# Proof of existence of Levi-Civita connection via the Koszul Formula

I'm currently self-studying the basic concepts of Riemannian Manifolds, and I'm stuck on the proof of the existence of the Levi-Civita connection that is presented in the book "Semi-Riemannian Geometry" by O'Neill, which is a coordinate-free approach.

Suppose $$(M,g)$$ is a Riemannian manifold, and let $$\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$$ be a connection. So far, I've been able to prove that, if $$\nabla$$ verifies the conditions of the Levi-Civita connection, then it must satisfy the Koszul Formula:

$$\langle\nabla_{X}Y,Z\rangle=\dfrac{1}{2}(X\langle Y, Z\rangle+Y\langle Z, X\rangle-Z\langle X, Y\rangle -\langle Y,[X, Z]\rangle-\langle Z,[Y, X]\rangle+\langle X,[Z, Y]\rangle).$$

So now, I'm trying to prove existence by using the formula: more precisely, with this result that I proved previously:

If $$\omega:\mathfrak{X}(M)\to \mathcal{F}(M)$$ is a differential $$1$$-form, then there exists a unique vector field $$V$$ such that for any other vector field $$X$$, we have $$\omega(X)=\langle V,X\rangle$$.

My reasoning is the following: fix two vector fields $$X,Y$$, and let $$\omega_{X,Y}:\mathfrak{X}(M)\to \mathcal{F}(M)$$ be the map

$$\omega_{X,Y}(Z)=\dfrac{1}{2}(X\langle Y, Z\rangle+Y\langle Z, X\rangle-Z\langle X, Y\rangle -\langle Y,[X, Z]\rangle-\langle Z,[Y, X]\rangle+\langle X,[Z, Y]\rangle).$$

If I were able to prove that $$\omega_{X,Y}$$ is $$\mathcal{F}(M)$$-linear (which, knowing the Theorem to be true, it must be), then by using the preceding result I can define a unique vector field $$\nabla_{X}Y$$ such that $$\omega_{X,Y}(Z)=\langle \nabla_{X}Y,Z \rangle$$. Thus, I would have a well defined map $$\nabla$$ satisfying the Koszul Formula, and it would be (after some property checking) the Levi-Civita connection.

The problem is that, while it's easy to see that $$\omega_{X,Y}(Z_{1}+Z_{2})=\omega_{X,Y}(Z_{1})+\omega_{X,Y}(Z_{2})$$, I haven't been able to prove that $$\omega_{X,Y}(fZ)=f\omega_{X,Y}(Z)$$. In reality, what I've gotten is that

$$\omega_{X,Y}(fZ)=f\omega_{X,Y}(Z)-\dfrac{1}{2}\langle (Xf)Y+(Yf)X,Z \rangle$$,

but I'm not sure that the second summand is $$0$$. Is my attempt correct so far, or am I missing something?

• Lie groups/algebra? Apr 13 '20 at 19:07
• This appeared to me as part of something unrelated to Lie groups (even though my plan is to study isometric actions of a Lie group over a Riemannian manifold in some time). Apr 13 '20 at 19:11

Just checked and $$\omega_{X,Y}$$ is in fact $$C^\infty$$ linear. It seems like the terms you picked up were from the 4th and 6th term, the first and second terms also give extra terms:
$$X(\langle Y,fZ\rangle ) = X(f\langle Y,Z\rangle) =X(f)\langle Y,Z \rangle + fX(\langle Y,Z\rangle)$$
• I see! That was the mistake, i forgot that $X$ is a derivation, so I didn't get those terms. Thank you!! Apr 13 '20 at 19:36