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 $U$.

Can we extend $g$ to a metric $\tilde g$ on $M$ such that $\| \tilde g-g_0\|_{C^1} = \mathcal{o}(1)$ will hold

everywhereon $M$? ( i.e. bounded by something which tends to zero when $\epsilon \to 0$).

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 $1$.

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 $\mathbb{R}^d$.