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?

  • $\begingroup$ It is a connection defined on $T^*M$ $\endgroup$ – Arctic Char Jul 8 '20 at 19:47
  • $\begingroup$ But a specific one? $\endgroup$ – User1 Jul 8 '20 at 19:58
  • $\begingroup$ Yes, the one defined by your equation. It is also called the dual connection $\endgroup$ – Arctic Char Jul 8 '20 at 19:59
  • 1
    $\begingroup$ Oh, so the connection is defined by the second equation in my question? $\endgroup$ – User1 Jul 8 '20 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

$$X \left((\alpha ,v)\right)=(\nabla_X\alpha,v) +(\alpha,\nabla_X v)$$

From here by picking basis one forms and vectors you can arrive at the formula you had. So in short it is the induced connection on the cotangent bundle of the Levi-Civita connection.


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