {\partial \vec e^{}{}_{\gamma}}{\partial x^{\beta}} = \Gamma^{\alpha}{}_{\gamma \beta} \vec e^{}{}_{\alpha}$$ so it's pretty easy to interpret it as the $\alpha^{th}$ component of the partial $\frac {\partial \vec e^{}{}_{\gamma}}{\partial x^{\beta}}$, in the same basis. Can I understand the Christoffel symbol equivalently as something like the rate at which a change in the $x^{\beta}$ coordinate "changes" the $\vec e^{}{}_{\gamma}$ into $\vec e^{}{}_{\alpha}$? In other words, it's how much a small change in the coordinate pushes $\vec e^{}{}_{\gamma}$ into the direction of $\vec e^{}{}_{\alpha}$. Maybe the lack of rigor in my language is unsafe, but is something like this the case?

Meanwhile, the $v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}$ term in the covariant derivative is a weighted sum of Christoffel symbols, with each symbol in the sum weighted by the vector component $v^{\gamma}$. But if I can understand the Christoffel symbol as I wrote above, then each term in that sum is just the way $\vec e^{}{}_{\gamma}$ changes to point in the direction of $\vec e^{}{}_{\alpha}$ after a small change in the coordinate, weighted by how much of the vector $\vec v$ was already pointing in the $\vec e^{}{}_{\gamma}$ direction; after a small change, some of $v^{\gamma}$ now counts toward the $\vec e^{}{}_{\alpha}$ component, and the $v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}$ term is basically an expression of that quantity. This seems to me to capture exactly what you'd want a term that corrects for a changing basis to do.

Does this make sense, and is it a helpful way to think? I'm having trouble seeing whether it would generalize to a manifold on which neighboring tangent spaces can't be identified with one another (in fact I may have to ask about tangent spaces in general on here soon!). Thanks in advance for any input!

 

[Orodruin]

The Christoffel symbol ΓαγβΓαγβ\Gamma^{\alpha}{}_{\gamma \beta} is presented in the derivative of a basis vector (in this case a from the coordinate tangents):

∂→eγ∂xβ=Γαγβ→eα​

Be careful with index placement for the lower indices of the connection coefficients. Although they are symmetric for the Euclidean space and for the Levi-Civita connection on a Riemannian manifold, the general connection is not torsion free.

Can I understand the Christoffel symbol equivalently as something like the rate at which a change in the xβxβx^{\beta} coordinate "changes" the →eγe→γ\vec e^{}{}_{\gamma} into →eαe→α\vec e^{}{}_{\alpha}?

This to me sounds just like a rewriting of the first, so yes.

Does this make sense, and is it a helpful way to think? I'm having trouble seeing whether it would generalize to a manifold on which neighboring tangent spaces can't be identified with one another (in fact I may have to ask about tangent spaces in general on here soon!). Thanks in advance for any input!

In the general manifold, as you might have guessed, there is no unique way of connecting tangent spaces. Instead, this is done by specifying a connection on the tangent bundle. This connection is in essence a definition of what it means for a vector field to ”change” in different directions. The connection coefficients has the same interpretation given this definition of ”change”. Note that there is no unique connection on a manifold. However, if you require metric compatibility and that the connection is torsion free, then there is only a single possible connection - the Levi-Civita connection.

 

[physlosopher]

This connection is in essence a definition of what it means for a vector field to ”change” in different directions. The connection coefficients has the same interpretation given this definition of ”change”.

Thanks for this! It makes sense to me how a connection is established in an embedded manifold, such as the surface of a sphere in 3D Euclidean space. I can look at the $\vec e^{}{}_{\phi}$ and $\vec e^{}{}_{\theta}$ basis vectors and note how they change with the coordinates with reference back to the three Cartesian basis vectors, which are the same at each point on the 2D manifold. But I'm having trouble conceptualizing this without reference to the Cartesian coordinates, and thus without reference to a space in which the manifold is embedded. For example, at a given point on the spherical surface there are two basis vectors, and thus a 2D tangent space, correct? But at a point a small distance away, there is a different pair of vectors and a different tangent space is spanned. I guess what I'm not seeing is how a relationship can be established between one of those bases and the other - how could a basis vector change, even by an infinitesimal amount, to point slightly in the direction of a vector that isn't part of its tangent space? Do we need some other piece of information, such as the shapes of curves that can inhabit the manifold (the coordinate curves, perhaps)? "Shape" seems to rely on the notion of an external vector space to define directionality, or on embedding, so it feels like I'm running in circles.

Maybe I need to look at the math more carefully on the spherical surface, because I guess you can write down the connection coefficients for that manifold in spherical polar coordinates, which wouldn't reference 3D Cartesian coordinates (at least, explicitly). It's also very possible that I need to look at a more robust treatment of the tangent space at a point on the general manifold - the text I'm using sort of brushes over them.

 

[Orodruin]

I guess what I'm not seeing is how a relationship can be established between one of those bases and the other - how could a basis vector change, even by an infinitesimal amount, to point slightly in the direction of a vector that isn't part of its tangent space?

This is what I tried to convey. This is why you need the connection and there is no unique connection given a manifold, ie, there is no unique way of defining how nearby tangent spaces connect to each other - it can be done in many ways. In this way, the connection defines the geometry you are working with and the same manifold can have several different geometries. It is sufficient to define how the covariant derivative acts on a complete set of basis fields to fully define the connection. If you add some constraints on what the connection should satisfy, then you might end up with a unique connection.

Do we need some other piece of information, such as the shapes of curves that can inhabit the manifold (the coordinate curves, perhaps)?

What you need is the connection. It is what connects nearby tangent spaces and through it defines the geometry of your space.

As an example, the Levi-Civita connection with the stabdard metric is not the only possible connection on the sphere. You could, for example, define a connection that defines ”not changing” from point to point as keeping a vector’s length and angle to the meridians. Such a connection would imply that compass needles were always parallel transported when moving around the Earth (making the simplification that magnetic north/south coincides with the poles) and only has a single non-zero connection coefficient.

 

[physlosopher]

You could, for example, define a connection that defines ”not changing” from point to point as keeping a vector’s length and angle to the meridians.

This is a really helpful thought, thank you. And my apologies for not understanding earlier!

I think this may resolve a related confusion I've been having with regards to parallel transport as well. If a geodesic is defined as a curve that parallel transports its own tangent vector, I had wondered how (a segment of) a great circle on the surface of a sphere could count as parallel transporting its tangent vector (in fact I wondered how any path on a sphere could count). But we can define the connection to ensure this, correct? This way, even though in 3D space the tangent vector to the equator points in opposite directions on opposite sides of the sphere, we define a connection such that the tangent vector hasn't changed?

And to clarify, because I think this is where I became confused, the connection is a separate concept from the connection coefficients/Christoffel symbols, correct? Thanks once again for the help!

 

[Orodruin]

But we can define the connection to ensure this, correct?

This is how the geodesic is defined. The ”does not change” part refers to the concept of ”change” implied by the connection. Different connections may have different geodesics. The geodesics of the connection I mentioned are curves of constant bearing, not great circles, which are geodesics of the Levi-Civita connection.

This way, even though in 3D space the tangent vector to the equator points in opposite directions on opposite sides of the sphere, we define a connection such that the tangent vector hasn't changed?

It would be more accurate to state that the tangent vector has not changed along the geodesic. In the general curved manifold there is no way of uniquely comparing vectors in different tangent spaces as the parallel transport will be path dependent.

And to clarify, because I think this is where I became confused, the connection is a separate concept from the connection coefficients/Christoffel symbols, correct?

Not really. The connection coefficients can be used to define the connection and if you know the connection you can compute the connection coefficients. Specifying the connection coefficients is therefore the same as defining the connection. It is just a way of defining what ”change” means. If you are specifying the connection coefficients you are in essence specifying whether or not and if so how the coordinate basis ”changes”.

If you impose that parallel transport preserves inner products and that the connection is torsion free, then there is a unique connection whose coefficients you can compute from the metric. This is called the Levi-Civita connection and its connection coefficients are typically called Christoffel symbols. In other words, this connection is the only possible definition of ”change” that satisfies the given conditions.

 

[physlosopher]

Specifying the connection coefficients is therefore the same as defining the connection. It is just a way of defining what ”change” means.

Very helpful, thank you. With your help I think I've figured out where my confusion was originating. I'm pretty sure I was smuggling over some intuition from tangent spaces defined using a position vector, and picturing 2D manifolds as being embedded in 3D Euclidean space (or at least an affine space, I think).

To check that I'm on the right track: the tangent spaces are defined at each point on the manifold, and the connection (or connection coefficients) relate them by defining what it is for a vector field to "change" from a point with one tangent space to a point with another? This also defines what it is not to change, which gives us geodesics, correct?

I think I was implicitly looking for the connection to tell me how one tangent basis looked relative to another against a background space in which the manifold is embedded. I was wondering how connection coefficients for a 2D manifold could tell me how the bases change in 3D (and thus through a third dimension external to the manifold), but this just isn't what they do, correct? They instead take tangent spaces that are already defined on the manifold and relate them.

A (hopefully) final question: since there isn't a unique connection, does picking one impose added structure on the manifold?

Thanks again for the help, it's really shortening the half-life of my confusion!

 

[Orodruin]

To check that I'm on the right track: the tangent spaces are defined at each point on the manifold, and the connection (or connection coefficients) relate them by defining what it is for a vector field to "change" from a point with one tangent space to a point with another?

Correct, but only for ”nearby” tangent spaces, ie, it in essence defines local directional derivatives. For points that are separated, the parallel transport generally depends on the path.

This also defines what it is not to change, which gives us geodesics, correct?

Yes. Geodesics are by definition curves whose tangent vectors do not ”change” along the curve itself.

A (hopefully) final question: since there isn't a unique connection, does picking one impose added structure on the manifold?

Yes. However, note the uniqueness of the connection once you impose the conditions of the Levi-Cevita connection.

 

[physlosopher]

Thanks again, all your responses are really helpful. I feel like I've got a better grip on this now.

One last thought, in line with my hope at the onset of this thread of having a working geometrical intuition about the connection coefficients: considering how the connection works on a manifold, can I interpret $\Gamma^{k}{}_{ij}$ as identifying vectors in neighboring tangent spaces with one another? Thus, could I understand $\Gamma^{k}{}_{ij} > 0$ as saying something like "vector components in $\vec e^{}{}_{i}$ count a little bit toward the $\vec e^{}{}_{k}$ component after a small displacement in $x^{j}$"?

 

[Orodruin]

No. The connection coefficients define the ”change” in the basis vectors given a connection. Just as in Euclidean space, $\Gamma^i_{jk}$ tells you the i component of the change of the kth basis vector in the direction of the jth basis vector. It should not be surprising since this is the definition of what it geometrically means to ”change” is given by the connection. Introducing a connection just formalises this meaning.

 

[physlosopher]

Can I then think of the connection coefficients as defining an identity between the basis vectors in neighboring tangent spaces? My problem earlier with thinking about them in terms of changing basis vectors on a sphere, for example, was that the connection coefficients for polar coordinates on a sphere, $\Gamma^{\theta}{}_{\theta\phi}$ and so on, describe how those two basis vectors change relative to each other, but don't seem to fully describe how they change because they can't account for the changes toward or away from the radial basis vector, which on an embedded sphere in 3D Euclidean space is happening as we move in $\theta$ or $\phi$. To clear up this confusion I tried to stop thinking in terms of a third dimension, and stopped trying to understand how the connection coefficients on that 2D manifold could tell me by themselves how to take the pair of basis vectors at one point and draw the pair at a neighboring point. Instead I started thinking something like "basis vectors are already defined all over the sphere, and the connection coefficients provide the relationship (identity?) between the basis vectors at neighboring points."

And modifying the question in my last post, could I instead think about $v^{k}\Gamma^{i}{}_{kj}$ (just the one term, not the sum over k) as being the amount by which the kth component of $\vec v$ will map onto the ith component in the basis in the neighboring tangent space an infinitesimal step in $x^{j}$ away?

 

[Orodruin]

they can't account for the changes toward or away from the radial basis vector,

There isno radial basis vector. We aretalking anout the tangent spaces of the sphere, which are two-dimensional, not ther embedding into $\mathbb R^3$.

Instead I started thinking something like "basis vectors are already defined all over the sphere, and the connection coefficients provide the relationship (identity?) between the basis vectors at neighboring points."

Indeed, the basis fields are already there. The connection coefficients just tell you what theor directional derivatives are.

And modifying the question in my last post, could I instead think about vkΓikjvkΓikjv^{k}\Gamma^{i}{}_{kj} (just the one term, not the sum over k) as being the amount by which the kth component of →vv→\vec v will map onto the ith component in the basis in the neighboring tangent space an infinitesimal step in xjxjx^{j} away?

Careful with the order of the lower indices again. It (correcting for the indices) is the i component of the ”change” in the vector field $\vec v$ due to the change in the basis when you go in the j direction.

 

[physlosopher]

Thanks again!

So a simple example I was thinking about was the unit circle in polar coordinates. Being a 1D manifold, it only has a single connection coefficient, correct? This is $\Gamma^{\phi}{}_{\phi\phi} = 0$. This is a statement that the single basis vector $\vec e_{\phi}$ doesn't change as we move in $\phi$, correct? But I can only understand this statement as making sense if it's actually identifying the basis vector at one point with the basis vector at the "next" point. This is the way in which it defines what change is. Furthermore, this understanding would let me understand how parallel transport is to be defined on that manifold: a $\phi$ component in one tangent space does not change as an effect of a changing basis (because there isn't one) as we move infinitesimally in $\phi$. Is this a reasonable way of thinking?

 

[Orodruin]

Being a 1D manifold, it only has a single connection coefficient, correct?

While it is true that you can always find such a coordinate system (assuming metric compatibility), given a coordinate system, several different connections are possible.

Edit: To clarify, I am here talking about coordinates such that the connection coefficient is zero. It is true that there is a single connection coefficient.

But I can only understand this statement as making sense if it's actually identifying the basis vector at one point with the basis vector at the "next" point.

Not identifying, but ”connecting” them with a meaning of what it means for a vector (or tensor) to have a non-zero directional derivative. In essence, you put up a number of reasonable conditions for a directional derivative to satisfy and it then turbs out that there are several different such directional detivatives (unless you go on to specify further conditions).

 

[physlosopher]

Edit: To clarify, I am here talking about coordinates such that the connection coefficient is zero. It is true that there is a single connection coefficient.

Thanks for the clarification, that does make sense.

Not identifying, but ”connecting” them with a meaning of what it means for a vector (or tensor) to have a non-zero directional derivative.

Right, that makes sense. So we don't need something as strong as identification of basis vectors at neighboring points in order to define what it is for a tensor to change from point to point.

Thanks again for all your responses, they've been extremely helpful!

 

[Adam Marsh]

For those running across this thread who might be helped by a picture, I thought I'd add the below.

So the covariant derivative $\nabla_{v}w$ can be described as "the difference between $w$ and its parallel transport in the direction $v$," and the relation $\nabla_{v}w=\check{\Gamma}\left(v\right)\vec{w}+\mathrm{d}\vec{w}\left(v\right)$ can be viewed as roughly saying that "the change in $w$ under parallel transport is equal to the change in the frame relative to its parallel transport plus the change in the components of $w$ in that frame."

Here the notation is such that the following are all equivalent:
$$\begin{aligned}\nabla_{e_{\mu}}w & =\mathrm{d}\vec{w}\left(e_{\mu}\right)+\check{\Gamma}\left(e_{\mu}\right)\vec{w}\\
\nabla_{\mu}w^{\nu} & =\mathrm{d}{w^{\nu}}\left(e_{\mu}\right)+\Gamma^{\nu}{}_{\lambda}\left(e_{\mu}\right)w^{\lambda}\\
\nabla_{\mu}w^{\nu} & =\partial_{\mu}w^{\nu}+\Gamma^{\nu}{}_{\lambda\mu}w^{\lambda} \end{aligned}$$

This means we can describe the Christoffel symbols $\Gamma^{\nu}{}_{\lambda\mu}$ as "measuring the $\nu$th component of the difference between $e_{\lambda}$ and its parallel transport in the direction $e_{\mu}$."

More on this approach here.