Let $\pi:E\to M$ be a rank $k$ vector bundle over the compact manifold $M$ and let $i:M\hookrightarrow E$ denote the zero-section. Then we have a splitting of the restriction of $TE$ to the zero-section (see here): $$TE|_M\cong TM\oplus E.$$
Now suppose that we are given metrics $g_{TM}$ on the bundle $TM$ and $g_E$ on the bundle $E$. By the above splitting this yields a metric $g^i$ on $TE|_M$.
Question: Can we extend the metric $g^i$ to a global metric on $TE$?
That is, is there a metric $g$ on the bundle $TE$ such that its restriction $g|_M$ to the zero section coincides with $g^i$?
Thanks for any insights.