A manifold $M$ is said to be spherically symmetric if, in terms of geodesic polar coordinates $(r,\theta)\in (0,\infty)\times \mathbb{S}^{n-1}\equiv M\setminus \{o\}$ the Riemannian metric $ds^2$ of $M$ can be written as $$ds^2 = dr^2 + f(r)^2d\theta^{2}$$
For a function $f\in C^{2}([0,R])$, $R\in (0,\infty]$ with $f(0)=0$, $f'(0)=1$, $f\vert_{(0,R)}>0$ and there exists a point $o\in M$ for which the exponential mapping is a diffeomorphism of $T_{o}M$ onto $M$.
If $u\in C^{2}(M)$, i.e, $u:M\to\mathbb{R}$ a function of class $C^{2}$, how can write a taylor expansion for the function $u$ in some point $y=(r,\theta)\in M$, using the condition of that manifold is spherically symmetric?
My idea was, define a function $\varphi:\mathbb{R}\to M$, and then the composition $u\circ\varphi:\mathbb{R}\to\mathbb{R}$. To that end, one may write down the Taylor expansion of $u\circ\varphi$ in a local chart around $p=\varphi(0)$,
$$u(\varphi(t))=u(p)+\partial_i u \dot \varphi^i(0)\cdot t+\ldots$$
But I still do not see that it uses the condition of symmetry of the manifold....
But, more precissely how can form a 'taylor expansion' (or maybe a directional derivate) in the direction of the vector $\rho$, where $\rho=d(o, y)$, with $y=(\rho,\theta)$
Thanks!!