Orthonormal Frame as a function

Question

Let $M$ be a smooth manifold. We know that the frame at a point $p\in M$ can be defined as an isomorphism $f:\mathbb{R}^n\longrightarrow T_pM$. Is there a way of defining an orthonormal frame in a similar way, as a map?

Answer 1

You need a Riemannian metric $g$ on $M$ for this to make sense.

An orthonormal frame is just one in which the vectors are orthonormal. Since the vectors of a frame $f$ correspond to the image under $f$ of the standard basis of $\mathbb R^n$, this is equivalent to $f : (\mathbb R^n, \delta) \to (T_p M, g)$ being an isometry of inner product spaces. (Here $\delta$ is just the standard Euclidean metric.)