# Levi-Civita connection on a sphere in the vielbein formalism

+ 4 like - 0 dislike
1786 views

I am trying to learn the vielbein formalism and have a question for the example of the Riemann sphere $S^2$. I am afraid my question is rather elementary, as it seems to be a simple sign error. Still, could someone help me figure this out?

On the sphere with coordinates $(x,y,z) = (\cos ϕ \sin θ, \sin ϕ \sin θ, \cos θ)$ and metric $ds^2 = dθ^2 + \sin θ^2 dϕ^2$, we can define the zweibein

$$e_θ = ∂_θ , \quad e_ϕ = \frac{1}{\sin θ} ∂_ϕ$$

The Levi-Civita connection for the metric is torsion-free, which means

$$\nabla_{e_ϕ} e_θ - \nabla_{e_θ} e_ϕ = [e_ϕ,e_θ]$$

A separate calculation shows that $\nabla_{e_θ}e_ϕ = 0$, so we can use this formula to quickly calculate the connection form $ω_{ab}$:

$$\nabla_{e_ϕ} e_θ = [e_ϕ,e_θ] = -∂_θ(\frac{1}{\sin θ})·∂_ϕ = \cot θ · e_ϕ \equiv ω_{θ\ \ }^{\ \ ϕ}(e_ϕ) e_ϕ$$

Unfortunately, this calculation seems to be wrong, because it contradicts the statement

$$ω^{ϕ\ \ }_{\ \ θ}(e_ϕ) = \cot θ$$

that I found in some lecture notes (formula (2.345)). The connection form is antisymmetric, so one of the two values should be $-\cot θ$, but I can't decide which.

Can somebody help me find the source of this sign discrepancy?

Ooh, it appears to be a matter of convention for the connection form $\omega$! It looks like physicists use the notation $\nabla_X e_b = \omega^a_{\ \ b}(X) e_a$ whereas mathematicians tend to use the notation $\nabla_X e_i = \sum_j \omega^j_i(X) e_j$. Consequently, physicists write the Cartan structure equations as $\Omega^a_{\ \ b} = d\omega^a_{\ \ b} + \omega^a_{\ \ c} \wedge \omega^c_{\ \ b}$ whereas mathematicians write them as $\Omega^{j}_i = d\omega^j_i - \sum_k \omega^k_i \wedge \omega^j_k$ with a different sign. I tried to derive the formalism myself, and accidentally picked a convention that is that is neither the mathematician's nor the physicists' convention. Will put this into a real answer soon.

+ 3 like - 0 dislike

The calculation is correct, but it appears that there are different conventions in use for the connection 1-form $\omega$. In the question, the definition for $ω$ used to the very right of the $\equiv$-sign in the question is not one of the standard conventions, that's why the signs differ.

Apparently, physicists tend use the notation $\nabla_X e_b = \omega^a_{\ \ b}(X) e_a$ (example), whereas mathematicians tend to use the notation $\nabla_X e_i = \sum_j \omega^j_i(X) e_j$ (example [pdf]).

Consequently, physicists write the Cartan structure equations for the curvature as $\Omega^a_{\ \ b} = d\omega^a_{\ \ b} + \omega^a_{\ \ c} \wedge \omega^c_{\ \ b}$ whereas mathematicians write them as $\Omega^{j}_i = d\omega^j_i - \sum_k \omega^k_i \wedge \omega^j_k$ with a different sign.

By the way, instead of calculating the commutator, the connection 1-form can also be calculated from the Cartan structure formula for the dual frame. Using the dual frame $θ^θ = dθ$ and $θ^ϕ = \sin θ \, dϕ$, the equations read
$$0 = dθ^θ + ω^θ_{\ ϕ} \wedge θ^ϕ = d(dθ) - \sin θ\, ω^ϕ_{\ θ} \wedge dϕ = -\sin θ\, ω^ϕ_{\ θ} \wedge dϕ$$

$$0 = dθ^ϕ + ω^ϕ_{\ θ} \wedge θ^θ = d(\sin θ\, dϕ) - dθ \wedge ω^ϕ_{\ θ} = dθ \wedge (\cos θ\, dϕ - ω^ϕ_{\ θ}) .$$
The first equation implies that the form $ω^ϕ_{\ θ}$ is a multiple of the form $dϕ$. The second equation implies that the prefactor is $\cos θ$, so we have
$$ω^ϕ_{\ θ} = \cos θ\, dϕ = \cot θ·θ^ϕ .$$

answered Feb 15, 2015 by (775 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.