Coordinate free Christoffel symbols

[kostas230]

I've been trying to come up with a oordinate free formula of Christoffel symbols. For the Christoffel symbols of the first kind it's really easy. Since
$$\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right)$$

we can easily generalize the formula:

$$\Gamma\left(X,Y,Z\right) = \frac{1}{2}\left(Yg\left(X,Z\right)+Zg\left(X,Y\right)-Xg\left(Y,Z\right)\right)$$

How can we generalize in this way the Christoffel symbols of the second kind $${\Gamma^\lambda}_{\mu\nu} = g^{\lambda\sigma}\Gamma_{\lambda\mu\nu}$$

I'm thinking that a way of doing it would be through some kind of contraction, but I'm not sure how since the Christoffel symbols of the first kind are not tensors to begin with.

 

[Matterwave]

How did you arrive at your formula? As the Christoffel symbols tell you how to take a co variant derivative expressly given a certain coordinate system, and are not tensors, I don't see how you can get a coordinate independent expression for them.

In your equation, assuming that a Christoffel symbol is a tensor like object, your expression on the left would be a scalar, while your expression on the right looks like a vector to me.

 

[kostas230]

While you are correct by saying that the Christoffel symbols give you coordinate expressions for the covariant derivative, the Christoffel symbols are defined by the equation:
$$\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right)$$

In the same way, I define:

$$\Gamma\left(X,Y,Z\right) = \frac{1}{2}\left(Yg\left(X,Z\right)+Zg\left(X,Y\right)-Xg\left(Y,Z\right)\right).$$

The right term is a scalar and we can easily retrieve the original Christoffel symbols:

$$\Gamma\left(\partial_\lambda ,\partial_\mu , \partial_\nu \right) = \frac{1}{2}\left( \partial_\nu g\left(\partial_\mu , \partial_\lambda \right) + \partial_\mu g\left(\partial_\nu , \partial_\lambda \right) - \partial_\lambda g\left(\partial_\nu , \partial_\mu \right)\right) = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right) = \Gamma_{\lambda\mu\nu}$$

 

[Matterwave]

Can you maybe fix that last expression? I'm seeing a bunch of latex code...

 

[kostas230]

Fixed. Sorry, it's 1:50 am and I'm half asleep. xD
I must have missed a parenthesis or a letter somewhere.

 

[Matterwave]

Ah, I see what you mean. But of course, such an expression as you have on the right is as well explicitly not coordinate independent, so I'm not sure why you prefer such an expression? I think it makes the issue more confusing actually...

 

[kostas230]

I'm just poking things. :P

I'm only curious if a similar definition can be given for the Christoffel symbols of the second kind. Also, I fail to see why the 2nd expression is not coordinate independent, since $X,Y,Z$ are arbitrary vector fields. In the third expression I just considered some coordinate system and showed that you can retrieve the original Christoffel symbols.

 

[Matterwave]

I'm just poking things. :P

I'm only curious if a similar definition can be given for the Christoffel symbols of the second kind. Also, I fail to see why the 2nd expression is not coordinate independent, since $X,Y,Z$ are arbitrary vector fields. In the third expression I just considered some coordinate system and showed that you can retrieve the original Christoffel symbols.

There's no way you can have a coordinate independent expression for a coordinate dependent object. That would defeat the purpose of calling such an object "coordinate dependent"!

Why do we know the right hand side is coordinate dependent? Let's look at 3-D flat space. In some coordinates (rectilinear coordinates) the right hand side is 0, while in other coordinates (e.g. polar coordinates) the right hand side is not 0. A coordinate independent expression can not be 0 in some coordinates and non-0 in other coordinates.

In a different way to say it, a tensor expression like $T(X,Y)$ must be able to be evaluated at a single point. But the expression on the right of your equation explicitly depends on how the vectors $X,Y,Z$ change as we move around (since you have partial derivatives there).

 

[kostas230]

Oh, I see where's the problem. Thank you!

 

[Matterwave]

No problem. :)

 

[Incnis Mrsi]

There's no way you can have a coordinate independent expression for a coordinate dependent object. That would defeat the purpose of calling such an object "coordinate dependent"!

Actually there is a way to specify covariant derivative in a coodinate-free manner, because it is controlled by an affine connection in the tangent bundle, so named Levi-Civita connection. But one must pass from the manifold to its tangent bundle, that is also a manifold but of double dimension.

 

[Matterwave]

Actually there is a way to specify covariant derivative in a coodinate-free manner, because it is controlled by an affine connection in the tangent bundle, so named Levi-Civita connection. But one must pass from the manifold to its tangent bundle, that is also a manifold but of double dimension.

Of course there is a way to specify a co variant derivative in a coordinate-free manner, a co variant derivative is a legitimate tensor operation, and is coordinate independent.

The expression $\nabla_X Y$ is already coordinate independent just as $Y$ is a coordinate independent vector.

The OP wanted a coordinate independent way to express specifically a Christoffel symbol, which is not a tensor.

 

[Incnis Mrsi]

Specially for Matterwave: Christoffel symbol is a coordinate expression of Levi-Civita connection, a case of an affine connection in a bundle. If you get a point x ∈ M and a vector q ∈ T_xM, then, in local coordinates, dq^λ = Γ^λ_μνq^μdx^ν specify (dim M)-dimensional tangent flat to the connection at the point (x, q) in (2 dim M)-dimensional total space of the bundle TM. Sorry, Ī’m not sure that use μ and ν in proper order, but the thought is expressed clearly enough.

If you forget coordinates (x^ν, q^λ) but remember the bundle structure and the field of tangent flats, then you’ll obtain an invariant representation of Christoffel symbol, like of any other affine connection.

 

[Matterwave]

Specially for Matterwave: Christoffel symbol is a coordinate expression of Levi-Civita connection, a case of an affine connection in a bundle. If you get a point x ∈ M and a vector q ∈ T_xM, then, in local coordinates, dq^λ = Γ^λ_μνq^μdx^ν specify (dim M)-dimensional tangent flat to the connection at the point (x, q) in (2 dim M)-dimensional total space of the bundle TM. Sorry, Ī’m not sure that use μ and ν in proper order, but the thought is expressed clearly enough.

If you forget coordinates (x^ν, q^λ) but remember the bundle structure and the field of tangent flats, then you’ll obtain an invariant representation of Christoffel symbol, like of any other affine connection.

I don't understand. You say the Christoffel symbols are a "coordinate expression" of the Levi-Civita connection, which of course I agree with, but then you say that you can express them in an "invariant representation" (which I assume you mean coordinate-independent), without showing how such a construction is constructed. Can you elaborate?

 

[Incnis Mrsi]

Actually Ῑ tried to explain how to construct a thing called “Ehresmann connection” (didn’t know this eponym, deemed it a common differentially-geometric knowledge) from Christoffel symbol components.

Several things are essentially the same:

• Covariant derivative operator for vector fields;
• Affine connection, presented as:
  • a connection form, in local coordinates expressed with one upper and two lower indices (e.g. Christoffel symbol);
  • an Ehresmann connection, that in this case requires explicitly the total space of TM;
  • (possibly more options).

We can encode the connection data (Ῑ abused terminology when called it “Christoffel symbol”) in terms of tensor fields on TM, not on M. Must Ῑ write explicitly now an Ehresmann connection is encoded in an (1,1) tensor field and decoded from it?

 

[Matterwave]

Actually Ῑ tried to explain how to construct a thing called “Ehresmann connection” (didn’t know this eponym, deemed it a common differentially-geometric knowledge) from Christoffel symbol components.

Several things are essentially the same:

• Covariant derivative operator for vector fields;
• Affine connection, presented as:
  • a connection form, in local coordinates expressed with one upper and two lower indices (e.g. Christoffel symbol);
  • an Ehresmann connection, that in this case requires explicitly the total space of TM;
  • (possibly more options).

We can encode the connection data (Ῑ abused terminology when called it “Christoffel symbol”) in terms of tensor fields on TM, not on M. Must Ῑ write explicitly now an Ehresmann connection is encoded in an (1,1) tensor field and decoded from it?

I have little experience with the language of connections in forms, but from my understanding, a connection form is not the same thing as a "Christoffel symbol". They might serve a similar purpose, but it would not be right to call them Christoffel symbols. In addition, the wikipedia article seems to suggest that the connection forms are basis dependent as well.