Because I am interested how to extend results with anayltic continuation on manifolds, I am reading Affine Osserman connections and their Riemann extensions*. In theorem 12 they say in the first sentence that at each point $p\in M$, where $(M,g)$ is a semi-Riemannian manifold, the metric and the curvature tensor $\mathcal{R}$ are analytic functions. And I'm not able to justify this statement for myself. Here are my thoughts:
A function on a manifold is analytic at a point if it is an analytic function with respect to every chart $(U,\varphi)$ around $p\in M$. I hope this is the right way to talk about analyticity on a manifold.
Now I wanted to look at the metric in a fixed chart as a polynomial, namely $$g(\cdot)=\sum_{i,j=1}^n g_{ij}(\cdot) dx^i(\cdot)\otimes dx^j(\cdot).$$ So, I look at $g$ as a polynomial function that gives me for every point the inner product on the tangent spcae and in the same way I can look at $\mathcal{R}$ as a polynomial in the components of the metric and it's derivatives up to order 2.
Now I would argue that polynomials are analytic and therefore $g$ and $\mathcal{R}$ are anayltic at every $p\in M$.
I am sure that I overlook something. For example do I have to be concerned about the $dx^i\otimes dx^j$? Or what happens if I change the charts? Since the transition functions are not analytic in general do I need to assume a real-analytic structure or is this not necessary?