If $u\in C^{2}(\mathbb{R}^{n})$, then the taylor expansion, if we take the change of variable $z=(x-y)/\epsilon$, $x,y\in\mathbb{R}^{n}$,
$$u(y) - u(x) = u(x+\epsilon z) - u(x) = \epsilon\nabla u(x)\cdot z + \frac{1}{2} z^{T}\cdot H u(x)\cdot z + O(\|z\|^3) \quad(*)$$
Now, I want to generalized the above. If we take a riemannian manifold $M$, then (the only taylor expansion that I find), the Taylor expansion of $u$ around a point $p\in M$,
$$u(\exp_{p}(v)) \approx u(p) + \langle \nabla_{M} u(p), v\rangle_{p} \qquad ,v\in T_{p}M, v \approx 0 \qquad(**)$$
The above expansion does not depend on the choice of the coordinate system at $p$.
In the expression $(**)$ for $M=\mathbb{R}^{n}$, the $\exp_{p}(v)$ is the identity. So,
$$u(v) \approx u(p) + \langle \nabla u(p), v\rangle_{p} \qquad ,v\in T_{p}M\sim \mathbb{R}^{n}, v \approx 0 $$
Therefore, to me it has sense a taylor expansion as $(**)$.
So there exist a taylor expansion as $(**)$ of a manifold?