Local description of the Levi-Civita connection and the Christoffel symvols

Question

I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. I have some questions regarding the local description of the Levi-Civita connection $\nabla^g$. First I will give you some definitions.

1. For a vector bundle $E\to M$ the graded $C^\infty$-module of "$E$-valued differential forms on $M$, $\Omega^\bullet (M,E)$, is defined by $\Omega^k(M,E)=\Gamma^\infty(M,E)\otimes_{C^\infty(M)}\Omega^k(M)\cong\Gamma^\infty(M,\bigwedge^kT^*M\otimes E)$.

2. A connection on $E$ is a linear map $\nabla:\Gamma^{\infty}\to\Omega^\bullet (M,E)$ satisfying the Leibniz rule $\nabla(s\otimes\omega)=(\nabla s)\otimes\omega+s\otimes\mathrm{d}\omega \quad \mathrm{for}\,s\in\Gamma^{\infty}(M,E), \omega\in \Omega^\bullet(M).$

3. Define the operator $\nabla_X:=\iota_X\circ\nabla+\nabla\circ\iota_X$, where $\iota_X$ is given by $\iota_X(s\otimes\omega)=s\otimes\iota_X\omega$ with $\iota_X\omega$ the contraction of forms.

4. Let $(M,g)$ be a Riemannian manifold. Then the Levi-Civita connection $\nabla^g$ on $TM$ is the unique affine, torsion-free connection satisfying: $\quad g(\nabla^gX,Y)+g(X,\nabla^gY)=\mathrm{d}(g(X,Y)) \quad \forall\,\,X,Y\in\mathfrak{X}(M,\mathbb{R}).$

My questions:

• It was written that locally, $\nabla^g=d+\alpha$, where $\alpha\in\Omega^1(U,\mathrm{End}\,TM)$ is the connection 1-form on chart $U\in M$. Why does $\nabla^g$ looks like this?

• Then the Christoffel symbols where defined by $\Gamma^k_{ij}\,dx^i\otimes\partial_k:=\nabla^g\partial_j$. To show that $\Gamma^\bullet_{ij}=\Gamma^\bullet_{ji}$, I tired to show that $\nabla^g_i\partial_j=\Gamma^k_{ij}\partial_k$

$\qquad\nabla^g_i\partial_j=\iota_j(\Gamma^k_{ij}\,dx^i\otimes\partial_k)+\nabla^g(\iota_i\partial_k)=\Gamma^k_{ij}\,dx^i(\partial_i)\otimes\partial_k+\nabla^g(\iota_i\partial_k)=\Gamma^k_{ij}\,dx^i\partial_k+\qquad\nabla^g(\iota_i\partial_k)$ $\qquad$what happens with the right summand?

• Moreover it was mentioned that $g(\nabla^g_ZX,Y)+g(X,\nabla^g_ZY)=Z(g(X,Y)$ has locally the form $\,g_{jm}\Gamma^m_{li}+g_{im}\Gamma^m_{lj}=\partial_lg_{ij}$. To show this, I tried to write the metric in local coordinates $g(X,Y)=g_{ij}X^iX^j$ and $\nabla^g_ZX=\nabla_{Z^j\partial_j} Y^k\partial_k = Z^j \nabla_j (Y^k\partial_k)$ (how can I rewrite $\nabla_j (Y^k\partial_k)$)

Thanks for your help.