I am an undergrad student learning Riemannian geometry. My question is about whether you have a nice orthonormal frame in the following sence.
Let $(M, g)$ be a Riemannian manifold, with $\bigtriangledown$ the Levi-Civita Connection, and $p$ an arbitrary point of $M$. I would like to ask whether an orthonormal frame $(X_i)_i$ around $p$ exists which satisfies $\bigtriangledown _{X_i} X_j =0$ at $p$ for all $i$, and $j$.
This kind of things appears in the proof of Bochner Formula in the pdf below.
https://www.math.uh.edu/~minru/Riemann09/bochnerhodge.pdf
I understand that such a frame should be convenient, but unfortunately I am not sure if it exists. Here's what I've thought:
Probably one way of dealing with this is normal coordinates. (As a matter of fact, the author refers to it in the first page of the pdf.) I fully understand that extending an orthonormal BASIS at $p$ makes a normal coordinate $(x_i)_i$, which satisfies $g_{i, j}(p) = \delta _{i, j}$ because an orthonormal basis was chosen at $p$, and also satisfies $\Gamma_{i, j}^k = 0$, or $\bigtriangledown_{\partial _i}\partial _j = 0$ since it is normal.
However, when making the frame $(\partial _i)$ into an orthonormal FRAME, there is a problem. That is, I am planning to do some Gram–Schmidt orthonormalization around $p$ to generate a new orthonormal frame $(X_i)$, but I cannot assert $\bigtriangledown _{X_i}X_j = 0$ at $p$ because, in writing down $\bigtriangledown _{X_i}X_j$ in terms of the coorinate $(x_i)$, it is not simply a "function-linear combination" of $\bigtriangledown _{\partial _i}\partial _j = 0$, but there are some other terms of the form $f(\partial _i g) \partial _j$, where $f$, and $g$ are functions.
Therefore, In the end, I post this question to ask the existence of orthonormal frame $(X_i)_i$ around a point exists which satisfies $\bigtriangledown _{X_i} X_j =0$ at the point. Thank you in advance for any help.