This is not really a specific question but rather a flawed argument and I am trying to spot where are the flaws, I apologize in advance.
Given a n-dimensional smooth manifold $\mathcal{M}$ consider the principal frame bundle $F\mathcal{M}$, this bundle has $GL(n,\mathbb{R})$ structure group right?
Assume now that this frame bundle admits a reduction of the structure group to $O(n,\mathbb{R})$. If this is possible, then at any given point $p\in\mathcal{M}$ one can define a metric tensor. How exactly? Can we define such metric tensor smoothly over an open neighborhood of $p$ or only at $p$ itself? Can we define it the tensor over the whole manifold?
Assuming that it is possible to define such a smooth metric tensor in an open neighborhood of $p$, can we find a chart such that the tensor has canonical components .i.e diag(1,1,...,1) in the coordinate bases of that chart over the whole neighborhood?
Thank you very much in advance!
