Let $M$ be a smooth closed manifold, and let $g\_0$ be a Riemannian metric on $M$. 


Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-g\_0|\_U\|\_{C^1} < \epsilon$ on $U$.



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> Can we extend $g$ to a metric $\tilde g$ on $M$ such that $\| \tilde g-g\_0\|\_{C^1} = \mathcal{o}(1)$ will hold **everywhere** on $M$? ( i.e. bounded by something which tends to zero when $\epsilon \to 0$).
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I am fine with shrinking $U$ if that's necessary. (So the "extension" $\tilde g$ needs to coincide with $g$ on *some* small neighbourhood of $p$).


**Edit:**


I am actually interested in a more refined version of this problem- where the **size of $U$ depends on $\epsilon$:**


Specifically, $U\_{\epsilon}=B\_{\epsilon}(p)$ where $B\_{\epsilon}(p)$ is the $\epsilon$-ball around $p$ w.r.t. the given metric $g$. So, now we are *given* the estimate
$\| g-g\_0|\_{U\_{\epsilon}}\|\_{C^1} < \epsilon$ on $U$, and we ask if we can extend $g$ (perhaps after shrinking $U\_{\epsilon}$) to a metric $\tilde g$ which is $o(1)$-close to $g\_0$ in the $C^1$ sense. 


Deane Yang gave a construction where the error $\| \tilde g-g\_0\|\_{C^1} \approx 1$, so it does not tend to zero with $\epsilon$. 


His construction is to take $\tilde{g} = (1-\chi)g\_0 + \chi g$, where 
$\chi$ is a smooth function that is identically $1$ on a neighbourhood of $p$ and compactly supported on $U\_{\epsilon}$. 
This means that $|d\chi| \approx \frac{1}{\text{diam}(U\_{\epsilon})}$ on average, so


$$
|d\tilde{g} - dg\_0|\le |d\chi| |g-g\_0| + |\chi||dg-dg\_0| \le (\frac{1}{\text{diam}(U\_{\epsilon})}+1)\epsilon.
$$
Since $\text{diam}(U) \approx \epsilon$, we get an error of order $1$.




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If that helps, I am interested in the special case where $M=\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ is the $d$-dimensional torus, and where $g\_0$ is a metric on $\mathbb{T}^d$ which is isometric to one of the standard flat metrics induced on $\mathbb{T}^d$ by the Euclidean metric on $\mathbb{R}^d$.



Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?
Let $\chi$ be a smooth function that is identically $1$ on a neighborhood of $p$ and compactly supported on $U$. Let $\tilde{g} = (1-\chi)g\_0 + \chi g$. 


