# Christoffel symbols, dual space

I am confused with the definition of Christoffel symbols for the dual space.

Let $$M$$ be some manifold, $$x_i$$ local coordinates

The Christoffel symbols are defines as

$$\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k$$

where $$\nabla$$ is the Levi Civita connection on $$M$$.

Now I read that

$$\nabla_{\partial_i} dx_j = - \Gamma^j_{ik} dx_k$$

but what is $$\nabla$$ here? It can't be the Levi-Civita connection right?

• It is a connection defined on $T^*M$ Commented Jul 8, 2020 at 19:47
• But a specific one? Commented Jul 8, 2020 at 19:58
• Yes, the one defined by your equation. It is also called the dual connection Commented Jul 8, 2020 at 19:59
• Oh, so the connection is defined by the second equation in my question? Commented Jul 8, 2020 at 20:09

Whenever one has a connection $$\nabla$$ on $$TM$$ there is an induced connection on the dual bundle $$T^*M$$ (which we will also call $$\nabla$$ because the abuse of notation is consistent), and it defined in such a way that the connection satisfies a sort of product rule when taking the covariant derivative of $$(\alpha,v)=\alpha(v)$$.
$$\nabla_X (\alpha,v) = (\nabla_X\alpha,v) +(\alpha,\nabla_X v)$$
But then recall that the action of the covariant derivative on a function $$\nabla_X(f)=X(f)$$ so we have
$$X \left((\alpha ,v)\right)=(\nabla_X\alpha,v) +(\alpha,\nabla_X v)$$