I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself.
Suppose that $M$ is a smooth compact manifold of dimension $m$ and $g$ is a smooth Riemann metric on $M$. $\newcommand{\ve}{{\varepsilon}}$ $\newcommand{\pa}{\partial}$ Suppose that $(g^\ve)_{\ve>0}$ is a family of smooth Riemann metrics satisfying the following properties.
A. We know that for any $p\in M$ there exists an open neighborhood $U\ni p$ and local coordinates $x^1,\dotsc, x^m$ on $U$ such that
$$ g^\ve_{ij} \to g_{ij} $$
uniformly on the compacts of $U$, where
$$ g^\ve =\sum_{i,j} g^\ve_{ij}dx^idx^j,\;\;g=\sum_{i,j}g_{ij}dx^idx^j. $$
B. With $p$, $U$ and $(x^i)$ as above, denote (Edited following Deane Yang's inquiry.) The note by $K^\ve_{ij}$ Gr_2(TM)$ the sectional curvature bundle of Grassmanians of $g^\ve$ along 2$-planes in the plane spanned by tangent bundle. The sectional curvature $\pa_{x^i},\pa_{x^j}$, K^\ve$ can then be viewed as a function $U\to\mathbb{R}$. K^\ve: Gr_2(TM)\to\mathbb{R}$. We know that for any $i\neq j$ there exists a smooth function $K_{ij}^0:U\to\mathbb{R}$ K^0: Gr_2(TM)\to \mathbb{R}$ such that
$$ K_{ij}^\ve\to K_{ij}^0 $$K^\ve\to K^0$ uniformlyon the compacts of $U$..
Question. Can we conclude that the function $K_{ij}^0$ K^0$ in B is the sectional curvature of $g$ along the plane spanned by $\pa_{x^i},\pa_{x^j}$?g$?
