## Modification and Trivialization of a connection

Let $E$ be a Dirac bundle with metric $h^E$, and $\nabla^E$ be a metric connection on it. We modify this connection by zero order terms containing a Killing field $X$:

$\nabla^{E,X}_v s = \nabla^E_v s + \langle X , v \rangle s$

Now we use Paralleltransport along geodesic and get

$\nabla^{E,X} = d + \omega$, where $\omega$ the connection one-form.

Question: Do we have $h^E(\partial_i s, s') = \partial_i(h^E(s,s')) - h^E(s,\partial_i s')$ although $\nabla^{E,X}$ is not metric?

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This looks too much like homework. You should be able to answer your question using a straightforward calculation. Also, your notation is inconsistent (you seem to be using $\partial_i$ to mean the connection in the last line. – Deane Yang Oct 6 at 14:14