Error-term in the expansion of a riemannian metric in normal coordinates Let $(M,g)$ be a Riemannian metric and $\exp_x^{-1}:B_r(x) \to B_r(0) \subset \mathbb{R}^n$ be a normal coordinate system around $x \in M$ with respect to some orthonormal basis $e_1,\ldots, e_n$. The coefficients $g_{ij}$ of the metric tensor in $tz$, $t \in \mathbb{R}$ and $z \in \mathbb{R}^n$ is then given by
$$
t^2 g_{ij}(t):= t^2g_{ij}(tz) =< Y_i(t), Y_j(t)>
$$
where the $Y_i(t)$ are the unique solution of the Jacobi field equations
$$
Y_\alpha(t)'' + R\left(\gamma'(t),Y_\alpha(t)\right)\gamma'(t) \quad (*)
$$
along the geodesic $\gamma(t):=\exp_x t z$ with initial condition $Y_i(0)=0$ and $Y'_i(0)=e_i$. For example in the "Lectures on Differential Geometry" by Schoen and Yau this is used to compute the first few terms in the expansion of the $g_{ij}$;
$$
\bar{g}_{ij}(x)= \left(\delta_{\alpha \beta} - \dfrac{1}{3}R_{i kl j}x^{l} x^{k}\right) + lower Order
$$
I am having trouble understanding what this "lower Order" means. How do I use relation $(*)$ to prove that the reminder actually is smaller than the other terms in the expansion for $|x|$ small?
 A: The idea is first to show this lemma:

If $\gamma$ is a curve, $Y$ a smooth vector field along $\gamma$, and if $\forall k \geqslant 0$, $Y_k$ is the unique parallel vector field along $\gamma$ such that $Y^k(0) := Y^{(k)}(0) = \left(\nabla_\gamma^{(k)}Y\right)(0)$, where $\nabla_\gamma$ is the covariant derivative along $\gamma$, then:
$$
Y(t) \underset{t\to 0}{=} \sum_{j=0}^k \frac{t^j}{j!}Y^j(t) + o(t^k)
$$

This is just the Taylor formula applied in any parallel frame along $\gamma$. Then, one can compute, using this formula,
$$
g_{ij}(t) = \frac{1}{t^2} \langle Y_i(t) , Y_j(t)\rangle
$$
by developping the scalar product with Taylor formula (careful with indexes: there will be $Y_i^k$!) and compute the Taylor expansion of $g_{ij}$ at any orders in terms of the different covariant derivatives of $Y_i$ and $Y_j$ (there are many mixed terms). These last terms can be computed thanks to the Jacobi equation. For example, for $k = 2$:
$$
Y^2(0) = -R(\gamma'(0),Y(0))\gamma'(0) = 0,
$$
for $k = 3$:
$$
Y^3(0) = -R(\gamma'(0),Y'(0))\gamma'(0),
$$
for $k=4$:
$$
Y^4(0) = - \left(\nabla_{\gamma'(0)}R\right)(\gamma'(0),Y'(0))\gamma'(0) + \ldots ,
$$
etc. Note that we use several times the geodesics equation. One can show that the terms in the Taylor expansion of $g_{ij}$  are universal polynomials in the coefficients of $R$, $\nabla R$, $\nabla \nabla R$... and $Y'(0)$, $Y''(0)$, $Y'''(0)\ldots$
