Big oh Notation in Riemannian Geometry

Question

I am studying Lee's Book Riemannian Geometry 2ed and on p.328 he writes $g_{ij}=\delta_{ij}+ {O}(r^2)$, where $r$ is the radial distance function in a normal neighborhood. How can I prove it?

Actually I can not understand the definition of "Big oh" notation for functions $f,g:U\subset M \to R$. What $f(x)={O}(g(x))$ means in this case?

Thanks!

Answer 1

Since we have normal coordinates centered at $p$ we have

$g_{ij}(p)=\delta_{ij} $ and $\partial_k g_{ij}(p)=0$.

So, it follows by Taylor expansion formula.