Expressing a connection of a vector bundle at a point

Let $$P\to X$$ be a rank $$n$$ vector bundle with a metric over a smooth manifold $$X$$, and let $$\nabla$$ be a connection on $$P$$ that is compatible with the metric. Fix a point $$x\in X$$. Then is it true that we can choose an orthonormal basis $$e_1,\dots,e_n$$ of $$P_x$$ so that $$\nabla=\sum_k \nabla_{e_k}\otimes de_k$$ at $$x$$? I am reading Morgan's book on Seiberg-Witten theory, and in the proof of Proposition 5.1.5, the writer chooses such a basis, but I can't see how this can be done.

• Your statement makes no sense, as it confuses an orthonormal frame for $P$ with an orthonormal frame for $X$. Moreover, please say what $de_k$ is supposed to mean. See the comments below. Jul 25, 2022 at 21:08

I will assume that the $$\{e_i\}$$ are a frame for the tangent bundle, as this is the only way in which the expression $$\nabla_{e_i}$$ could make sense. The connection $$1$$-form $$A$$ is defined by $$\nabla e_i=\sum e_j\otimes A^j_i$$. The connection coefficients $$\Gamma^k_{ij}$$ are defined by $$\nabla_{e_j}e_i=\sum\Gamma^k_{ij}e_k$$. We want a basis such that $$\nabla e_i=\sum_j\nabla_{e_j}e_i\otimes \theta^j$$, denoting by $$\theta^j$$ the coframe. Using the above definitions, this amounts to $$\nabla e_i=\sum_j(\sum_k\Gamma^k_{ij}e_k)\otimes\theta^j=\sum_ke_k\otimes(\sum_j\Gamma^k_{ij}\theta^j)$$ This means that we want a basis so that the connection $$1$$-form is related to the connection coefficients via $$A^j_i=\sum_k\Gamma^j_{ki}\theta^k$$ This is the relation between the connection $$1$$-form and the connection coefficients, for a metric connection (e.g. on https://en.wikipedia.org/wiki/Connection_form#Example:_the_Levi-Civita_connection ). So it should indeed be possible to find such a frame.
• What does $\theta^j = de_j$ even mean? Jul 25, 2022 at 16:49
• Notation for the coframe associated to the frame $e_j$ which is used on wikipedia. Jul 25, 2022 at 18:57
• Surely, the fact that we have the expression $\nabla_{e_i}$ appearing in the OP means that the $\{e_i\}$ are in fact a frame for the tangent bundle, not for an arbitrary vector bundle? Jul 25, 2022 at 19:47
• No, the OP says explicitly that they're an orthonormal frame for $P_x$. I think you need to read questions more carefully. Jul 25, 2022 at 20:49