TITLE: Christoffel symbols on a Lie group in Riemann normal coordinates
QUESTION [11 upvotes]: Consider a coordinate patch around the identity element in a  Lie group given by the exponential mapping (Riemann normal coordinates).  We have  a Levi-Civita connection corresponding to the bi-invariant metric associated with the Killing form.
What are the Christoffel symbols for this metric in the Riemann normal coordinates?
Here is a refinement of this question:  The Christoffel symbols will have a Taylor expansion in the Riemann coordinates  with the coefficients being some tensors constructed out of the Lie algebra structure constants. The first nonvanishing tensor is a 4-tensor constructed out of two factors of the structure constants. That one is easy to compute – this is related to the curvature 4-tensor at the identity. The question is – are the other tensors known in any closed form?

REPLY [3 votes]: The Riemann normal coordinates on a group manifold are obtained by the application
of the exponential map to its Lie algebra.
This description (and notation) is based on the appendices of the following article by Marinov and terentyev.
http://physics.technion.ac.il/~site/wisw/memoriam/marinov/pub/Mar46.pdf
It is assumed that the group is semisimple.
I have written explicit expressions up to the third order. The Einstein summation convention is implemented.
The invariant metric on a semisimple Lie group manifold has the form:
$g_{ab}(\xi) = M_a^p(\xi) M_b^q(\xi) G_{pq}$
where: $G_{pq}$ is the Cartan-Killing form of the group Lie algebra.
The vielbeins $M_a^b(\xi)$ have the form:
$M_b^a(\xi) = \int_0^1 Ad(exp(\tau\xi))_b^a d\tau$
The Christoffel symbols are are given by:
$\Gamma_{ab}^c (\xi) = L_p^c (\xi)(\partial_a M_b^p +\partial_b M_a^p)$
where:  $L_a^b(\xi)$ are the inverse vielbeins:
$L_a^b(\xi)M_b^c(\xi) = \delta_a^c$
given by:
$L_b^a(\xi) = \frac{\partial(log(exp(\eta)exp(\xi))^a)}{\partial \eta^b}|_{\eta=0}$
Using the Baker Campbell Hausdorff formula, we obtain the following expression for the inversse vielbeins 
$L_b^a(\xi) = \delta_b^a +\frac{1}{2}C_{bc}^a\xi^c+\frac{1}{12}C_{bc}^pC_{pd}^a\xi^c\xi^d +\frac{1}{720}C_{bn}^m C_{mt}^r C_{rp}^q C_{qs}^a\xi^n \xi^t \xi^p \xi^s + ...$
Using the Hadamard formula, we obtain the following expression for the vielbeins 
$M_b^a(\xi) = \delta_b^a -\frac{1}{2}C_{bc}^a\xi^c +\frac{1}{6}C_{bc}^pC_{pd}^a\xi^c\xi^d -\frac{1}{24}C_{bn}^m C_{mt}^r C_{rp}^a \xi^n \xi^t \xi^p +\frac{1}{120}C_{bn}^m C_{mt}^r C_{rp}^q C_{qs}^a\xi^n \xi^t \xi^p \xi^s + ...$