Christoffel connection in Kaluza Klein Theory

[off-diagonal]

Now I'm just start to study the Kaluza-Klein theory from http://arxiv.org/abs/grqc/9805018.
I try to calculate the Einstein Field equations in 5 dimensional vacuum space-time.

we start with 5D metric tensor,

$$\hat{g}_{AB}=\begin{pmatrix} g_{\alpha}_{\beta}+k^{2}\phi^{2}A_{\alpha}A_{\beta} & k\phi^{2}A_{\alpha}\\ k\phi^{2}A_{\beta}&\phi^{2} \end{pmatrix}$$

where A,B indices run from 0,1,2,3,4 and $$\alpha,\beta$$ run from 0,1,2,3

Next, I have to calculate Christoffel connection from:
$$\displaystyle \hat{\Gamma}^{A}_{BC}=\frac{1}{2}\hat{g}^{AD}(\partial_{B}{\hat{g}_{CD}}+\partial_{C}{\hat{g}_{BD}}-\partial_{D}{\hat{g}_{BC}})$$

Kaluza propose the extra condition called "cylindrical condition" that says all derivative involve with the fifth coordinate must vanish. so we can conclude that
$$\partial_4{\hat{g}_{AB}}= \partial_4{\hat{\Gamma}^{A}_{BC}}=0$$
then all the connection would be
$$\displaystyle\hat{\Gamma}^{4}_{44},\hat{\Gamma}^{4}_{4\alpha},\hat{\Gamma}^{\alpha}_{44},\hat{\Gamma}^{\alpha}_{4\beta},\hat{\Gamma}^{4}_{\alpha\beta},\hat{\Gamma}^{\alpha}_{\beta\gamma}$$

Here's is my questions, I confuse about the calculation of the last two connection $$\hat{\Gamma}^{4}_{\alpha\beta},\hat{\Gamma}^{\alpha}_{\beta\gamma}$$
for an example:
$$\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma D}(\partial_{\alpha}{\hat{g}_{\beta D}}+\partial_{\beta}{\hat{g}_{\alpha D}}-\partial_{D}{\hat{g}_{\alpha\beta}})$$

I'm confused about index sum "D". Can I just replace index D with 4 dimensional (Greek)index or I have to sum it to 4D index plus the fifth-D ones.
$$\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma D}(\partial_{\alpha}{\hat{g}_{\beta D}}+\partial_{\beta}{\hat{g}_{\alpha D}}-\partial_{D}{\hat{g}_{\alpha\beta}})=\frac{1}{2}[\hat{g}^{\sigma \lambda}(\partial_{\alpha}{\hat{g}_{\beta \lambda}}+\partial_{\beta}{\hat{g}_{\alpha\lambda}}-\partial_{\lambda}{\hat{g}_{\alpha\beta}})+\hat{g}^{\sigma 4}(\partial_{\alpha}{\hat{g}_{\beta 4}}+\partial_{\beta}{\hat{g}_{\alpha 4}})$$
or just
$$\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma\lambda}(\partial_{\alpha}{\hat{g}_{\beta \lambda}}+\partial_{\beta}{\hat{g}_{\alpha\lambda}}-\partial_{\lambda}{\hat{g}_{\alpha\beta}})$$

which one is correct?
otherwise you may be just told me a book or paper that the explicit form of all possible connections in Kaluza-Klein theory were expressed. I would be appreciate.

 

[Mentz114]

If the EM field is going to act on a charged test body, then there must be some Christoffel symbols involving the 5th dimension. So I'd guess your first conjecture is right.

Have a look at this paper. It has a bit on the kinematics of K-K theory ( but that is not it's main interest ).