I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth function $f$ such that the Ricci tensor $R_{ij}$ of the metric $g_{ij}$ satisfies $$R_{ij}+\nabla_i\nabla_j f=\rho g_{ij}$$ for some constant $\rho>0$.
In the proof, it asserts that (P.1164): Suppose $|\nabla f|^2=0$ on some nonempty open set of $M$. Since any gradient shrinking Ricci soliton is analytic in harmonic coordinates, it follows that $|\nabla f|^2=0$ on $M$.
I would like to understand why "any gradient shrinking Ricci soliton is analytic in harmonic coordinates". Also, I don't understand why "any gradient shrinking Ricci soliton is analytic in harmonic coordinates" would imply that $|\nabla f|^2=0$ on $M$. Thank you very much.
Thomas Ivey has shown in his paper (Local existence of Ricci solitons, Manuscripta Mathematica 91, 151–162 (1996)) that any Ricci soliton is analytic in harmonic coordinates. If you don't have access to the paper you can get a version in the following link http://iveyt.people.cofc.edu/papers/locnew.pdf.
With respect to the other question, as suggested by @IsaacSolomon in his comment, if a function vanishes on an open set and it is analytic then it vanishes globally.