Equivalence for Christoffel symbol and Koszul formula I am trying to show to define a Levi-civita connection, it's equivalent to define Christoffel symbols or define Koszul formula.
$$ 2g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X)$$
One direction is easy, given Koszul formula, take $X=\frac{\partial}{\partial x^i},Y=\frac{\partial}{\partial x^j}, Z=\frac{\partial}{\partial x^k}$ and compute.
For the other direction, I set $X=x^i\partial_i,Y=y^j\partial_i,Z=z^k\partial_k$ and try to show LHS and RHS of Koszul formula are equal. After expressing both sides explicitly, I think I need to use some other conditions to cancel some terms before I can plug in the definition of Christoffel symbols. Could you give some reference or a detailed illustration? Thanks.
 A: This is proved in my Riemannian Manifolds book. (See the proof of Theorem 5.4.)
A: Actually, you should add $\frac{1}{2}$ in RHS of formula, and it should have been
$$\langle\nabla_XY,Z\rangle = \frac{1}{2}(X\langle Y,Z\rangle+Y\langle X,Z\rangle-Z\langle X,Y\rangle+\langle[X,Y],Z\rangle-\langle[X,Z],Y\rangle-\langle[Y,Z],X\rangle)$$


*

*$\langle\cdot,\cdot\rangle=g(\cdot,\cdot)$

*You can find it on http://en.wikipedia.org/wiki/Levi-Civita_connection

It is accessible to use the local frame to prove the formula. However, we can prove it by the axioms, which also can be founded on http://en.wikipedia.org/wiki/Levi-Civita_connection.
Below is the proof by local coordinate(sketch)
Assume $X=X_i\frac{\partial}{\partial x_i},~~Y=Y_i\frac{\partial}{\partial x_i},~~Z=Z_i\frac{\partial}{\partial x_i}$, then
$$[X,Y]=(X_j\frac{\partial Y_i}{\partial x_j}-Y_j\frac{\partial X_i}{\partial x_j})\frac{\partial}{\partial x_i}$$
$$[X,Z]=(X_j\frac{\partial Z_i}{\partial x_j}-Z_j\frac{\partial X_i}{\partial x_j})\frac{\partial}{\partial x_i}$$
$$[Y,Z]=(Y_j\frac{\partial Z_i}{\partial x_j}-Z_j\frac{\partial Y_i}{\partial x_j})\frac{\partial}{\partial x_i}$$
and
$$\nabla_XY=(X_iY_j\Gamma_{ij}^k+X_i\frac{\partial Y_k}{\partial x_i})\frac{\partial}{\partial x_k}$$
Noticing there are three facts


*

*$\langle X,Y\rangle=g_{ij}X_iY_j$;

*$\Gamma_{ij}^k=\frac{1}{2}g^{kl}(\frac{\partial g_{il}}{\partial x_j}+\frac{\partial g_{jl}}{\partial x_i}-\frac{\partial g_{ij}}{\partial x_l})$ 

*$(g_{ij})(g^{kl})=I$
Then take the local presentation into formula, use the three facts, and you will find $$\text{LHS=RHS}$$
Because the work needs patience and carefulness, you know $\cdots$.
