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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}$?g$?

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# On a family of $C^0$-convergent Riemann metrics

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 by $K^\ve_{ij}$ the sectional curvature of $g^\ve$ along the plane spanned by $\pa_{x^i},\pa_{x^j}$, viewed as a function $U\to\mathbb{R}$. We know that for any $i\neq j$ there exists a smooth function $K_{ij}^0:U\to\mathbb{R}$ such that

$$K_{ij}^\ve\to K_{ij}^0$$

uniformly on the compacts of $U$.

Question. Can we conclude that the function $K_{ij}^0$ in B is the sectional curvature of $g$ along the plane spanned by $\pa_{x^i},\pa_{x^j}$?